Function spaces for decoupling

TL;DR

This paper introduces new function spaces that reformulate decoupling inequalities and improve classical estimates, with invariance properties under specific operators.

Contribution

The authors define novel function spaces $\\mathcal{L}_{W,s}^{q,p}$ that connect decoupling inequalities with Hardy spaces for Fourier integral operators.

Findings

Spaces are invariant under Euclidean half-wave propagators.

Achieve improvements in fractional integration theorems.

Enhance local smoothing estimates.

Abstract

We introduce new function spaces $\mathcal{L}_{W,s}^{q,p}(\mathbb{R}^{n})$ that yield a natural reformulation of the $\ell^{q}L^{p}$ decoupling inequalities for the sphere and the light cone. These spaces are invariant under the Euclidean half-wave propagators, but not under all Fourier integral operators unless $p=q$, in which case they coincide with the Hardy spaces for Fourier integral operators. We use these spaces to obtain improvements of the classical fractional integration theorem and local smoothing estimates.

Equations

$$W^{s(p),p}({\mathbb{R}}^n)\subseteq {\mathcal{H}}_{FIO}^p({\mathbb{R}}^n)\subseteq W^{-s(p),p}({\mathbb{R}}^n),$$

$$s(p):=\frac{n-1}{2}\Big{|}\frac{1}{2}-\frac{1}{p}\Big{|}$$

$$\|f\|_{\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})}=\|\rho(D)f\|_{L^{p}(\mathbb{R}^{n})}+\Big{(}\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{W^{s,p}(\mathbb{R}^{n})}^{q}\mathrm{d}\omega\Big{)}^{1/q}<\infty.$$

$$W^{n(\frac{1}{p_1}-\frac{1}{p_2}),p_1}({\mathbb{R}}^n)\subseteq L^{p_2}({\mathbb{R}}^n)$$

$$\Vert f{\Vert }_{W^{-s(p),p}({\mathbb{R}}^n)}\lesssim \Vert f{\Vert }_{{\mathcal{H}}_{\mathrm{dec}}^{p,2;0}({\mathbb{R}}^n)}\lesssim \Vert f{\Vert }_{W^{s(p),p}({\mathbb{R}}^n)},$$

$$\Vert f{\Vert }_{W^{-\epsilon ,p}({\mathbb{R}}^n)}{\lesssim }_{\epsilon }\Vert f{\Vert }_{{\mathcal{H}}_{\mathrm{dec}}^{p,2;0}({\mathbb{R}}^n)}$$

$$\Vert {\phi }_{\omega }(D)f{\Vert }_{{\mathcal{H}}^1({\mathbb{R}}^n)}\eqsim \Vert {\phi }_{\omega }(D)f{\Vert }_{H_{\omega }^1({\mathbb{R}}^n)},$$

$$A_{\omega ,\sigma }(x):={\sigma }^2(\omega \cdot x)\omega +\sigma {\Pi }_{\omega }^{\perp }x,$$

$$A_{\omega ,\sigma }={\sigma }^{P_{\omega }}=\exp (\log (\sigma )P_{\omega }).$$

$$B_{\tau }^{\omega }(x):=\{y\in {\mathbb{R}}^n\mid |x-y{|}_{\omega }<\tau \}$$

$$B_{\tau }(x):=\{y\in {\mathbb{R}}^n\mid |x-y|<\tau \}$$

$$|B_{\tau }^{\omega }(x)|={\tau }^{n+1}|B_1(0)|.$$

$$|x{|}_{\omega }\le |\omega \cdot x{|}^{1/2}+|{\Pi }_{\omega }^{\perp }x|\le 2|x{|}_{\omega },$$

$$|x|\le |x{|}_{\omega }\text{ if and only if }x\in B_1^{\omega }(0)=B_1(0).$$

$$1=|A_{\omega ,1/\sigma }(x)|=\sqrt{{\sigma }^{-4}|\omega \cdot x{|}^2+{\sigma }^{-2}|{\Pi }_{\omega }^{\perp }x{|}^2}$$

$$|x{|}_{\omega }=\sigma =\sqrt{{\sigma }^{-2}|\omega \cdot x{|}^2+|{\Pi }_{\omega }^{\perp }x{|}^2}.$$

$$|x{|}_{\omega }\le |(\omega \cdot x)\omega {|}_{\omega }+|{\Pi }_{\omega }^{\perp }x{|}_{\omega }=|\omega \cdot x{|}^{1/2}+|{\Pi }_{\omega }^{\perp }x|.\text{qed}$$

$$M_{\omega }f(x):=\underset{\tau >0}{\sup }{\text{fint}}_{B_{\tau }^{\omega }(x)}|f(y)|\mathrm{d}y$$

$$\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{\infty}|M_{\omega}F(\cdot,\sigma)(x)|^{q}\frac{\mathrm{d}\sigma}{\sigma}\Big{)}^{\frac{p}{q}}\mathrm{d}x\Big{)}^{\frac{1}{p}}\leq C\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{\infty}|F(x,\sigma)|^{q}\frac{\mathrm{d}\sigma}{\sigma}\Big{)}^{\frac{p}{q}}\mathrm{d}x\Big{)}^{\frac{1}{p}}$$

$$\mathrm{supp}(\hat{f})\subseteq \{\xi \in {\mathbb{R}}^n\mid |\xi {|}_{\omega }\le \kappa {\sigma }^{-1}\}.$$

$$|f(x-y){|}^r\le C(1+{\sigma }^{-1}|y{|}_{\omega }{)}^{n+1}{M}_{\omega }(|f{|}^r)(x).$$

$$Mf(x):=\underset{\tau >0}{\sup }{\text{fint}}_{B_{\tau }(x)}|f(y)|\mathrm{d}y$$

$$|f(x-y){|}^r\lesssim (1+{\sigma }^{-1}|y|{)}^{n}M(|f{|}^r)(x)$$

$${\int }_0^{\infty }\Psi (\sigma \xi {)}^2\frac{\mathrm{d}\sigma }{\sigma }=1(\xi \ne 0).$$

$$\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{\infty}|\Psi(\sigma D)^{2}f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\Big{)}^{p/2}\mathrm{d}x\Big{)}^{1/p}<\infty,$$

$${\int }_0^{\infty }{\Psi }_0(\sigma {)}^2\frac{\mathrm{d}\sigma }{\sigma }=1.$$

$${\int }_0^{\infty }{\Psi }_{\omega }(A_{\omega ,\sigma }\xi {)}^2\frac{\mathrm{d}\sigma }{\sigma }=1(\xi \ne 0).$$

$$|{\mathcal{F}}^{-1}({\Psi }_{\omega })(A_{\omega ,1/\sigma }y)|\le C_N(1+{\sigma }^{-1}|y{|}_{\omega }{)}^{-N}$$

$$\|f\|_{H^{p}_{\omega}(\mathbb{R}^{n})}:=\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{\infty}|\Psi_{\omega}(A_{\omega,\sigma}D)^{2}f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\Big{)}^{p/2}\mathrm{d}x\Big{)}^{1/p}<\infty.$$

$$H_{\omega }^p({\mathbb{R}}^n)=L^p({\mathbb{R}}^n)\text{ if }p\in (1,\infty ),$$

Full text

The research leading to these results has received funding from the Norwegian Financial Mechanism 2014-2021, grant 2020/37/K/ST1/02765. This research was funded in part by the National Science Center, Poland, grant 2021/43/D/ST1/00667. Yung was partially supported by Future Fellowship FT200100399 from the Australian Research Council.

1. Introduction

This article provides a bridge between the theory of decoupling inequalities in Fourier analysis on the one side, and that of invariant spaces for wave propagators and more general Fourier integral operators on the other.

1.1. Setting

In recent years, the theory of decoupling inequalities has developed into an active and highly fruitful part of harmonic analysis. Starting with seminal work by Bourgain and Demeter [4], which in turn has its origins in ideas of Wolff [33], decoupling inequalities have found applications both in number theory and to partial differential equations, leading for example to a proof of the main conjecture in the Vinogradov mean-value theorem [5], and yielding significant progress towards the local smoothing conjecture for the Euclidean wave equation (see e.g. [4, 12, 2]).

In a separate development, a scale of invariant spaces $(\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n}))_{1\leq p\leq\infty}$ for wave propagators and more general Fourier integral operators (FIOs) was introduced in [13], extending a pioneering construction due to Smith [28] for $p=1$. These spaces satisfy the following Sobolev embeddings into the $L^{p}$ scale, for $1<p<\infty$:

[TABLE]

with the appropriate modifications involving the local Hardy space $\mathcal{H}^{1}(\mathbb{R}^{n})$ and $\mathrm{bmo}(\mathbb{R}^{n})$ for $p=1$ and $p=\infty$, respectively. Here and throughout, we write

[TABLE]

for $1\leq p\leq\infty$. These embeddings allow one to recover the optimal $L^{p}$ regularity of FIOs due to Seeger, Sogge and Stein [27], and the accompanying sharp fixed-time $L^{p}$ regularity for wave equations. On the other hand, the Hardy spaces for FIOs have been applied to both wave equations with rough coefficients [11, 14] and to nonlinear wave equations [23, 20] in ways that would not be possible by working on $L^{p}(\mathbb{R}^{n})$ directly.

Very recently, it was realized that invariant spaces for wave propagators are also connected to decoupling inequalities. First, in [22] it was shown that, loosely speaking, $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ is the largest space of initial initial data for which one can get local smoothing estimates by relying on the $\ell^{p}$ decoupling inequality for the light cone. This yields new local smoothing estimates that improve upon those in the local smoothing conjecture for $p\geq 2(n+1)/(n-1)$. This connection was subsequently extended to other decoupling inequalities in [23, 20], thereby yielding improved local smoothing estimates for both constant-coefficient and variable-coefficient wave equations.

However, this development only concerns applications of decoupling inequalities to wave equations, and it shows that the Hardy spaces for Fourier integral operators are natural spaces of initial data in this regard. In the present article, we aim to develop the connection between decoupling inequalities and invariant spaces for wave propagators further, by introducing function spaces that capture the decoupling inequalities for both the sphere and the light cone in their full generality. We subsequently use these function spaces to obtain improved fractional integration theorems, local smoothing estimates, and well-posedness results for nonlinear wave equations.

1.2. Main results

We introduce a collection of function spaces $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$, for $p,q\in[1,\infty)$ and $s\in\mathbb{R}$. Here $p$ is a spatial integrability parameter, $s$ a smoothness parameter, and $q$ measures integrability with respect to angular localizations. More precisely, cf. Definition 4.1, for $1<p<\infty$ the space $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ consists of all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that

[TABLE]

Here $\rho(D)$ is a relatively unimportant low-frequency cut-off, and the $\varphi_{\omega}$, for $\omega$ in the unit sphere $S^{n-1}$, are Fourier multipliers which localize to a paraboloid in the direction of $\omega$ (for their exact properties see Section 3.1). For $p=1$ one replaces $W^{s,p}(\mathbb{R}^{n})$ by $(1+\Delta)^{-s/2}\mathcal{H}^{1}(\mathbb{R}^{n})$ in (1.2).

By combining parabolic frequency localizations with the implicit Littlewood–Paley decomposition of $L^{p}(\mathbb{R}^{n})$, (1.2) involves a dyadic-parabolic decomposition which goes back to [10] and which plays a key role in the proof of the optimal $L^{p}$ regularity of FIOs in [27]. On a more intrinsic level, angular localization is one of the fundamental tenets of microlocal analysis, and it allows one to deal with the phenomenon of propagation of singularities which is inherent to hyperbolic equations and FIOs. Accordingly, $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ coincides with $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ when $p=q$, and it is invariant under FIOs of order zero. More generally, $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is invariant under the Euclidean half-wave propagators $e^{it\sqrt{-\Delta}}$ for all $1\leq p,q<\infty$ and $s,t\in\mathbb{R}$, cf. Theorem 5.1. On the other hand, variable-coefficient propagation of singularities does not distinguish between the position and momentum variables, and therefore $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is not invariant under general FIOs if $p\neq q$, as we show in Section 5.2.

Although the spaces $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ do not have the same invariance properties as $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for $p\neq q$, several key results persist. For example, (1.1) holds with $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ replaced by $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ for more general values of $p$ and $q$, and in particular for all $1<p<\infty$ if $q=2$ (see Theorem 6.2). In fact, the latter case has additional features beyond those of $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, which allow us to improve the classical fractional integration theorem

[TABLE]

for $1<p_{1}\leq 2\leq p_{2}<\infty$ (see Theorem 6.3 and Remark 6.4). Outside of this range, we obtain improved mapping properties for FIOs of negative order, cf. Corollary 6.5 and Remark 6.6. We also note that the spaces $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ behave in the natural way under complex interpolation and duality, as is shown in Section 4.3.

These are some of the basic properties of the spaces $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$, and how they relate to those of $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. On the other hand, it is not immediately clear how these spaces relate to the theory of Fourier decoupling. For example, decoupling inequalities are typically formulated using a discrete decomposition of functions that have Fourier support in a fixed compact set, whereas (1.2) is only of interest at high frequencies. However, after rescaling and observing that the continuous decomposition in (1.2) is equivalent to a discrete one on dyadic frequency annuli, the $\ell^{2}$ decoupling inequality for the sphere can be reinterpreted as an improvement over the Sobolev embeddings for $\mathcal{H}_{\mathrm{dec}}^{p,2;s}(\mathbb{R}^{n})$, for functions with highly localized frequency support. More precisely, for all $1<p<\infty$ and $f\in\mathcal{H}_{\mathrm{dec}}^{p,2;0}(\mathbb{R}^{n})$ one has

[TABLE]

and these Sobolev exponents cannot be improved for general $f$. On the other hand, the $\ell^{2}$ decoupling inequality for the sphere is equivalent to the inequality

[TABLE]

for all $2<p\leq 2(n+1)/(n-1)$ and $\varepsilon>0$, if $\mathrm{supp}(\widehat{f}\,)\subseteq\{\xi\in\mathbb{R}^{n}\mid R-1\leq|\xi|\leq R+1\}$ for some $R\geq 2$ (see Corollary 7.4). In Corollary 7.8 we reinterpret the $\ell^{q}$ decoupling inequality for the cone in a similar manner, as an improved embedding for functions in $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n+1})$ with frequency support near the light cone.

This reformulation of the decoupling inequality for the light cone seems to indicate that $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n+1})$ is a natural space for the solutions to the Euclidean wave equation. Such a connection was not made in [22, 23, 20], where the relevant function spaces were only used for the initial data. On the other hand, just as in those cases, we obtain local smoothing estimates for the Euclidean wave equation using $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ as a space of initial data. Moreover, when combined with the fractional integration theorem for $\mathcal{H}_{\mathrm{dec}}^{p,2;s}(\mathbb{R}^{n})$, one simultaneously obtains improved local smoothing estimates, as well as improvements of the classical Strichartz estimates for suitable initial data (see Theorem 8.1 and Remark 8.3). Just as in [23, 20], we proceed to apply these local smoothing estimates to nonlinear wave equations with initial data outside of $L^{2}$-based Sobolev spaces, in Section 8.2.

1.3. Techniques

Apart from the specific results and the connection to decoupling theory, a significant difference between the present article and earlier contributions in this direction concerns the techniques that we use.

Namely, in [28, 13] the Hardy spaces for FIOs were defined using conical square functions over the cosphere bundle $S^{*}\mathbb{R}^{n}=\mathbb{R}^{n}\times S^{n-1}$, which allows one to incorporate the theory of tent spaces to prove various fundamental properties of the spaces. Such a conical square function characterization over the cosphere bundle is implicitly contained in (1.2) when $p=q$, due to Fubini’s theorem, but when $p\neq q$ this argument breaks down. More generally, the fact that FIOs are typically not bounded on $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ indicates that one cannot expect to apply the same techniques to $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ when $p\neq q$.

On the other hand, [23] introduced Besov spaces adapted to the half-wave group using a similar norm as in (1.2), for general $p$ and $q$, albeit with Besov spaces replacing Sobolev spaces over $L^{p}(\mathbb{R}^{n})$. The latter does not make a difference on dyadic frequency annuli, so for our applications to decoupling theory one could also use the spaces in [23]. On the other hand, working with Besov spaces does not allow one to recover the optimal fixed-time $L^{p}$ regularity for wave equations, and it does not yield improvements of the fractional integration theorem in (1.3). Moreover, on a technical level, for Besov spaces it typically suffices to obtain estimates on dyadic frequency annuli, as opposed to having to deal with all frequency scales simultaneously through a square function, as is necessary in our setting. In fact, [23] does not provide a full function space theory, focusing instead on applications to nonlinear wave equations, but (simpler versions of) the techniques in this article can also be used to derive properties of the adapted Besov spaces from [23].

So, instead of relying on existing techniques, we develop new tools and make connections to other areas to prove our main results. For example, we incorporate the theory of parabolic Hardy spaces from [6, 7], by observing that it is convenient to replace $W^{s,p}(\mathbb{R}^{n})$ in (1.2) by $(1+\Delta)^{-s/2}H^{p}_{\omega}(\mathbb{R}^{n})$, where $H^{p}_{\omega}(\mathbb{R}^{n})$ is a parabolic Hardy space associated to a family of dilations which is anisotropic in the direction of $\omega$. For $1<p<\infty$ one has $L^{p}(\mathbb{R}^{n})=H^{p}_{\omega}(\mathbb{R}^{n})$, but in Proposition 2.6 we show that also

[TABLE]

due to the fact that $\varphi_{\omega}(D)$ localizes to a paraboloid in the direction of $\omega$. One can then use anisotropic Calderón–Zygmund theory to prove embeddings and invariance properties of $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$.

As in [13], we rely on wave packet transforms that lift functions on $\mathbb{R}^{n}$ to phase space $T^{*}\mathbb{R}^{n}$ minus the zero section, parametrized in spherical coordinates as $\mathbb{R}^{n}\times S^{n-1}\times(0,\infty)$. This allows us to embed $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ into a larger, but simpler and more established, space of functions on phase space. We can then derive various properties of $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ from those of the encompassing space. However, unlike in [13], we do not use a tent space norm on $T^{*}\mathbb{R}^{n}$, working instead with an $L^{q}(S^{n-1};L^{p}(\mathbb{R}^{n};L^{2}(0,\infty)))$ norm which arises naturally from (1.2), via the Littlewood–Paley decomposition of $L^{p}(\mathbb{R}^{n})$. This in turn means that we cannot rely on tools such as the atomic decomposition of tent spaces. Instead, to prove the fundamental Theorem 3.4, we use both the boundedness of the vector-valued Hardy–Littlewood maximal function over $S^{n-1}$, as well as the boundedness of anisotropic maximal functions in each separate direction $\omega\in S^{n-1}$. Theorem 3.4 then allows us to deduce properties of $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ from those of $L^{q}(S^{n-1};L^{p}(\mathbb{R}^{n};L^{2}(0,\infty)))$.

1.4. Organization

This article is organized as follows. In Section 2 we collect background on the anisotropic dilations on $\mathbb{R}^{n}$ and the corresponding norms, on specific parabolic Hardy spaces, and on Fourier integral operators. In Section 3 we then introduce the parabolic frequency localizations and the associated wave packet transforms, and we prove various properties of these transforms. In Section 4 we introduce the spaces $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ and we derive many of their basic properties. The fact that they are invariant under the Euclidean wave propagators, but not under general FIOs, is shown in Section 5. Section 6 then contains the Sobolev embeddings and fractional integration theorems for $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$, while Section 7 connects these spaces to decoupling inequalities. Finally, in Section 8 we obtain local smoothing estimates using $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$, as well as well-posedness results for nonlinear wave equations.

1.5. Notation and terminology

The natural numbers are ${\mathbb{N}}=\{1,2,\ldots\}$, and $\mathbb{Z}_{+}:={\mathbb{N}}\cup\{0\}$. Throughout this article we fix $n\in{\mathbb{N}}$ with $n\geq 2$.

For $\xi\in\mathbb{R}^{n}$ we write $\langle\xi\rangle=(1+|\xi|^{2})^{1/2}$, and $\hat{\xi}=\xi/|\xi|$ if $\xi\neq 0$. We use multi-index notation, where $\partial_{\xi}=\nabla_{\xi}=(\partial_{\xi_{1}},\ldots,\partial_{\xi_{n}})$ and $\partial^{\alpha}_{\xi}=\partial^{\alpha_{1}}_{\xi_{1}}\ldots\partial^{\alpha_{n}}_{\xi_{n}}$ for $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ and $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}_{+}^{n}$. Moreover, $\partial_{x\eta}^{2}\Phi$ is the mixed Hessian of a function $\Phi$ of the variables $x$ and $\eta$.

The Fourier transform of a tempered distribution $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is denoted by $\mathcal{F}f$ or $\widehat{f}$, and its inverse Fourier tranform by $\mathcal{F}^{-1}f$. If $f\in L^{1}(\mathbb{R}^{n})$ and $\xi\in\mathbb{R}^{n}$, then $\mathcal{F}f(\xi)=\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}f(x)\mathrm{d}x$. The standard distributional pairing between $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ and $g\in\mathcal{S}(\mathbb{R}^{n})$ is denoted by $\langle f,g\rangle_{\mathbb{R}^{n}}$, and we write $\varphi(D)$ for the Fourier multiplier with symbol $\varphi$.

The volume of a measurable subset $B\subseteq\Omega$ of a measure space $\Omega$ will be denoted by $|B|$, and its indicator function by $\mathbf{1}_{B}$. The space of bounded linear operators between Banach spaces $X$ and $Y$ is $\mathcal{L}(X,Y)$, and $\mathcal{L}(X):=\mathcal{L}(X,X)$.

We write $f(s)\lesssim g(s)$ to indicate that $f(s)\leq Cg(s)$ for all $s$ and a constant $C>0$ independent of $s$, and similarly for $f(s)\gtrsim g(s)$ and $g(s)\eqsim f(s)$.

2. Preliminaries

In this section we first introduce a family of norms, associated with groups of dilations. We then define parabolic Hardy spaces associated with these dilations, and finally we collect some background on a specific class of Fourier integral operators.

2.1. A family of norms

In this subsection, we define a collection of norms on $\mathbb{R}^{n}$, and the associated metrics. We then derive some of their basic properties.

For $\omega\in S^{n-1}$, $x\in\mathbb{R}^{n}$ and $\sigma>0$, set

[TABLE]

where $\Pi_{\omega}^{\perp}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the orthogonal projection onto the complement of the span of $\omega$. Note that, with $P_{\omega}(x):=x+(\omega\cdot x)\omega$, one has

[TABLE]

This implies that $(A_{\omega,\sigma})_{\sigma>0}$ is a group of transformations as in [6], and we will rely on the theory developed there in what follows.

Let $|x|_{\omega}$ be the unique $\sigma_{x}>0$ such that $|A_{\omega,1/\sigma_{x}}(x)|=1$. Then $|\cdot|_{\omega}$ is a norm on $\mathbb{R}^{n}$ (see [6, Section 1.4]), and $|A_{\omega,\sigma}x|_{\omega}=\sigma|x|_{\omega}$ for all $\sigma>0$. Let $\mathbb{R}^{n}_{\omega}$ be the metric measure space obtained by endowing $\mathbb{R}^{n}$ with the associated metric and the standard Lebesgue measure. For $\tau>0$, we write

[TABLE]

for the ball in $\mathbb{R}^{n}_{\omega}$ around $x\in\mathbb{R}^{n}$ with radius $\tau$, and

[TABLE]

for the corresponding ball in $\mathbb{R}^{n}$. Note that $B_{\tau}^{\omega}(x)$ is a convex set with respect to the Euclidean metric, given that $B_{\tau}^{\omega}(0)$ is the inverse image of $B_{1}(0)$ under the linear map $A_{\omega,1/\tau}$. For the same reason, $B^{\omega}_{1}(x)=B_{1}(x)$. Moreover, using that $A_{\omega,\tau}$ has determinant $\tau^{n+1}$, it follows that

[TABLE]

In particular, $\mathbb{R}^{n}_{\omega}$ is a doubling metric measure space.

We will often work with the equivalent expression for $|\cdot|_{\omega}$ provided by the following lemma.

Lemma 2.1**.**

Let $\omega\in S^{n-1}$ and $x\in\mathbb{R}^{n}$. Then

[TABLE]

and

[TABLE]

Proof.

Set $\sigma:=|x|_{\omega}$. Then, by definition and by (2.1),

[TABLE]

and

[TABLE]

This implies (2.4). Moreover, trivial estimates yield $|x|_{\omega}\geq|x|_{\omega}^{-1}|\omega\cdot x|$ and $|x|_{\omega}\geq|\Pi_{\omega}x|$, which proves the second inequality in (2.3).

For the other inequality, it follows from (2.5) that, for all $y\in\mathbb{R}^{n}$, one has $|y|_{\omega}=|\Pi_{\omega}^{\perp}y|$ if $\omega\cdot y=0$. Moreover, $|y|_{\omega}=|y|_{\omega}^{-1}|\omega\cdot y|$ and $|y|_{\omega}=|\omega\cdot y|^{1/2}$ if $\Pi_{\omega}^{\perp}y=0$. Hence one may use that $|\cdot|_{\omega}$ is a norm to write

[TABLE]

It follows from Lemma 2.1 that the metric associated with $|\cdot|_{\omega}$ is essentially the restriction to the subspace $\mathbb{R}^{n}\times\{\omega\}\subseteq S^{*}\mathbb{R}^{n}$ of the metric on the cosphere bundle from [13, 28]. Similarly, the square of the metric associated with $|\cdot|_{\omega}$ essentially coincides with the quasi-distance in [11].

Since $\mathbb{R}^{n}_{\omega}$ is a doubling metric measure space, it is natural to consider the (centered) Hardy–Littlewood maximal function $M_{\omega}$ on $\mathbb{R}^{n}_{\omega}$, given by

[TABLE]

for $x\in\mathbb{R}^{n}$ and $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$. We record the following standard lemma concerning the boundedness of the vector-valued extension of $M_{\omega}$ to $L^{p}(\mathbb{R}^{n};L^{q}(\mathbb{R}_{+}))$, for $p,q\in(1,\infty)$. Here and throughout, we write $\mathbb{R}_{+}$ for the measure space $(0,\infty)$, endowed with the Haar measure $\mathrm{d}\sigma/\sigma$.

Lemma 2.2**.**

Let $p,q\in(1,\infty)$. Then there exists a $C\geq 0$ such that

[TABLE]

*for all $w\in S^{n-1}$ and $F\in L^{p}(\mathbb{R}^{n};L^{q}(\mathbb{R}_{+}))$. *

Proof.

For a fixed $\omega\in S^{n-1}$, the statement is a special case of [30, Section II.5.14] or [25, Lemma 2.8]. But by rotation, the resulting constant is independent of $\omega$.∎

In later sections, we will combine Lemma 2.2 with the following pointwise variant of [25, Lemma 2.3].

Lemma 2.3**.**

Let $r,\kappa>0$. Then there exists a $C\geq 0$ such that the following holds for all $\omega\in S^{n-1}$, $\sigma>0$ and $f\in L^{\infty}(\mathbb{R}^{n})$ with

[TABLE]

For all $x,y\in\mathbb{R}^{n}$ one has

[TABLE]

Proof.

After replacing $f(x)$ by $\sigma^{n+1}f(A_{\omega,\sigma}x)$ and using (2.2), one may suppose that $\sigma=1$. The proof is then completely analogous to that of the corresponding inequality for the classical Hardy–Littlewood maximal function (see e.g. [32, Theorem 1.3.1]), using the mean value theorem and the fact that balls in $\mathbb{R}^{n}_{\omega}$ are convex with respect to the Euclidean metric.

∎

Remark 2.4**.**

As was just indicated, Lemma 2.3 is an anisotropic version of a lemma involving the classical (centered) Hardy-Littlewood maximal function $M$, given by

[TABLE]

for $x\in\mathbb{R}^{n}$ and $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$. More precisely, using notation as in Lemma 2.3,

[TABLE]

for all $x,y\in\mathbb{R}^{n}$, $\sigma>0$ and $f\in L^{\infty}(\mathbb{R}^{n})$ with $\mathrm{supp}(\widehat{f}\,)\subseteq\{\xi\in\mathbb{R}^{n}\mid|\xi|\leq\kappa\sigma^{-1}\}$. Also recall that Lemma 2.2 holds with $M_{\omega}$ replaced by $M$ (see [30, Theorem II.1.1]).

2.2. Hardy spaces

In this subsection we collect some basics on specific parabolic Hardy spaces. We refer to [6, 7] for the general theory of parabolic Hardy spaces.

Throughout this article, fix a real-valued and radial $\Psi\in C^{\infty}_{c}(\mathbb{R}^{n})$ with $\mathrm{supp}(\Psi)\subseteq\{\xi\in\mathbb{R}^{n}\mid|\xi|\in[1/2,2]\}$, such that

[TABLE]

Recall that $H^{p}(\mathbb{R}^{n})$, for $p\in[1,\infty)$, consists of those $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that

[TABLE]

and this quantity is equivalent to $\|f\|_{H^{p}(\mathbb{R}^{n})}$. For $p\in(1,\infty)$ one has $H^{p}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$, and $H^{1}(\mathbb{R}^{n})$ is the classical real Hardy space.

We define the parabolic Hardy spaces in an analogous manner, using the dilations from (2.1). In fact, to compare the classical Hardy spaces and the parabolic Hardy spaces, it will be convenient to make a specific choice of $\Psi$. Namely, we will assume that $\Psi(\xi)=\Psi_{0}(|\xi|)$, where $\Psi_{0}\in C^{\infty}_{c}(\mathbb{R})$ is real-valued, with $\Psi_{0}(\sigma)=0$ for $\sigma\notin[1/2,2]$, such that

[TABLE]

Then $\Psi$ has the required properties.

Next, for $\omega\in S^{n-1}$ and $\xi\in\mathbb{R}^{n}$, set $\Psi_{\omega}(\xi)=\Psi_{0}(|\xi|_{\omega})$. Then $\Psi_{\omega}\in C^{\infty}_{c}(\mathbb{R}^{n})$, due to the fact that $\xi\mapsto|\xi|_{\omega}$ is smooth on $\mathbb{R}^{n}\setminus\{0\}$ (see [6, Lemma 1.5]). Moreover, $\mathrm{supp}(\Psi_{\omega})\subseteq\{\xi\in\mathbb{R}^{n}\mid|\xi|_{\omega}\in[1/2,2]\}$, and

[TABLE]

We write $\Psi_{\omega}(A_{\omega,\sigma}D)$ for the Fourier multiplier with symbol $\xi\mapsto\Psi_{\omega}(A_{\omega,\sigma}\xi)$. Note that $\Psi_{\omega}(A_{\omega,\sigma}D)$ has kernel $\sigma^{-(n+1)}\mathcal{F}^{-1}(\Psi_{\omega})(A_{\omega,1/\sigma}y)$, for $y\in\mathbb{R}^{n}$. Moreover,

[TABLE]

for each $N\geq 0$, where $C_{N}\geq 0$ is a constant independent of $\omega$, $\sigma$ and $y$. The fact that the constant is independent of $\omega$ follows by rotation.

Definition 2.5**.**

Let $p\in[1,\infty)$ and $\omega\in S^{n-1}$. Then $H^{p}_{\omega}(\mathbb{R}^{n})$ consists of those $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that

[TABLE]

In fact, this is not how the parabolic Hardy spaces were originally defined in [6, 7]. However, it follows from [25] that, up to norm equivalence, the relevant definitions are equivalent.

We note that $H^{p}_{\omega}(\mathbb{R}^{n})\subseteq L^{p}(\mathbb{R}^{n})$ for all $w\in S^{n-1}$ and $p\in[1,\infty)$. Moreover,

[TABLE]

with equivalence of norms (see [7, Theorem 1.2]).

We conclude this subsection with the following proposition, which relates the parabolic and classical Hardy space norms of a function with frequency support inside a paraboloid.

Proposition 2.6**.**

Let $p\in[1,\infty)$ and $\kappa_{1},\kappa_{2}\geq 1$. Then there exists a $C>0$ such that the following holds. Let $\omega\in S^{n-1}$ and $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ satisfy

[TABLE]

Then $f\in H^{p}(\mathbb{R}^{n})$ if and only if $f\in H^{p}_{\omega}(\mathbb{R}^{n})$, in which case

[TABLE]

Proof.

For $p\in(1,\infty)$, the statement in fact follows directly from (2.14). However, our argument applies for general $p\in[1,\infty)$, so there is no need to rely on (2.14).

Fix $\omega\in S^{n-1}$ and $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ with the prescribed support properties. Since $H^{p}(\mathbb{R}^{n})$ and $H^{p}_{\omega}(\mathbb{R}^{n})$ are both contained in $L^{p}(\mathbb{R}^{n})$, we may suppose in the remainder that $f\in L^{p}(\mathbb{R}^{n})$, which will allow us to apply Lemma 2.3.

We first make a preliminary observation which will also be useful later on. Namely, we claim that, if $\xi\in\mathbb{R}^{n}$ is such that $|\xi|\geq\kappa_{1}^{-1}$ and $|\hat{\xi}-\omega|\leq\kappa_{2}|\xi|^{-1/2}$, then one has

[TABLE]

To prove this claim, note that the first statement follows from (2.4). On the other hand, one has $|\omega\cdot\xi|^{1/2}\leq|\xi|^{1/2}$ and, by assumption,

[TABLE]

Now (2.3) yields the right-most inequality in (2.16). For the remaining inequality, we can combine the first inequality in (2.16) with (2.3) to write

[TABLE]

which proves the claim.

Next, for a general $\xi\in\mathbb{R}^{n}$ and $\tau>0$, one has $\Psi(\tau\xi)=0$ unless $|\xi|\in[(2\tau)^{-1},2\tau^{-1}]$. Similarly, for $\sigma>0$ one has $\Psi(A_{\omega,\sigma}\xi)=0$ unless $|A_{\omega,\sigma}\xi|_{\omega}\in[1/2,2]$, i.e. unless $|\xi|_{\omega}\in[(2\sigma)^{-1},2\sigma^{-1}]$. Hence (2.16) implies that

[TABLE]

whenever $\Psi_{\omega}(A_{\omega,\sigma}D)\Psi(\tau D)f\neq 0$.

Now fix $r\in(0,\min(p,2))$. We claim that, with notation as in (2.6) and (2.8),

[TABLE]

and

[TABLE]

for implicit constants independent of $\omega$, $f$, $\sigma,\tau>0$ and $x\in\mathbb{R}^{n}$. To see why this is true, first consider (2.18), and let $N>(n+1)(1+\tfrac{1}{r})$. We may suppose that $\sigma$ and $\tau$ satisfy (2.17). Then, by (2.16),

[TABLE]

Moreover, $f\in L^{p}(\mathbb{R}^{n})$, by assumption, and then $\Psi(\tau D)f\in L^{\infty}(\mathbb{R}^{n})$, by a standard Sobolev embedding. Hence (2.13), Lemma 2.3 and (2.3) yield

[TABLE]

The argument for (2.19) is analogous, although one relies on (2.9) instead of Lemma 2.3, which is allowed because of (2.16). This proves the claim.

Now suppose that $f\in H^{p}(\mathbb{R}^{n})$, and set $a:=(8(4+\kappa^{2}))^{-1}$ and $b:=8(1+\kappa)^{2}$. For $\sigma>0$ and $x\in\mathbb{R}^{n}$, (2.10), (2.18) and Hölder’s inequality yield

[TABLE]

Hence

[TABLE]

Finally, we can use that $M_{\omega}$ is bounded on $L^{p/r}(\mathbb{R}^{n};L^{2/r}(0,\infty))$, cf. Lemma 2.2:

[TABLE]

where in the final step we used (2.11). This shows that $f\in H^{p}_{\omega}(\mathbb{R}^{n})$, with the required norm bound. The proof of the reverse inequality is analogous, relying on (2.19) and the boundedness of the classical Littlewood–Paley maximal function. ∎

2.3. Fourier integral operators

Here we collect some background on Fourier integral operators. We refer to [16, 8] for the general theory of Fourier integral operators and the associated notions from symplectic geometry. On the other hand, in this article we will mostly work with concrete oscillatory integral representations for such operators, and this subsection contains the relevant definitions.

For $m\in\mathbb{R}$ and $\rho,\delta\in[0,1]$, recall that Hörmander’s symbol class $S^{m}_{\rho,\delta}$ consists of all $a\in C^{\infty}(\mathbb{R}^{2n})$ such that

[TABLE]

for all $\alpha,\beta\in\mathbb{Z}_{+}^{n}$. In [13] a slightly different symbol class was considered, the elements of which have additional regularity when differentiated in the radial direction in the fiber variable. This class, denoted by $S^{m}_{\rho,\delta,1}$, consists of all $a\in C^{\infty}(\mathbb{R}^{2n})$ such that

[TABLE]

for all $\alpha,\beta\in\mathbb{Z}_{+}^{n}$ and $\gamma\in\mathbb{Z}_{+}$. Note that $S^{m}_{1/2,1/2,1}$ contains $S^{m}_{1,1/2}$ but is strictly contained in $S^{m}_{1/2,1/2}$, which is the critical symbol class for the calculus of Fourier integral operators, in several respects.

In the following definition, and throughout the rest of this article, $o:=\mathbb{R}^{n}\times\{0\}\subseteq\mathbb{R}^{2n}$ denotes the zero section.

Definition 2.7**.**

Let $m\in\mathbb{R}$, $a\in S^{m}_{1/2,1/2,1}$ and $\Phi\in C^{\infty}(\mathbb{R}^{2n}\setminus o)$ be such that $\Phi$ is real-valued and positively homogeneous of degree $1$ in the $\eta$-variable, and $\det\partial^{2}_{x\eta}\Phi(x,\eta)\neq 0$ for $(x,\eta)\in\mathrm{supp}(a)\setminus o$. Set

[TABLE]

for $f\in\mathcal{S}(\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$. Then $T$ is a Fourier integral operator of order $m$ and type $(1/2,1/2,1)$ in standard form. If, additionally, the following conditions hold:

1. (1)

$\sup_{(x,\eta)\in\mathbb{R}^{2n}\setminus o}|\partial_{x}^{\beta}\partial_{\eta}^{\alpha}\Phi(x,\hat{\eta})|<\infty$ for all $\alpha,\beta\in\mathbb{Z}_{+}^{n}$ with $|\alpha|+|\beta|\geq 2$; 2. (2)

$\inf_{(x,\eta)\in\mathbb{R}^{2n}\setminus o}|\det\partial^{2}_{x\eta}\Phi(x,\eta)|>0$; 3. (3)

$(\partial_{\eta}\Phi(x,\eta),\eta)\mapsto(x,\partial_{x}\Phi(x,\eta))$ is a well-defined bijection on $\mathbb{R}^{2n}\setminus o$,

then we say that $T$ is associated with a global canonical graph.

Remark 2.8**.**

If (3) holds, then $(\partial_{\eta}\Phi(x,\eta),\eta)\mapsto(x,\partial_{x}\Phi(x,\eta))$ is in fact a homogeneous canonical transformation on $\mathbb{R}^{2n}\setminus o$, and the canonical relation of $T$ is the graph of this transformation.

If (1) and (2) hold, then (3) holds if and only if $\eta\mapsto\partial_{x}\Phi(x,\eta)$ is a bijection on $\mathbb{R}^{n}\setminus\{0\}$ for each $x\in\mathbb{R}^{n}$, by Hadamard’s global inverse function theorem [19, Theorem 6.2.8]. Another application of the global inverse function theorem then shows that condition (3) is superfluous for $n\geq 3$. Moreover, (3) holds if $\Phi(x,\eta)=x\cdot\eta+\phi(\eta)$ for some $\phi\in C^{\infty}(\mathbb{R}^{n}\setminus\{0\})$ which is positively homogeneous of degree $1$. This case will be considered frequently by us, and it is characterized by the property that $\partial_{x}\Phi(x,\eta)=\eta$ for all $(x,\eta)\in\mathbb{R}^{2n}\setminus o$.

Recall that a compactly supported Fourier integral operator of order $m$ and type $(1,0)$, associated with a local canonical graph, can, modulo an operator with a Schwartz kernel which is a Schwartz function, be expressed as a finite sum of operators which in appropriate coordinate systems are Fourier integral operators in standard form (see e.g. [29, Proposition 6.2.4]), with symbol $a\in S^{m}_{1,0}$. In this case, the symbol $a$ has compact support in the $x$-variable, and thus (1) and (2) hold on the support of $a$, while the map in (3) is a locally well-defined homogeneous canonical transformation. By contrast, for operators associated with a global canonical graph as in Definition 2.7, the symbols are not required to have compact spatial support, but the conditions on the phase function hold uniformly on all of $\mathbb{R}^{2n}\setminus o$.

3. Wave packet transforms

In this section we introduce the parabolic frequency localizations which appear in the definition of the function spaces for decoupling, and we use them to define associated wave packets and wave packet transforms. We then prove some properties of these transforms, most notably the boundedness of specific operators on phase space associated with them.

3.1. Wave packets

Throughout the rest of this article, we fix a family $(\varphi_{\omega})_{\omega\in S^{n-1}}\subseteq C^{\infty}(\mathbb{R}^{n})$ with the following properties:

1. (1)

For all $\omega\in S^{n-1}$ and $\xi\neq 0$, one has $\varphi_{\omega}(\xi)=0$ if $|\xi|<\frac{1}{4}$ or $|\hat{\xi}-\omega|>|\xi|^{-1/2}$. 2. (2)

For all $\alpha\in\mathbb{Z}_{+}^{n}$ and $\beta\in\mathbb{Z}_{+}$, there exists a $C_{\alpha,\beta}\geq 0$ such that

[TABLE]

for all $\omega\in S^{n-1}$ and $\xi\neq 0$. 3. (3)

The map $(\xi,\omega)\mapsto\varphi_{\omega}(\xi)$ is measurable on $\mathbb{R}^{n}\times S^{n-1}$, and

[TABLE]

for all $\xi\in\mathbb{R}^{n}$ with $|\xi|\geq 1$.

Remark 3.1**.**

For the construction of such a collection, see e.g. [21, Section 3.1]. In fact, the functions constructed there have slightly different support properties than in (1). However, the construction in [21, Section 3.1] also yields functions satisfying (1), if one shrinks the support of the function $\varphi$ used there, and slightly modifies the definition of $\varphi_{\omega}$ itself. Moreover, instead of (3.1), from [21, Remark 3.3] one obtains

[TABLE]

for a standard symbol $m\in S^{0}(\mathbb{R}^{n})$. However, the proof of [21, Lemma 3.2] also shows that $\sqrt{m}\in S^{0}(\mathbb{R}^{n})$, so that one may replace $\varphi_{\omega}$ by $\sqrt{m}\varphi_{\omega}$ to arrive at (3.1).

It is relevant to note that the exact support properties of the $\varphi_{\omega}$ are not relevant for this article; all the arguments go through if the conditions in (1) are modified up to fixed constants. In our setting it is useful to note that, by (2.4) and (2.16),

[TABLE]

for all $\omega\in S^{n-1}$.

Next, we use the collection $(\varphi_{\omega})_{\omega\in S^{n-1}}$ to construct wave packets. For $\omega\in S^{n-1}$, $\sigma>0$ and $\xi\in\mathbb{R}^{n}$, set $\psi_{\omega,\sigma}(\xi):=\Psi(\sigma\xi)\varphi_{\omega}(\xi)$, where $\Psi\in C^{\infty}_{c}(\mathbb{R}^{n})$ is as in (2.10). Also let

[TABLE]

These functions have the following properties, analogous to those in [13, Lemma 4.1].

Lemma 3.2**.**

For all $\omega\in S^{n-1}$ and $\sigma>0$, one has $\psi_{\omega,\sigma}\in C^{\infty}_{c}(\mathbb{R}^{n})$. Each $\xi\in\mathrm{supp}(\psi_{\omega,\sigma})$ satisfies $\frac{1}{2}\sigma^{-1}\leq|\xi|\leq 2\sigma^{-1}$ and $|\hat{\xi}-\omega|\leq 2\sqrt{\sigma}$. For all $\xi\in\mathbb{R}^{n}$ with $|\xi|\geq 1$, one has

[TABLE]

For all $\alpha\in\mathbb{Z}_{+}^{n}$ and $\beta\in\mathbb{Z}_{+}$, there exists a constant $C=C(\alpha,\beta)\geq 0$ such that

[TABLE]

for all $\omega\in S^{n-1}$, $\sigma>0$ and $\xi\in\mathbb{R}^{n}$. Also, for each $N\geq 0$ there exists a $C_{N}\geq 0$ such that, for all $\omega\in S^{n-1}$, $\sigma>0$ and $x\in\mathbb{R}^{n}$,

[TABLE]

In particular, $\{\sigma^{\frac{n-1}{4}}\mathcal{F}^{-1}(\psi_{\omega,\sigma})\mid\omega\in S^{n-1},\sigma>0\}\subseteq L^{1}(\mathbb{R}^{n})$ is uniformly bounded. Finally, $\rho\in C^{\infty}_{c}(\mathbb{R}^{n})$ is such that $\rho(\xi)=0$ if $|\xi|\geq 1$, and $\rho(\xi)=1$ if $|\xi|\leq 1/4$.

Proof.

The proof is completely analogous to that of [13, Lemma 4.1]. The first few and the last two statements follow from properties (1), (2) and (3) of $(\varphi_{\omega})_{\omega\in S^{n-1}}$, and those of $\Psi$. One then obtains (3.4) from integration by parts, using also the equivalent expression for $|\cdot|_{\omega}$ from Lemma 2.1. Finally, since $\rho$ is radial, to see that $\rho$ is smooth it suffices to show that all its derivatives vanish where $\rho(\xi)=0$. ∎

3.2. Wave packet transforms

In this subsection we introduce wave packet transforms that lift functions on $\mathbb{R}^{n}$ to

[TABLE]

endowed with the measure $\mathrm{d}x\mathrm{d}\omega\frac{\mathrm{d}\sigma}{\sigma}$. Note that the map $(x,\omega,\sigma)\mapsto(x,\sigma^{-1}\omega)$ identifies $S^{*}_{+}\mathbb{R}^{n}$ with $\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\{0\})$, i.e. phase space minus the zero section.

To properly define our wave packet transforms, it will be convenient to work with a class of test functions on $S^{*}_{+}\mathbb{R}^{n}$, and the associated distributions. As in [13, Section 2.2], we let $\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$ consist of those $F\in L^{\infty}(S^{*}_{+}\mathbb{R}^{n})$ such that

[TABLE]

is an element of $L^{\infty}(S^{*}_{+}\mathbb{R}^{n})$ for all $N\geq 0$, endowed with the topology generated by the corresponding weighted $L^{\infty}$-norms. Here $\Upsilon:(0,\infty)\to(0,1]$ is given by

[TABLE]

Let $\mathcal{J}^{\prime}(S^{*}_{+}\mathbb{R}^{n})$ be the space of continuous linear $G:\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})\to{\mathbb{C}}$, endowed with the topology induced by $\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$. We denote the duality between $F\in\mathcal{J}^{\prime}(S^{*}_{+}\mathbb{R}^{n})$ and $G\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$ by $\langle F,G\rangle_{S^{*}_{+}\mathbb{R}^{n}}$. If $G\in L^{1}_{\mathrm{loc}}(S^{*}_{+}\mathbb{R}^{n})$ is such that

[TABLE]

defines an element of $\mathcal{J}^{\prime}(S^{*}_{+}\mathbb{R}^{n})$, then we write $G\in\mathcal{J}^{\prime}(S^{*}_{+}\mathbb{R}^{n})$.

We will typically work with the function space

[TABLE]

on $S^{*}_{+}\mathbb{R}^{n}$, for $p,q\in[1,\infty]$, consisting of all $F\in L^{1}_{\mathrm{loc}}(S^{*}\mathbb{R}^{n})$ such that

[TABLE]

It is then relevant to note that

[TABLE]

for all $p,q\in[1,\infty]$, and the inclusion is dense if $p,q\in[1,\infty)$. Moreover,

[TABLE]

for all $p,q\in[1,\infty]$.

We can now define our wave packet transform, in a similar manner as in [13]. For $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ and $(x,\omega,\sigma)\in S^{*}_{+}\mathbb{R}^{n}$, set

[TABLE]

Next, for $F\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$, set

[TABLE]

These transforms have the following properties.

Proposition 3.3**.**

The following statements hold:

1. (1)

$W:L^{2}(\mathbb{R}^{n})\to L^{2}(S^{*}_{+}\mathbb{R}^{n})$* is an isometry;* 2. (2)

$W:\mathcal{S}(\mathbb{R}^{n})\to\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$* and $W:\mathcal{S}^{\prime}(\mathbb{R}^{n})\to\mathcal{J}^{\prime}(S^{*}_{+}\mathbb{R}^{n})$ are continuous;* 3. (3)

$V:\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n})$* is continuous, and $V$ extends uniquely to a continuous map $V:\mathcal{J}^{\prime}(S^{*}_{+}\mathbb{R}^{n})\to\mathcal{S}^{\prime}(\mathbb{R}^{n})$;* 4. (4)

$\langle VF,g\rangle_{\mathbb{R}^{n}}=\langle F,Wg\rangle_{S^{*}_{+}\mathbb{R}^{n}}$* for all $F\in\mathcal{J}^{\prime}(S^{*}_{+}\mathbb{R}^{n})$ and $g\in\mathcal{S}(\mathbb{R}^{n})$, and $\langle Wf,G\rangle_{S^{*}_{+}\mathbb{R}^{n}}=\langle f,VG\rangle_{\mathbb{R}^{n}}$ for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ and $G\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$;* 5. (5)

$VWf=f$* for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$.*

Proof.

The statement and the proof are essentially contained in [13, Lemma 4.3]. More precisely, (1) follows from the definition of $\rho$ in (3.2) and from the properties of $\Psi$ and the $\varphi_{\omega}$, given that we have assumed that the surface measure $\mathrm{d}\omega$ on $S^{n-1}$ is unit normalized. Moreover, it is proved in [13, Lemma 4.3] that $W:\mathcal{S}(\mathbb{R}^{n})\to\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$ and $V:\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n})$ are continuous. The remaining statements follow from this, upon confirming that $\langle Wf,G\rangle_{S^{*}_{+}\mathbb{R}^{n}}=\langle f,VG\rangle_{\mathbb{R}^{n}}$ for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ and $G\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$. ∎

3.3. Operators on phase space

In this subsection we prove a key result about the boundedness of certain operators on function spaces over $S^{*}_{+}\mathbb{R}^{n}$.

For the proof of Proposition 4.5 below, it will be convenient for us to introduce wave packet transforms associated with other parabolic frequency localizations. More precisely, let $(\widetilde{\varphi}_{\omega})_{\omega\in S^{n-1}}\subseteq C^{\infty}(\mathbb{R}^{n})$ be a family with the same properties (1), (2) and (3) that $(\varphi_{\omega})_{\omega\in S^{n-1}}$ has, from Section 3.1. Let $\widetilde{\Psi}\in C^{\infty}_{c}(\mathbb{R}^{n})$ be such that $\mathrm{supp}(\widetilde{\Psi})\subseteq\{\xi\in\mathbb{R}^{n}\mid 1/2\leq|\xi|\leq 2\}$ and such that (2.10) holds with $\Psi$ replaced by $\widetilde{\Psi}$. For $\omega\in S^{n-1}$, $\sigma>0$ and $\xi\in\mathbb{R}^{n}$, set $\widetilde{\psi}_{\omega,\sigma}(\xi):=\widetilde{\Psi}(\sigma\xi)\widetilde{\varphi}_{\omega}(\xi)$, and write

[TABLE]

Now set, for $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ and $(x,\omega,\sigma)\in S^{*}_{+}\mathbb{R}^{n}$,

[TABLE]

Similarly, for $F\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$, write

[TABLE]

These transforms have the same properties as $W$ and $V$, from Proposition 3.3.

Theorem 3.4**.**

Let $p,q\in(1,\infty)$. Then

[TABLE]

boundedly.

Proof.

Let $p,q\in(1,\infty)$ and $F\in L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$. We have that

[TABLE]

where, for $(x,\omega,\sigma)\in S^{*}_{+}\mathbb{R}^{n}$,

[TABLE]

To estimate $I$, we start with by using the support properties of $\psi_{\omega,\sigma}$ and $\widetilde{\psi}_{\nu,\tau}$ and a change of variable to obtain the following:

[TABLE]

We now use the kernel estimates (3.4) to obtain the pointwise domination

[TABLE]

where $M_{\omega}$ is as in (2.6). We then apply, for a fixed $\omega\in S^{n-1}$, the vector-valued boundedness of $M_{\omega}$ on $L^{p}(\mathbb{R}^{n};L^{2}(\mathbb{R}_{+};\frac{d\sigma}{\sigma}))$ from Lemma 2.2.

This gives, for every $\omega\in S^{n-1}$ and $\alpha\in[1/4,4]$,

[TABLE]

where

[TABLE]

Taking $L^{q}(S^{n-1};d\omega)$ norm on both sides, we have that

[TABLE]

where $\mathcal{M}_{HL}$ denotes the Hardy-Littlewood maximal function on $S^{n-1}$ with value in the UMD lattice $L^{p}(\mathbb{R}^{n};L^{2}(\mathbb{R}_{+},\frac{d\sigma}{\sigma})))$.

By [24, Theorem 3], $\mathcal{M}_{HL}$ is bounded on $L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$. Combining this fact with the boundedness of the vector-valued anisotropic maximal functions on $L^{p}(\mathbb{R}^{n};L^{2}(\mathbb{R}_{+};\frac{d\sigma}{\sigma}))$ as before, we thus have that

[TABLE]

We now turn to $II$, and first notice that, by Lemma 3.2, the support properties of $\rho$ and $\widetilde{\psi}_{\nu,\tau}$ are such that

[TABLE]

Using the fact that $II$ is constant in $\omega$, and only depends on $\sigma$ via $\mathbf{1}_{[8,8e]}(\sigma)$, we thus have that

[TABLE]

Using the kernel estimates (3.4) and the vector-valued boundedness of $M_{\nu}$ as before, we then have that

[TABLE]

Similarly, the support properties of $\widetilde{\rho}$ and $\psi_{\omega,\sigma}$ are such that

[TABLE]

Using the kernel estimates (3.4) and the vector-valued boundedness of $M_{\omega}$ again, we thus have that

[TABLE]

Since $\widetilde{\rho}(D)$ is bounded on $L^{p}(\mathbb{R}^{n})$, we conclude that

[TABLE]

using Cauchy-Schwarz inequality in the last estimate. We finally turn to $IV$. Using the fact that this is term is constant in $\omega$, and only depends on $\sigma$ via $\mathbf{1}_{[8,8e]}(\sigma)$, we have that

[TABLE]

since $\widetilde{\rho}(D)$ and ${\rho}(D)$ are bounded on $L^{p}(\mathbb{R}^{n})$. Applying Hölder’s inequality, we thus have $\|IV\|_{L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})}\lesssim\|F\|_{L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})}$, which concludes the proof.

∎

4. Function spaces for decoupling

In this section we first define the function spaces for decoupling. We then derive some of their basic properties, including interpolation and duality theorems.

4.1. Definition

For simplicity of notation, we write $\mathcal{H}^{p}(\mathbb{R}^{n}):=L^{p}(\mathbb{R}^{n})$ for $p\in(1,\infty)$. Also, $\mathcal{H}^{1}(\mathbb{R}^{n})$ is the local real Hardy space, with norm

[TABLE]

for $f\in\mathcal{H}^{1}(\mathbb{R}^{n})$. Here $\rho\in C^{\infty}_{c}(\mathbb{R}^{n})$ is as in (3.2), although the specific choice of low-frequency cut-off is irrelevant. Moreover, $\mathcal{H}^{\infty}(\mathbb{R}^{n}):=\mathrm{bmo}(\mathbb{R}^{n})$ is the dual of $\mathcal{H}^{1}(\mathbb{R}^{n})$, and we write $\mathcal{H}^{s,p}(\mathbb{R}^{n}):=\langle D\rangle^{-s}\mathcal{H}^{p}(\mathbb{R}^{n})$ for all $p\in[1,\infty]$ and $s\in\mathbb{R}$.

Recall that $(\varphi_{\omega})_{\omega\in S^{n-1}}\subseteq C^{\infty}(\mathbb{R}^{n})$ is a fixed collection of parabolic frequency localizations, introduced in Section 3.1.

Definition 4.1**.**

Let $p\in[1,\infty]$, $q\in[1,\infty)$ and $s\in\mathbb{R}$. Then $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ consists of those $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that $\rho(D)f\in L^{p}(\mathbb{R}^{n})$, $\varphi_{\omega}(D)f\in\mathcal{H}^{s,p}(\mathbb{R}^{n})$ for almost all $\omega\in S^{n-1}$, and $(\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{\mathcal{H}^{s,p}(\mathbb{R}^{n})}^{q}\mathrm{d}\omega)^{1/q}<\infty$, endowed with the norm

[TABLE]

It is straightforward to see that $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})=\langle D\rangle^{-s}\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ for all $p\in[1,\infty]$, $q\in[1,\infty)$ and $s\in\mathbb{R}$. Note also that one may replace $\mathcal{H}^{s,p}(\mathbb{R}^{n})$ by $\langle D\rangle^{-s}H^{p}(\mathbb{R}^{n})$ in Definition 4.1, since each $\varphi_{\omega}$ vanishes near zero. Then Proposition 2.6 shows that in fact

[TABLE]

for all $p,q\in[1,\infty)$, $s\in\mathbb{R}$ and $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$.

Remark 4.2**.**

For all $p\in[1,\infty)$ and $s\in\mathbb{R}$, the space $\mathcal{H}^{p,p;s}_{\mathrm{dec}}(\mathbb{R}^{n})$ coincides with $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})=\langle D\rangle^{-s}\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, as defined in [13, 14]. This follows from equivalent characterizations of $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ proved in [21, 9]. These characterizations involve norms as in (4.1).

4.2. Basic properties

To derive some of the basic properties of the function spaces for decoupling, it will be convenient to characterize them using the wave packet transform $W$ from (3.8), and the spaces $L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$ from (3.5).

Lemma 4.3**.**

Let $p,q\in[1,\infty)$. Then an $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ satisfies $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ if and only if $Wf\in L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$, in which case

[TABLE]

In particular, $\mathcal{H}_{\mathrm{dec}}^{2,2;s}(\mathbb{R}^{n})=W^{s,2}(\mathbb{R}^{n})$ for all $s\in\mathbb{R}$, with equivalence of norms.

Proof.

The first statement follows from the Littlewood–Paley characterization of $\mathcal{H}^{p}(\mathbb{R}^{n})$, the support properties of $\Psi$ and the $\varphi_{\omega}$, and the triangle inequality. Then the second statement follows from Proposition 3.3 (1). ∎

We can now combine this lemma with Theorem 3.4 to derive various properties of the function spaces for decoupling.

For the following proposition, recall the definition of the transform $V$ from (3.9).

Proposition 4.4**.**

Let $p,q\in(1,\infty)$. Then $W:\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})\to L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$ is an isomorphism onto a complemented subspace, $V:L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})\to\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ is bounded, and

[TABLE]

is an isomorphism. In particular, $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is a Banach space for all $s\in\mathbb{R}$.

Proof.

The first statement follows from Lemma 4.3, Theorem 3.4 and Proposition 3.3 (5), since $WV$ is a bounded projection on $L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$ with image $W\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$. Next, note that $VF\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ for each $F\in L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$, by (3.7) and Proposition 3.3 (5). Hence the second statement also follows from Lemma 4.3 and Theorem 3.4. For the same reasons, $V:L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})\to\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ is surjective, which implies (4.2). The final statement in turn follows from (4.2), or alternatively because $W:\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})\to W\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ is an isomorphism onto a complemented subspace. ∎

Next, we show that the definition of $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is independent of the choice of parabolic frequency localizations. More precisely, let $(\widetilde{\varphi}_{\omega})_{\omega\in S^{n-1}}\subseteq C^{\infty}(\mathbb{R}^{n})$ be a family with the same properties (1), (2) and (3) that $(\varphi_{\omega})_{\omega\in S^{n-1}}$ has, from Section 3.1, and let $\widetilde{\rho}$ be the associated low-frequency cut-off, from (3.10).

Proposition 4.5**.**

Let $p,q\in(1,\infty)$ and $s\in\mathbb{R}$. Then there exists a $C>0$ such that the following holds for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$. One has $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ if and only if $\widetilde{\rho}(D)f\in L^{p}(\mathbb{R}^{n})$, $\widetilde{\varphi}_{\omega}(D)f\in\mathcal{H}^{s,p}(\mathbb{R}^{n})$ for almost all $\omega\in S^{n-1}$, and $(\int_{S^{n-1}}\|\widetilde{\varphi}_{\omega}(D)f\|_{\mathcal{H}^{s,p}(\mathbb{R}^{n})}^{q}\mathrm{d}\omega)^{1/q}<\infty$, in which case

[TABLE]

Proof.

It suffices to consider the case where $s=0$. We will use the operators $\widetilde{W}$ and $\widetilde{V}$ from (3.11) and (3.12). More precisely, one has $\widetilde{V}\widetilde{W}f=f$ for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$, as follows from Proposition 3.3 (5) with $W$ and $V$ replaced by $\widetilde{W}$ and $\widetilde{V}$. Hence Proposition 4.4 and Theorem 3.4 yield

[TABLE]

for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ for which the final quantity is finite. By Proposition 4.4 with $W$ replaced by $\widetilde{W}$, this proves one of the required inequalities. The other inequality follows by symmetry. ∎

Finally, we consider the Schwartz functions as a subset of $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$.

Proposition 4.6**.**

Let $p,q\in(1,\infty)$ and $s\in\mathbb{R}$. Then $\mathcal{S}(\mathbb{R}^{n})\subseteq\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ lies dense.

Proof.

It suffices to consider the case where $s=0$. By Proposition 3.3 (2) and (3.6), one has $Wf\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})\subseteq L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$ for all $f\in\mathcal{S}(\mathbb{R}^{n})$. By Lemma 4.3, this in turn implies that $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$.

To see that $\mathcal{S}(\mathbb{R}^{n})$ in fact lies dense in $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$, let $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ be given. Then $Wf\in L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$, by Lemma 4.3. Since $J(S^{*}_{+}\mathbb{R}^{n})\subseteq L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$ lies dense, there exists a sequence $(F_{j})_{j=0}^{\infty}\subseteq\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$ such that $F_{j}\to Wf$ in $L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$, as $j\to\infty$. Then, by Propositions 3.3 and 4.4, $(VF_{j})_{j=0}^{\infty}\subseteq\mathcal{S}(\mathbb{R}^{n})$ and

[TABLE]

as $j\to\infty$. ∎

4.3. Interpolation and duality

In this subsection we prove interpolation and duality properties of the function spaces for decoupling.

First, we prove that the function spaces for decoupling form a complex interpolation scale.

Proposition 4.7**.**

Let $p_{0},p_{1},p,q_{1},q_{2},q\in(1,\infty)$, $s_{0},s_{1},s\in\mathbb{R}$ and $\theta\in[0,1]$ be such that $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$, $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ and $s=(1-\theta)s_{0}+\theta s_{1}$. Then

[TABLE]

with equivalent norms.

Proof.

It suffices to consider the case where $s_{0}=s_{1}=s=0$. It is a basic fact about interpolation of vector-valued function spaces (see [17, Theorem 2.2.6]) that

[TABLE]

Hence one can use combine Theorem 3.4 with a result about interpolation of complemented subspaces (see [31, Theorem 1.17.1.1]) to conclude that

[TABLE]

Moreover, by Proposition 4.4,

[TABLE]

is an isomorphism for all $\tilde{p},\tilde{q}\in(1,\infty)$, which proves the required statement. ∎

Next, we show that the function spaces for decoupling have natural duality properties.

Proposition 4.8**.**

Let $p,q\in(1,\infty)$ and $s\in\mathbb{R}$. Then

[TABLE]

with equivalent norms, where the duality pairing is given by

[TABLE]

for $f\in\mathcal{H}_{\mathrm{dec}}^{p^{\prime},q^{\prime};-s}(\mathbb{R}^{n})$ and $g\in\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$, and by $\langle f,g\rangle_{\mathbb{R}^{n}}$ if $g\in\mathcal{S}(\mathbb{R}^{n})$.

Proof.

Since $\langle D\rangle^{s}$ commutes with $W$, it suffices to consider the case where $s=0$.

First let $f\in\mathcal{H}_{\mathrm{dec}}^{p^{\prime},q^{\prime};0}(\mathbb{R}^{n})$. Then, by Proposition 4.4, $Wf\in L^{q^{\prime}}_{\omega}L^{p^{\prime}}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$, and $Wg\in L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$ for all $g\in\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$. Hence Hölder’s inequality yields

[TABLE]

Moreover, by Proposition 3.3, if $g\in\mathcal{S}(\mathbb{R}^{n})$ then

[TABLE]

In particular, if $\langle Wf,Wg\rangle_{S^{*}_{+}\mathbb{R}^{n}}=0$ for all $g\in\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$, then $f=0$. So $\mathcal{H}_{\mathrm{dec}}^{p^{\prime},q^{\prime};0}(\mathbb{R}^{n})\subseteq(\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n}))^{*}$.

Conversely, let $l\in(\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n}))^{*}$. Then $l\circ V\in(L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n}))^{*}$, by Proposition 4.4. Moreover, by [17, Theorems 1.3.10 and 1.3.21],

[TABLE]

with the natural duality pairing. Hence there exists an $F\in L^{q^{\prime}}_{\omega}L^{p^{\prime}}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$ such that

[TABLE]

for all $G\in L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})$. Set $f:=VF$. Then $f\in\mathcal{H}_{\mathrm{dec}}^{p^{\prime},q^{\prime};0}(\mathbb{R}^{n})$, by Proposition 4.4. Also, Proposition 3.3 implies that

[TABLE]

for all $g\in\mathcal{S}(\mathbb{R}^{n})$. Finally, Propositions 3.3 and 4.4 now yield

[TABLE]

for all $G\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$. This concludes the proof, since Lemma 4.3 implies that

[TABLE]

where the supremum is over all $G\in\mathcal{J}(S^{*}_{+}\mathbb{R}^{n})$ with $\|G\|_{L^{q}_{\omega}L^{p}_{x}L^{2}_{\sigma}(S^{*}_{+}\mathbb{R}^{n})}\leq 1$. ∎

5. Invariance

In this section we show that $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is invariant under the Euclidean half-wave propagators, but that it is not invariant under general Fourier integral operators unless $p=q$.

5.1. Bounded operators

We first note that the function spaces for decoupling are invariant under the action of the Euclidean half-wave propagators.

Theorem 5.1**.**

Let $p,q\in(1,\infty)$ and $s\in\mathbb{R}$. Then there exist $C,N>0$ such that, for all $t\in\mathbb{R}$, one has $e^{it\sqrt{-\Delta}}\in\mathcal{L}(\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n}))$, with

[TABLE]

Proof.

First note that it suffices to consider the case where $s=0$. Moreover, since $\rho^{\prime}(D)e^{it\sqrt{-\Delta}}\in\mathcal{L}(L^{p}(\mathbb{R}^{n}))$ for all $t\in\mathbb{R}$ and $\rho^{\prime}\in C^{\infty}_{c}(\mathbb{R}^{n})$ with $\rho^{\prime}\equiv 1$ on $\mathrm{supp}(\rho)$, we may prove that

[TABLE]

for all $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$. In turn, to this end it suffices to show that

[TABLE]

for an implicit constant independent of $\omega$ and $f$. But the latter statement follows from the arguments in [11, Section 9], which rely on writing

[TABLE]

for a suitable $\chi_{\omega}\in C^{\infty}(\mathbb{R}^{n})$ with the property that $\chi_{\omega}\equiv 1$ on $\mathrm{supp}(\varphi_{\omega})$. Then $\chi_{\omega}(D)(e^{it(\sqrt{-\Delta}-w\cdot D)}-1)$ is an anisotropic Calderón–Zygmund operator, and an application of the Marcinkiewicz–Lizorkin multiplier theorem (see [34] for a version with values in a Banach space) yields the required conclusion. ∎

5.2. Unbounded operators

We recall from [13] (see also [20, Corollary 3.4] and Corollary 6.5 below) that $\mathcal{H}_{\mathrm{dec}}^{p,p;s}(\mathbb{R}^{n})=\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ is invariant under any FIO of order zero as in Definition 2.7, if the symbol has compact support in the spatial variable. In this section we show that, by contrast, for all $q\in(1,\infty)\setminus\{p\}$, the space $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is in general not invariant under such Fourier integral operators.

Let $\phi:\mathbb{R}^{2}\to\mathbb{R}^{2}$ denote the diffeomorphism

[TABLE]

We denote the inverse map by $\psi$. Clearly, we have

[TABLE]

We let $T$ denote the operation of pullback by $\psi$, followed by multiplication by a smooth cutoff function $k(x)$ that is equal to $1$ when $|x|\leq 4$ and zero when $|x|\geq 8$. That is,

[TABLE]

for $f\in\mathcal{S}(\mathbb{R}^{2})$ and $x\in\mathbb{R}^{2}$. Thus, $T$ is a properly supported FIO of order zero, associated to a canonical transformation, namely the transpose of $d\psi$ (which maps $T_{\psi(x)}^{*}\mathbb{R}^{2}$ to $T_{x}^{*}\mathbb{R}^{2}$) on $T^{*}\mathbb{R}^{2}$. In fact, $T$ is an operator as in Definition 2.7, with symbol $(2\pi)^{-2}k$ and phase function $\Phi(x,\eta)=\psi(x)\cdot\eta$. As such, it is bounded on $\mathcal{H}_{\mathrm{dec}}^{p,p;s}(\mathbb{R}^{2})=\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{2})$ for all $p\in(1,\infty)$ and $s\in\mathbb{R}$.

Theorem 5.2**.**

Let $p,q\in(1,\infty)$ be such that $p\neq q$. Then $T$ is not bounded on $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{2})$.

Remark 5.3**.**

For $s\in\mathbb{R}$, the operator $T_{s}:=\langle D\rangle^{s}T\langle D\rangle^{-s}$ has similar properties as $T$, and it is not bounded on $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{2})$. Moreover, it follows from the proof of Theorem 5.2 that one can use a similar construction to show, for all $n\geq 2$, that $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is not invariant under general FIOs of order zero.

Before getting into the technical details, we describe the idea. First, any Fourier integral operator $F$, including $T$, maps one wave packet, located at $(x,\omega)\in S^{*}\mathbb{R}^{n}$, to another (slightly distorted) wave packet, located at $\chi(x,\omega)$ where $\chi$ is the map on $S^{*}\mathbb{R}^{n}$ induced by the (homogeneous) canonical relation of $F$. Below, we choose $u$ as a sum of many wave packets all pointing in the same direction $\omega_{0}$. We shall see that the $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ norm sums the $L^{p}(\mathbb{R}^{n})$ norm of each wave packet in, essentially, a ‘little $\ell^{p}$ way’, thus the growth of the norm in $N$, the number of wave packets, will be $\sim N^{1/p}$. The image of these wave packets, however, will all have different frequency directions $\omega$. As a result, the $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{n})$ norm sums the $L^{p}(\mathbb{R}^{n})$ norm of each wave packet in a ‘little $\ell^{q}$ way’, thus the growth of the norm in $N$ of the image of $u$ will be $\sim N^{1/q}$. If $q<p$ then the norm of the image grows faster than the norm of $u$, demonstrating that the operator is not bounded. In the other case $q>p$, we just have to consider the opposite situation, that is, we consider $u$ that is a sum of many wave packets in different directions such that $Tu$ is a sum of wave packets all pointing in the same direction.

Proof of Theorem 5.2.

We first consider the case where $q<p$.

We choose a frequency scale $R>0$, which will be large and, eventually, taken to infinity. We let $\omega_{0}$ be the unit vector $(1,0)\in\mathbb{R}^{2}$ and let $u_{0}$ be the wave packet

[TABLE]

for $x\in\mathbb{R}^{2}$, where $\chi\in C^{\infty}_{c}(\mathbb{R})$ is supported on $[1/2,4]$ and identically $1$ on $[1,2]$. Thus, $\hat{u}_{0}$ is a symbol of order $1/4$ and type $(1/2,0)$ supported in a second dyadic region where $|\xi|\in[R,2R]$ and $\hat{\xi}$ is within111Throughout this subsection, we will use $C$ and $c$ for sufficiently large, resp. sufficiently small, constants, the value of which may change from line to line. $CR^{-1/2}$ of $\omega_{0}$.

We now choose an integer $N$ comparable to $R^{\alpha}$ for some fixed $0<\alpha<1/2$. We consider a function that consists of a sum of $N$ translates of $u_{0}$, along the line $x_{1}=1$. We define

[TABLE]

Let

[TABLE]

for $x\in\mathbb{R}^{2}$. We set

[TABLE]

In preparation for computing the $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{2})$ norm of $u$, we apply $\varphi_{\omega}(D)$ to $u$. As this operator is a Fourier multiplier, this gives us

[TABLE]

The product $\varphi_{\omega_{0}}(\xi)\varphi_{\omega}(\xi)$ vanishes for $|\omega-\omega_{0}|\geq CR^{-1/2}$, and is a symbol of order $1/2$ and type $(1/2,1)$ (that is, type $1$ in the radial direction and $1/2$ in orthogonal directions) with support contained in $\mathrm{supp}(\varphi_{\omega_{0}})$ for $|\omega-\omega_{0}|\leq CR^{-1/2}$. We apply a standard integration by parts argument, using the fact that the function $e^{i(x-y_{j,N})\cdot\xi}$ is invariant under the differential operators

[TABLE]

to obtain a uniform upper bound:

[TABLE]

for $|\omega-\omega_{0}|\leq CR^{-1/2}$ and $x\in\mathbb{R}^{2}$, where $C_{N}\geq 0$ is uniform for $\omega$ in this range. Essentially this is the same estimate as in (3.4), with $\sigma=R^{-1}$ and using (2.3), except that the symbol here is order $1/4$ greater than that in (3.4). Since the $\varphi_{\omega}(D)u_{j,N}$ are all decaying rapidly at scale $R^{-\beta}$, for any $\beta<1/2$ due to (5.1), the $L^{p}(\mathbb{R}^{2})$ norm of $\varphi_{\omega}(D)u_{j,N}$ is equal to the $L^{p}(\mathbb{R}^{2})$ norm of $\varphi_{\omega}(D)u_{j,N}$ restricted to a ball $B_{j,N}\subseteq\mathbb{R}^{2}$ of radius $R^{-\beta}$ centered at $y_{j,N}$, where $\beta$ is chosen so that $\alpha<\beta<1/2$, up to an $O(R^{-\infty})$ error. As these balls are all disjoint (using that $\beta>\alpha$ and the fact that the spacing between the centers of these balls is $\sim 1/N\sim R^{-\alpha}$), it follows that $\|\varphi_{\omega}(D)u\|_{L^{p}(\mathbb{R}^{2})}^{p}$ is equal, up to an $O(R^{-\infty})$ error, to the sum

[TABLE]

We therefore obtain an upper bound for the $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{2})$ norm of $u$, by using this bound in the region $|\omega-\omega_{0}|\leq CR^{-1/2}$ and noting that the integrand is zero for all other $\omega$. Using the definition of the norm of $\mathcal{H}_{\mathrm{dec}}^{p,q;0}(\mathbb{R}^{2})$ in Definition 4.1, we obtain

[TABLE]

where $C$ is independent of $u$ and $R$.

We now consider $v:=Tu$, which is

[TABLE]

for $x\in\mathbb{R}^{2}$.

We want to apply the Fourier multiplier $\varphi_{\omega}(D)v$ to $v$, so we take the Fourier transform of $v$. This is

[TABLE]

Let $v_{j,N}$ be the $j$th term in this sum. We scale the frequencies $\xi$ and $\eta$ by a factor of $R$ and write in the form

[TABLE]

For each fixed $\xi$, the phase function $f(x,\eta):=-x\cdot\xi+(\psi(x)-y_{j,N})\cdot\eta$ has a unique critical point at $x=x_{j,N}:=\psi^{-1}(y_{j,N})$ and $\xi=d_{x}\psi(x_{j,N})\cdot\eta$, which uniquely determines $\eta$ since $\psi$ is a diffeomorphism and therefore $d_{x}\psi$ is an invertible linear map. It is easy to check that the critical point is nondegenerate, with Hessian at the critical point given in block form by

[TABLE]

We notice that zero entry in all the $(\eta_{i},\eta_{j})$ positions, which is a crucial feature that will be exploited in the proof of Lemma 5.4.

Lemma 5.4**.**

The stationary phase expansion is valid for the integral (5.3) as $R\to\infty$.

Proof.

The proof is a bit subtle, as the amplitude $u(x):=k(x)\psi_{\omega_{0}}(R\eta)\chi(|\eta|)$ also depends on the parameter $R$, so we have to show that the remainder term in the asymptotic expansion vanishes to higher and higher powers of $R$. This requires careful estimation of derivatives of the amplitude $u$, tracking the dependence on $R$.

We begin by noting that $u$ satisfies estimates

[TABLE]

We denote the critical point by $(x_{0},\eta_{0})$, and introduce an $R$-dependent cutoff function $\tilde{\chi}(x,\eta)$ that is $1$ in a neighbourhood of $(x_{0},\eta_{0})$ of diameter $R^{-b}$ and is supported in a larger $R^{-b}$ neighbourhood of this point, with derivatives estimated by corresponding powers of $R^{b}$. As we shall see, we need to take $1/4<b<1/2$.

First we consider $u_{1}=u(1-\tilde{\chi})$, which therefore vanishes in a $R^{-b}$ neighbourhood of the critical point. Since the Hessian is nondegenerate, it follows that the derivative of the phase function has size $|\nabla f|\geq CR^{-b}$ on the support of $u_{1}$. Then Theorem 7.7.1 of [15] shows that, for any integer $k$, we can estimate the integral pointwise by

[TABLE]

Using the fact that the derivatives $|D^{\alpha}u_{1}|$ are bounded by $R^{|\alpha|/2}$ (using $b<1/2$) this is estimated by

[TABLE]

where the exponent tends to $-\infty$ as $k\to\infty$. Thus, $u_{1}$ contributes nothing to the expansion as $R\to\infty$.

Next we consider $u_{2}=u\tilde{\chi}$. In this case, we write $f(x,\eta)=f(x_{0},\eta_{0})+Q(x-x_{0},\eta-\eta_{0})/2+g(x,\eta)$, where $Q$ is the quadratic part of the Taylor series and $g$ vanishes to third order at $(x_{0},\eta_{0})$. We now view the $g$ term as belonging to the amplitude rather than the exponential, so we consider the integral

[TABLE]

Let $u_{3}$ be the amplitude in this expression. We use the identity (see [15, proof of Lemma 7.7.3])

[TABLE]

and we use estimate (7.6.7) of [15, Section 7.6]: for $s$ an integer larger than $n/2=1$,

[TABLE]

We need to estimate the powers of $R$ in the sum of squared norms on the RHS. We use the crucial fact that the second order differential operator $Q(D_{x},D_{\eta})$ only has first derivatives in the $\eta$ direction, according to (5.4). So, each application of $Q(D_{x},D_{\eta})$ involves either two $x$-derivatives or one $x$-derivative and one $\eta$-derivative. We notice that $x$-derivatives are harmless (in terms of accumulating powers of $R$) when they hit the $k$ factor, while if they hit the $\tilde{\chi}$ factor we incur $R^{b}$ and if they hit the $e^{iRg}$ factor they incur $R|\nabla g|$. Since $g$ vanishes to third order at $(x_{0},\eta_{0})$ and since $u_{3}$ is supported within distance $R^{-b}$ of $(x_{0},\eta_{0})$, we estimate $R|\nabla g|$ by $CR^{1-2b}$ (since $\nabla g$ still has second order vanishing at the critical point). On the other hand, a $\eta$-derivative incurs either $R^{1/2}$ if it hits the $\psi_{\omega_{0}}(R\eta)$ factor or $CR^{1-2b}$ if it hits the $e^{iRg}$ factor. Taking into account $b>1/4$, we incur at worst $R^{1/2}$ from the $\eta$-derivative.

Thus, each time we apply $Q(D_{x},D_{\eta})$ to $u_{3}$ we incur at worst $R^{1-2b}R^{1/2}$. This is an exponent strictly less than $1$ since $b$ has been chosen strictly greater than $1/4$. Since we gain a power $R^{-1}$ from each factor of $Q(D_{x},D_{\eta})$, this shows that the RHS of (5.7) is estimated by a power of $R$ with exponent tending to $-\infty$ linearly with $k$. We conclude that the ‘formal’ stationary phase expansion, which can be taken to be the expansion appearing on the LHS of (5.7), is a valid asymptotic expansion as $R\to\infty$. ∎

Continuation of the proof of Theorem 5.2. Lemma 5.4 shows that we can restrict attention to a finite number of terms of the expansion. We first consider the leading term in the expansion. The contribution to the Fourier transform of $\phi_{\omega}(D)v$ of the leading term is

[TABLE]

Its support is contained in the union of sets, over $j=1\dots N$,

[TABLE]

We now consider this function of $\xi$, assuming that $\omega$ lies in $O(R^{-1/2})$ neighbourhoods of the direction $\omega_{j,N}$ of $d\psi^{t}_{x_{j,N}}\omega_{0}$, for $j=1,\dots,N$. It is easy to compute that

[TABLE]

Notice that, as $j$ varies, these directions $\omega_{j,N}$ are separated by an amount at least $cN^{-1}\gg R^{-1/2}$. Let $S_{j,N}$ be the neighbourhood of $\omega_{j,N}$ in $S^{1}$ of size $cR^{-1/2}$, for a suitably small $c<1$. Then for $\omega$ in $S_{j,N}$, we have $\varphi_{\omega}(D)v=\varphi_{\omega}(D)v_{j}$ up to an error which is $O(R^{-\infty})$ in $L^{p}$, where $v_{j}=u_{j}\circ\psi$. We see that for $\omega\in S_{j,N}$, $\varphi_{\omega}(D)v$ is given by an inverse Fourier transform

[TABLE]

where $\hat{v}_{j}(\xi)$ is the $j$th summand in (5.8). We seek a lower bound on the $L^{p}$ norm of this function, for $\omega\in S_{j,N}$. Taking $c$ sufficiently small, there is a rectangle in $\eta$-space of uniform size $cR\times cR^{1/2}$ where the first term in the expansion of $\hat{v}_{j}(\xi)$ is at least $cR^{1/2}$. Moreover, the principal symbol is real and nonnegative, so its contribution to the integral (5.9) is non-oscillatory for $y=\psi(x)$ in a rectangle $X$ centred at $y_{j,N}$ of dimensions $cR^{-1}\times cR^{-1/2}$. The contribution of the leading term in the stationary phase expansion for $v_{j}$ thus has a real part at least $cR^{2}$ on this rectangle $X$, contributing at least $cR^{2}R^{-3/(2p)}$ to the $L^{p}$ norm.

Now when we consider the subleading terms in the asymptotic expansion, these give rise to an integral similar to (5.9) where the integrand, like the leading term, is a symbol of type $(1/2,1)$ as with (5.1), but with order lower, compared to the leading term, by at least $1/2$ (because it arises from a term on the RHS of (5.7) where at least one $\eta$-derivative hits $\psi_{\omega_{0}}$). Thus the inverse Fourier transform is bounded, similar to (5.1), by

[TABLE]

This contributes at most $CR^{2-\gamma}R^{-3/(2p)}$ to the $L^{p}$ norm, where $\gamma=0$ for the leading term but $\gamma>0$ for all subleading terms. This is a strictly smaller contribution compared to our lower bound for the leading term, and therefore can be neglected for $R$ sufficiently large.

We see that there is a lower bound of $cR^{2}R^{-3/(2p)}$ for the $L^{p}$ norm of $v$, when $\omega\in S_{j,N}$. Now taking an $L^{q}$ norm in $\omega$, we have shown a lower bound of the form

[TABLE]

Comparing this to (5.2) we see that pullback by the diffeomorphism $\psi$ is not bounded on $\mathcal{H}_{\mathrm{dec}}^{s,p,q}(\mathbb{R}^{n})$ for $q<p$. A very similar argument shows that diffeomorphisms are not bounded for $q>p$; we just need to act on a sum of wave packets pointing in different directions, such that pullback by the diffeomorphism maps them to wave packets pointing in the same direction. We omit the details. ∎

Remark 5.5**.**

This example makes clear that the same phenomenon will happen for any FIO whose canonical relation maps phase space points $(x_{j},\omega_{0})$ with varying base point $x_{j}$ and fixed direction $\omega_{0}$ to points with varying direction. For a concrete example, consider the FIO $T=e^{it\sqrt{L}}$, where $L$ is the divergence form operator

[TABLE]

in two space dimensions. The bicharacteristic flow associated with $\sqrt{L}$ is

[TABLE]

Letting $\omega=\xi_{1}/\xi_{2}$, we find that $x_{1}$ and $\omega$ satisfying the autonomous equations

[TABLE]

showing that $x_{1}$ and $\omega$ move along circles $\{x_{1}^{2}+\omega^{2}=\text{constant}\}$ at a speed depending only on the radius of the circle. It is thus easy to write down the exact solution and verify that for a fixed small positive time $t$, if $\omega(0)=0$ and $x_{1}(0)=c$ then $\omega(t)=\omega(t,c)$ varies with $c$. Threrefore, we can implement a similar example as above, to conclude that the FIO $e^{it\sqrt{L}}$ is not a bounded operator from $\mathcal{H}_{\mathrm{dec}}^{0,p,q}(\mathbb{R}^{n})$ for $q\neq p$. Again, we omit the details.

6. Embeddings

In this section we consider various embedding properties of $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$. To do so, we first prove some norm inequalities involving the parabolic frequency localizations from Section 3.1. We then consider inclusions involving the function spaces for decoupling and the classical Sobolev spaces $\mathcal{H}^{s,p}(\mathbb{R}^{n})$, and finally we prove fractional integration theorems for $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$.

6.1. Preliminary results

Some of the embedding properties involving the function spaces for decoupling are obvious. For example, by Hölder’s inequality, it is clear that as the parameters $q$ and $s$ grow, the space $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ gets smaller. In other words, for all $p,q,v\in[1,\infty)$ and $s,r\in\mathbb{R}$, one has

[TABLE]

if $s\geq r$ and $q\geq v$.

On the other hand, some of the other embeddings involving the function spaces for decoupling are more subtle, and to prove those we will rely in part on the following norm inequalities involving the functions $(\varphi_{\omega})_{\omega\in S^{n-1}}$ from Section 3.1.

Proposition 6.1**.**

Let $p\in[1,\infty)$ and $u\in(p,\infty)$, and set $s:=\frac{n+1}{2}(\frac{1}{p}-\frac{1}{u})$. Then there exists a $C\geq 0$ such that the following statements holds for all $\omega\in S^{n-1}$ and $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$.

1. (1)

If $\varphi_{\omega}(D)f\in\mathcal{H}^{p}(\mathbb{R}^{n})$, then $\langle D\rangle^{-s}\varphi_{\omega}(D)f\in L^{u}(\mathbb{R}^{n})$ and

[TABLE] 2. (2)

If $f\in H^{p}_{\omega}(\mathbb{R}^{n})$, then $\langle D\rangle^{-\frac{n-1}{4}}\varphi_{\omega}(D)f\in H^{p}(\mathbb{R}^{n})$ and

[TABLE]

Proof.

For (1), we write

[TABLE]

Here, for $\xi\in\mathbb{R}^{n}$,

[TABLE]

and

[TABLE]

where $\tilde{\varphi}_{\omega}(\xi)=1$ on the support of $\varphi_{\omega}(\xi)$. Then $m_{1}(D)$, being smoothing of order $2s$ non-isotropically, is bounded from $H^{p}_{\omega}(\mathbb{R}^{n})$ to $L^{u}(\mathbb{R}^{n})$. Moreover,

[TABLE]

for all $\alpha\in\mathbb{Z}_{+}^{n}$ and $\beta\in\mathbb{Z}_{+}$, so $m_{2}(D)$ is bounded from $L^{u}(\mathbb{R}^{n})$ to itself for all $1<u<\infty$. As a result,

[TABLE]

where we used Proposition 2.6 in the last inequality to pass between isotropic and non-isotropic versions of $\mathcal{H}^{p}$ if $p=1$. This proves (1).

To prove (2), one observes that

[TABLE]

for all $f\in H^{p}_{\omega}(\mathbb{R}^{n})$. Here the first inequality follows from Proposition 2.6, and the second from the fact that

[TABLE]

for all $\alpha\in\mathbb{Z}_{+}^{n}$ and $\beta\in\mathbb{Z}_{+}$, so that $\langle D\rangle^{-\frac{n-1}{4}}\varphi_{\omega}(D)$ is bounded on $H^{p}_{\omega}(\mathbb{R}^{n})$. ∎

6.2. Sobolev embeddings

Recall from Remark 4.2 that $\mathcal{H}_{\mathrm{dec}}^{p,p;s}(\mathbb{R}^{n})=\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ for all $p\in[1,\infty)$ and $s\in\mathbb{R}$. Hence the Sobolev embeddings for $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ from (1.1) immediately yield the following embeddings for $\mathcal{H}_{\mathrm{dec}}^{p,p;s}(\mathbb{R}^{n})$, for all $p\in[1,\infty)$ and $s\in\mathbb{R}$:

[TABLE]

The following theorem extends these embeddings to $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ for more general values of $q$.

Theorem 6.2**.**

Let $p\in[1,\infty]$, $q\in[1,\infty)$ and $s\in\mathbb{R}$. Then the following assertions hold.

1. (1)

If $q\leq\max(p,2)$, then

[TABLE] 2. (2)

If $q\geq\min(p,2)$, then

[TABLE]

Proof.

First note that (2) follows from (1) by duality, cf. Proposition 4.8. Moreover, in light of (6.1), we only need to prove (1) when $q=\max\{p,2\}$. This means we need to establish (1) when $2\leq p<\infty$ and $q=p$, which is already contained in (6.2), and to establish (1) when $1<p\leq 2$ and $q=2$, which we will do in the following.

Below we prove (1) when $p=1$ and $q=2$, and then appeal to interpolation.

Assume without loss of generality that $s=-s(1)=-\frac{n-1}{4}$. Our goal is to show that

[TABLE]

Since $\|\rho(D)f\|_{L^{1}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{1}(\mathbb{R}^{n})}\lesssim\|f\|_{\mathcal{H}^{1}(\mathbb{R}^{n})}$, we only need to bound the second term on the left hand side above. For $j\geq 1$, let $\Delta_{j}$ be the Littlewood-Paley projection onto $\{|\xi|\simeq 2^{j}\}$, and for $\omega\in S^{n-1}$, let

[TABLE]

Since

[TABLE]

(the term $\|f\|_{L^{1}}$ on the right hand side bounds $\sum_{-4\leq j\leq 0}\|\varphi_{\omega}(D)\Delta_{j}f\|_{\mathcal{H}^{-\frac{n-1}{4},1}}$), it suffices to show that

[TABLE]

Let $f$ be a local Hardy $\mathcal{H}^{1}$ atom, adapted to a (Euclidean) ball of radius $r$. By translation invariance we may assume that the ball is centered at [math]. First assume $0<r<1$, which is the main case. Then $\|f\|_{L^{\infty}}\leq r^{-n}$, and $\int f=0$. We will show that

[TABLE]

and

[TABLE]

(Here $B_{r}(0,\omega)$ is the non-isotropic ball $\{|x\cdot\omega|\leq r,|x\cdot\omega^{\perp}|\leq r^{1/2}\}$.) In addition, we will show that if $1<2^{j}<r^{-1}$, then

[TABLE]

which implies

[TABLE]

(because we can bound the $\ell^{2}$ norm by $\ell^{1}$ norm and apply Fubini). (6.4), (6.5) and (6.7) imply our desired estimate (6.3). We now prove (6.4), (6.5) and (6.6).

The proof of (6.4) is via $L^{2}$ theory. Cauchy-Schwarz and Plancherel gives

[TABLE]

where $R_{\omega,j}$ is the support of the multiplier for $T_{\omega,j}$; it is a dyadic-parabolic region of size $2^{j}\times(2^{j/2})^{n-1}$. Taking $L^{2}$ norm over $\omega\in S^{n-1}$, we have

[TABLE]

where in the last inequality we have used the fact that

[TABLE]

We may now bound $2^{-\frac{j}{2}(n-1)}$ trivially by $r^{\frac{n-1}{2}}$, and pull it outside the sum in $j$. Thus

[TABLE]

establishing (6.4).

The proofs of (6.5) and (6.6) are via kernel estimates. By rotation symmetry, we will assume $\omega=e_{1}$. For $j\geq 1$, let $m_{j}$ be the multiplier of $T_{\omega,j}$ and $K_{j}$ be the inverse Fourier transform of $m_{j}$. Then $m_{j}$ is supported in $\{|\xi_{1}|\simeq 2^{j},|\xi^{\prime}|\leq 2^{j/2}\}$, and

[TABLE]

It follows that

[TABLE]

for all positive integers $N$, where $\|x\|:=|x_{1}|^{1/2}+|x^{\prime}|$; furthermore,

[TABLE]

for all positive integers $N$. We will use (6.8) in the proof of (6.5), and use (6.9) (with $\alpha+|\beta|=1$) in the proof of (6.6).

To continue with the proof of (6.5), from (6.8), we have

[TABLE]

for all $x\in\mathbb{R}^{n}$ if $N>n+1$ (fix one such $N$ from now on). This is useful for $x\notin B_{10r}(0,\omega)$: in that case $\|x\|\geq 10r^{1/2}$, and

[TABLE]

(more precisely, the first inequality above is Cauchy-Schwarz, using that $\int|f|\leq 1$, and the second inequality holds if $\|x\|\geq 10r^{1/2}$, because then

[TABLE]

which upon taking the square root gives the second inequality). As a result, the left hand side of (6.5) is bounded by

[TABLE]

as desired.

Finally, the proof of (6.6) makes additional use of the cancellation condition $\int f=0$. To this end, suppose $1<2^{j}<r^{-1}$. Then

[TABLE]

because $\int f=0$. As a result,

[TABLE]

But for $|y|\leq r$,

[TABLE]

From (6.9), we have

[TABLE]

while

[TABLE]

This completes our proof of (6.6).

It remains to prove (6.3) when $f$ is a local Hardy atom adapted to $B_{r}(0)$ with $r>1$, which is easier. In that case, we will show that

[TABLE]

uniformly in $\omega\in S^{n-1}$, from which (6.3) follows. To prove (6.10), we split the integral into over $B_{10r}(0)$ and its complement. Then $L^{2}$ theory gives

[TABLE]

and kernel estimates give

[TABLE]

(which follows from (6.8) with $N$ replaced by $2N$ there, because $\|x\|\geq|x|^{1/2}$ if $|x|\geq 10$). It follows that

[TABLE]

for $N$ large, and

[TABLE]

if $x\notin B_{10r}(0)$. This shows

[TABLE]

as well, giving finally (6.10). Note we did not need any cancellation condition on $f$ in the above argument when $r>1$.

The above argument shows that if $\operatorname{Re}z=\frac{n-1}{4}$, then the operator

[TABLE]

is bounded from $H^{1}(\mathbb{R}^{n})$ into $L^{2}(S^{n-1};H^{1}(\mathbb{R}^{n}))$. In particular, it is bounded from $H^{1}(\mathbb{R}^{n})$ to $L^{2}(S^{n-1};L^{1}(\mathbb{R}^{n}))$, and one can check that the bound grows at most polynomially in the imaginary part of $z$. Furthermore, $T_{z}$ is bounded from $L^{2}(\mathbb{R}^{n})$ to $L^{2}(S^{n-1};L^{2}(\mathbb{R}^{n}))$ when $\textrm{Re}\,z=0$, by Plancherel, and this bound is independent of the imaginary part of $z$. Also note that if $\frac{1}{p}=\frac{1-\theta}{1}-\frac{\theta}{2}$, then we have

[TABLE]

(see e.g. [32, Theorem 2.4.7]) and

[TABLE]

by e.g. [3, Theorems 5.1.1 and 5.1.2]). Thus by Stein’s complex interpolation theorem, $T_{s(p)}$ is bounded from $L^{p}(\mathbb{R}^{n})$ to $L^{2}(S^{n-1};L^{p}(\mathbb{R}^{n}))$ for $1<p<2$. In other words,

[TABLE]

for $1<p<2$. Together with the estimate $\|\rho(D)f\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{W^{s(p),p}(\mathbb{R}^{n})}$ we obtain the assertion of Theorem 6.2 (1) for $1<p<2$. ∎

6.3. Fractional integration

In this subsection we consider embeddings between the spaces $\mathcal{H}_{\mathrm{dec}}^{p_{1},q;r}$ and $\mathcal{H}_{\mathrm{dec}}^{p_{2},q;s}(\mathbb{R}^{n})$.

More precisely, we consider fractional integration theorems, where one trades in the differentiability parameter $s$ to increase the integrability parameter $p$. In this regard, the classical result says that

[TABLE]

for all $p_{1},p_{2}\in[1,\infty]$ and $s\in\mathbb{R}$ with $p_{1}\leq p_{2}$. The following is an analog of this embedding involving the function spaces for decoupling, which in fact constitutes a strict improvement of (6.11) if $p_{1}\leq 2\leq p_{2}$ (and $(p_{1},p_{2})\neq(2,2)$).

Theorem 6.3**.**

Let $p_{1},p_{2},q\in[1,\infty)$ and $s\in\mathbb{R}$ be such that $p_{1}\geq p_{2}$. Then

[TABLE]

Proof of Theorem 6.3.

We may suppose that $s=0$. Recall that

[TABLE]

But for $r:=\frac{n+1}{2}(\frac{1}{p_{1}}-\frac{1}{p_{2}})$,

[TABLE]

by Proposition 6.1 (1). Hence

[TABLE]

Remark 6.4**.**

If $p_{1}\leq 2\leq p_{2}<\infty$, then one can recover (6.11) from Theorem 6.3, by combining (6.12) with Theorem 6.2:

[TABLE]

Here we used that $s(p_{1})+s(p_{2})=\frac{n-1}{2}(\frac{1}{p_{1}}-\frac{1}{p_{2}})$, because $p_{1}\leq 2\leq p_{2}$. In fact, by the sharpness of the embeddings in Theorem 6.2, in this case (6.12) is a strict improvement of (6.11), at least if $(p_{1},p_{2})\neq(2,2)$. On the other hand, for $p_{1},p_{2}\leq 2$ or $p_{1},p_{2}\geq 2$, again due to the sharpness of the embeddings in Theorem 6.2, (6.11) neither follows from (6.12), nor the other way around.

Although Theorem 6.3 only yields a strict improvement of (6.11) for $p_{1}\leq 2\leq p_{2}$, outside of this range we can nonetheless improve a classical result about the mapping properties of Fourier integral operators. Recall the definition of Fourier integral operators in standard form, from Definition 2.7.

Corollary 6.5**.**

Let $m\in\mathbb{R}$, and let $T$ be one of the following:

1. (1)

a Fourier integral operator of order $m$ and type $(1/2,1/2,1)$ in standard form, the symbol of which has compact support in the spatial variable; 2. (2)

a Fourier integral operator of order $m$ and type $(1/2,1/2,1)$ in standard form, associated with a global canonical graph; 3. (3)

a compactly supported Fourier integral operator of order $m$ and type $(1,0)$, associated with a local canonical graph.

Let $p,q\in[1,\infty)$ and $s\in\mathbb{R}$ be such that $p\leq q$. Then

[TABLE]

is bounded. In particular, suppose that one of the following conditions holds:

1. (a)

$1\leq p\leq q\leq 2$* and $m=\frac{1}{q}-\frac{1}{p}-2s(p)$;* 2. (b)

$2\leq p\leq q<\infty$* and $m=\frac{1}{q}-\frac{1}{p}-2s(q)$.*

Then $T:\mathcal{H}^{p}(\mathbb{R}^{n})\to L^{q}(\mathbb{R}^{n})$ is bounded.

Proof.

The first statement follows by combining Theorem 6.3 with the invariance of $\mathcal{H}_{\mathrm{dec}}^{q,q;s}(\mathbb{R}^{n})=\mathcal{H}^{s,q}_{FIO}(\mathbb{R}^{n})$ under the operator $T$. For $m=s=0$, this invariance is contained in [13, Theorem 6.10] in cases (2) and (3), and the techniques used there also allow one to treat operators as in (1). For general $m,s\in\mathbb{R}$, the mapping property $T:\mathcal{H}^{s+m,q}_{FIO}(\mathbb{R}^{n})\to\mathcal{H}^{s,q}_{FIO}(\mathbb{R}^{n})$ can be found in [20, Proposition 3.3 and Corollary 3.4] in cases (1) and (3). In case (2) it follows from [22, Proposition 3.3], after precomposing with the operator $\langle D\rangle^{-m}$.

The second statement follows from the first statement, combined with Theorem 6.2. Indeed, in both cases one has

[TABLE]

so that embedding $\mathcal{H}_{\mathrm{dec}}^{q,q;s(q)}(\mathbb{R}^{n})\subseteq L^{q}(\mathbb{R}^{n})$ concludes the proof.

∎

Remark 6.6**.**

In the case of an operator as in (1) with symbol in $S^{m}_{1,0}$, the final statement in Corollary 6.5 is well known (see e.g. [30, Section IX.6.15]), and from this one can derive the same statement for operators as in (3). Corollary 6.5 improves upon this result in several ways, since it allows for the larger class of $S^{m}_{1/2,1/2,1}$ symbols, removes the assumption that the symbol has compact spatial support, cf. (2), and yields stronger estimates through (6.13), due to the sharpness of the Sobolev embeddings in Theorem 6.2.

7. Decoupling

In this section we show that the decoupling inequalities for the sphere and the light cone are equivalent to norm bounds for certain functions in $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ and $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n+1})$, respectively. To do so, we first give an equivalent description of the $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ norm of functions with frequency support in a dyadic annulus.

7.1. A discrete description of the $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ norm

Throughout this section, for each $R\geq 2$, fix a maximal collection $V_{R}\subseteq S^{n-1}$ of unit vectors such that $|\nu-\nu^{\prime}|\geq R^{-1/2}$ for all $\nu,\nu^{\prime}\in V_{R}$. Then $V_{R}$ has approximately $R^{(n-1)/2}$ elements. Let $(\chi_{\nu})_{\nu\in V_{R}}\subseteq C^{\infty}(\mathbb{R}^{n}\setminus\{0\})$ be an associated partition of unity. More precisely, each $\chi_{\nu}$ is positively homogeneous of degree [math] and satisfies $0\leq\chi_{\nu}\leq 1$ and

[TABLE]

Moreover, $\sum_{\nu\in V_{R}}\chi_{\nu}(\xi)=1$ for all $\xi\neq 0$, and for all $\alpha\in\mathbb{Z}_{+}^{n}$ and $\beta\in\mathbb{Z}_{+}$ there exists a $C_{\alpha,\beta}\geq 0$ independent of $k$ such that, if $R/2\leq|\xi|\leq 2R$, then

[TABLE]

for all $\nu\in V_{R}$. Such a collection is straightforward to construct (see e.g. [30, Section IX.4]).

The following proposition, which gives an equivalent description of the $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ norm of a function with frequency support in a dyadic annulus, is the key tool to relate $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ to decoupling inequalities.

Proposition 7.1**.**

Let $p\in[1,\infty]$, $q\in[1,\infty)$ and $s\in\mathbb{R}$. Then there exists a $C>0$ such that the following holds. Let $f\in L^{p}(\mathbb{R}^{d})$ be such that $\mathrm{supp}(\widehat{f}\,)\subseteq\{\xi\in\mathbb{R}^{n}\mid R/2\leq|\xi|\leq 2R\}$ for some $R\geq 2$. Then

[TABLE]

Proof.

The case where $p=q$ is covered by [22, Proposition 4.1], at least for $R=2^{k}$ with $k\in{\mathbb{N}}$. For general $p$ and $q$, the statement follows from [23, Proposition 2.4], since $\|f\|_{L^{p}(\mathbb{R}^{n})}$ and the Besov space norm $\|f\|_{B^{0}_{p,q}(\mathbb{R}^{n})}$ are comparable, which in turn follows from the assumption that $f$ has frequency support in a dyadic annulus. ∎

Remark 7.2**.**

Let $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ be such that

[TABLE]

for some $R\geq 2$ and $\nu\in S^{n-1}$. Then Proposition 7.1 yields

[TABLE]

7.2. Decoupling for the sphere

For $\kappa>0$ and $R\geq 2\kappa$, let $S_{R,\kappa}$ be a $\kappa R^{-1}$-neighborhood of the sphere $S^{n-1}$ in $\mathbb{R}^{n}$, and let $V_{R}$ be as in the previous subsection. Then $\theta_{\nu}:=\mathrm{supp}(\chi_{\nu})\cap S_{R,\kappa}$, for each $\nu\in V_{R}$, is a curved rectangle of dimensions approximately $R^{-1/2}\times\dots\times R^{-1/2}\times R^{-1}$ pointing in the direction of $\nu$. Moreover, $S_{R,\kappa}=\cup_{\nu\in V_{R}}\theta_{\nu}$, and the $\theta_{\nu}$ have finite overlap.

This observation allows us to formulate the decoupling inequality for the sphere. More precisely, for $p,q\in[2,\infty)$, set

[TABLE]

Then the $\ell^{q}$ decoupling inequality for the sphere, from [4], is as follows.

Theorem 7.3**.**

Let $p,q\in[2,\infty)$ and $\kappa,\varepsilon>0$. Then there exists a $C\geq 0$ such that

[TABLE]

for all $R\geq 2\kappa$ and $f\in\mathcal{S}(\mathbb{R}^{n})$ with $\mathrm{supp}(\widehat{f}\,)\subseteq S_{R,\kappa}$.

We can reinterpret this decoupling inequality as a bound for the $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ norm of functions with frequency support which is highly localized in a radial sense. For $p\in(1,\infty)$, set

[TABLE]

and note that $\alpha(p)=\alpha(p^{\prime})=d(p,q)-\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})$ for $p\geq 2$.

Corollary 7.4**.**

Let $p\in(1,\infty)$, $q\in[2,\infty)$, $s\in\mathbb{R}$ and $\kappa,\varepsilon>0$. Then there exists a $C\geq 0$ such that the following holds for all $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ with $\mathrm{supp}(\widehat{f}\,)\subseteq\{\xi\in\mathbb{R}^{n}\mid R-\kappa\leq|\xi|\leq R+\kappa\}$ for some $R\geq 2\kappa$.

1. (1)

If $p\geq 2$, then

[TABLE] 2. (2)

If $p\leq 2$, then

[TABLE]

Proof.

For (1), we may assume without loss of generality that $s=\alpha(p)+\varepsilon$. Moreover, by slightly enlarging $\kappa$ and by applying Proposition 4.6, we may suppose that $f\in\mathcal{S}(\mathbb{R}^{n})$.

Now set $f_{R}(y):=R^{-n}f(y/R)$ for $y\in\mathbb{R}^{n}$, and recall that the $\chi_{\nu}$ are positively homogeneous of order zero. Then two changes of variables, Theorem 7.3 and Proposition 7.1 combine to yield

[TABLE]

This proves (1).

By duality, (2) follows from (1). More precisely, Propositions 4.8 and 4.6 yield $\|f\|_{\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})}\eqsim\sup|\langle f,g\rangle_{\mathbb{R}^{n}}|$, where the supremum is taken over all $g\in\mathcal{S}(\mathbb{R}^{n})$ with $\mathrm{supp}(\widehat{g})\subseteq\{\xi\in\mathbb{R}^{n}\mid R-2\kappa\leq|\xi|\leq R+2\kappa\}$ and $\|g\|_{\mathcal{H}_{\mathrm{dec}}^{p^{\prime},q^{\prime};-s}(\mathbb{R}^{n})}\leq 1$. Hence (1) yields

[TABLE]

Remark 7.5**.**

For $p\neq 2$, the estimates in (1) and (2) of Corollary 7.4 improve upon (1) and (2) of Theorem 6.2, which hold without any assumptions on the frequency support of $f$. Indeed, for all $p\in(2,\infty)$ one has

[TABLE]

On the other hand, Corollary 7.4 yields less norm control than in the parabolically localized setting of Remark 7.2.

Also note that Corollary 7.4 is equivalent to Theorem 7.3, in the sense that the same rescaling argument as above allows one to deduce Theorem 7.3 from (7.3).

Remark 7.6**.**

The general decoupling inequality in [4], for compact $C^{2}$ hypersurfaces with positive definite second fundamental form, follows from Theorem 7.3, by decomposing the surface into a finite number of small pieces. Hence Corollary 7.4 can also be used to reformulate this more general decoupling inequality.

7.3. Decoupling for the cone

For $R>0$, let $W_{R}$ and $(\chi_{\omega})_{\omega\in W_{R}}$ be collections with the same properties as $V_{R}$ and $(\chi_{\nu})_{\nu\in V_{R}}$, from Section 7.1, but with $n$ replaced by $n+1$. More precisely, $W_{R}\subseteq S^{n}$ is a maximal collection of unit vectors in $\mathbb{R}^{n+1}$ such that $|\omega-\omega^{\prime}|\geq R^{-1/2}$ for all $\omega,\omega^{\prime}\in W_{R}$, and $(\chi_{\omega})_{\omega\in W_{R}}\subseteq C^{\infty}(\mathbb{R}^{n+1}\setminus\{0\})$ is an associated partition of unity of functions that are positively homogeneous of order zero.

For $\kappa>0$, let $\Gamma_{R,\kappa}\subseteq\mathbb{R}^{n+1}$ be a $\kappa R^{-1}$-neighborhood of the truncated light cone

[TABLE]

Then we can formulate the $\ell^{q}$ decoupling inequality for the cone as follows, using the exponent $d(p,q)$ from (7.1).

Theorem 7.7**.**

Let $p,q\in[2,\infty)$ and $\kappa,\varepsilon>0$. Then there exists a $C\geq 0$ such that

[TABLE]

for all $R\geq 2\kappa$ and $f\in\mathcal{S}(\mathbb{R}^{n})$ with $\mathrm{supp}(\widehat{f}\,)\subseteq\Gamma_{R,\kappa}$.

Proof.

For $\omega\in W_{R}$, set $\theta_{\omega}:=\mathrm{supp}(\chi_{\omega})\cap\Gamma_{R,\kappa}$. Then $\Gamma_{R,\kappa}=\cup_{\omega\in W_{R}}\theta_{\omega}$, and the $\theta_{\omega}$ have finite overlap. Fix $\omega=(\omega_{1},\ldots,\omega_{n+1})\in W_{R}$ such that $\theta_{\omega}\neq\emptyset$, and suppose that only the first and last coordinates of $\omega$ are nonzero. We claim that $\theta_{\omega}$ is contained in a slab of dimensions approximately $R^{-1}\times R^{-1/2}\times\dots\times R^{-1/2}\times 1$, pointing along the light cone. By rotation, the required statement is then equivalent to the standard formulation of the $\ell^{q}$ decoupling inequality for the cone, from [4].

To prove the claim, let $\zeta=(\xi,\tau)\in\theta_{\omega}$, with $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$. Then, by assumption on $\omega$ and because $\zeta\in\mathrm{supp}(\chi_{w})$, one has

[TABLE]

Moreover, $|\zeta|\eqsim 1$, since $\zeta\in\Gamma_{R,\kappa}$. Hence

[TABLE]

It also follows from (7.4) that $\omega$ points in the direction of the light cone. By combining all this, one sees that $\theta_{\omega}$ is as claimed. ∎

We will reinterpret this decoupling inequality as a bound for the $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n+1})$ norm of functions with frequency support near the light cone. More precisely, let

[TABLE]

be a thickened version of the full light cone. For $p\in(1,\infty)$, let $\alpha(p)$ be as in (7.2).

Corollary 7.8**.**

Let $p\in(1,\infty)$, $q\in[2,\infty)$, $s\in\mathbb{R}$ and $\kappa,\varepsilon>0$. Then there exists a $C\geq 0$ such that the following holds for all $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n+1})$ with

[TABLE]

for some $R\geq 2$.

1. (1)

If $p\geq 2$, then

[TABLE] 2. (2)

If $p<2$, then

[TABLE]

Proof.

Apply a rescaling argument and duality, as in the proof of Corollary 7.4. ∎

8. Regularity for wave equations

In this section we obtain new regularity results for the Euclidean wave equation. We first prove local smoothing estimates involving the function spaces for decoupling, and then we indicate how these estimates can applied to nonlinear wave equations with initial data outside of $L^{2}$-based Sobolev spaces.

8.1. Local smoothing

The following is our main result on local smoothing involving the function spaces for decoupling.

Theorem 8.1**.**

Let $p\in(2,\infty)$, $q\in[2,\infty)$, $s\in\mathbb{R}$ and $\varepsilon>0$. Then there exists a $C>0$ such that

[TABLE]

for all $f\in\mathcal{H}_{\mathrm{dec}}^{p,q;s+\alpha(p)+\varepsilon}(\mathbb{R}^{n})$.

Proof.

For $p=q$, the statement is contained in [22, Theorem 4.4]. For $q=2$, it is a consequence of [23, Theorem 1.1] and the standard embedding $W^{s+\alpha(p)+\varepsilon,p}(\mathbb{R}^{n})\subseteq B^{s+\alpha(p)+\varepsilon/2}_{p,p}(\mathbb{R}^{n})$. However, the proof of both these results in fact directly extends to general $q$, upon using the $\ell^{q}$ decoupling inequality for the light cone. ∎

Remark 8.2**.**

The local smoothing conjecture for the Euclidean wave equation stipulates that for each $\varepsilon>0$ one has

[TABLE]

for an implicit constant independent of $f\in W^{s+\sigma(p)+\varepsilon,p}(\mathbb{R}^{n})$. Here $\sigma(p):=0$ for $2<p\leq 2n/(n-1)$, and $\sigma(p):=2s(p)-1/p$ for $p\geq 2n/(n-1)$. By the Sobolev embeddings for $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ in Theorem 6.2 (1), (8.1) improves upon (8.2) for $p\geq 2(n+1)/(n-1)$ and $2\leq q\leq p$. Moreover, by the sharpness of the Sobolev embedding in Theorem 6.2 (2), (8.1) in fact yields a strict improvement of (8.2) for such $p$ and $q$. On the other hand, for $2<p<2(n+1)/(n-1)$ and $q\in[2,p]$, (8.1) neither follows from (8.2), nor vice versa. In particular, due to the sharpness of the Sobolev embedding in Theorem 6.2 (2), (8.1) yields sharper estimates than (8.2) for certain initial data (see also Remark 7.2). Here it is relevant to note that the exponents $\alpha(p)$ in (8.1) and $\sigma(p)$ in (8.2) are sharp, for all $2<p<\infty$ and $2\leq q\leq p$. In the case of (8.1), this follows from Theorem 6.3 (6.1) and [22, Theorem 5.3].

We also note that, at least when restricted to dyadic frequency annuli, $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ is the largest space of initial data for which one can obtain local smoothing when applying the $\ell^{q}$ decoupling inequality in the manner in which it is typically used (for more on this see [22, 23]).

Remark 8.3**.**

By Theorem 6.3, one has $W^{1/2,2}(\mathbb{R}^{n})\subsetneq\mathcal{H}_{\mathrm{dec}}^{p,2;0}(\mathbb{R}^{n})$ for $p=2(n+1)/(n-1)$. Given that $\alpha(p)=0$, Theorem 8.1 therefore almost yields a strict improvement of the classical Strichartz estimate

[TABLE]

In fact, by the sharpness of the Sobolev embeddings in Theorem 6.3, Theorem 8.1 already complements (8.3). More precisely, for this specific $p$, Theorem 8.1 yields sharper estimates than (8.3) for a large class of initial data, while also allowing for initial data in $L^{p}$-based Sobolev spaces.

8.2. Nonlinear wave equations

In this subsection we indicate how our results can be applied to nonlinear wave equations with initial data in $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ for $p>2$. Such initial data might be referred to as “slowly decaying”, due to the fact that such a function may decay slower at infinity than an $L^{2}(\mathbb{R}^{n})$ function does. On the other hand, even for compactly supported initial data, the results presented here show that, assuming additional integrability beyond that of an $L^{2}(\mathbb{R}^{n})$ function, one can obtain well-posedness statements for rougher initial data than one obtains from Strichartz estimates (for more on this see [20, Section 1.3]).

Our results and proofs for nonlinear wave equations are analogous to those in [23] (see also [26, 20]). In particular, here we only will consider the cubic nonlinear wave equation in dimension $n=2$, although a similar approach can be used in other dimensions and for different nonlinearities.

Consider the Cauchy problem for the cubic nonlinear wave equation on $\mathbb{R}^{2}\times\mathbb{R}$:

[TABLE]

Our main result concerning (8.4) is as follows.

Theorem 8.4**.**

Let $q\in[2,\infty]$ and $\varepsilon,T>0$. Then (8.4) is quantitatively well posed with initial data space

[TABLE]

and solution space

[TABLE]

Moreover, (8.4) is also quantitatively well posed with initial data space

[TABLE]

and solution space

[TABLE]

Our notion of quantitative well-posedness is taken from [1], and the definition is recalled in the proof below. Via a fixed-point argument, it implies that there exists a $\delta=\delta(t_{0})>0$ such that, if $\|(u_{0},u_{1})\|_{X}<\delta$, then (8.4) has a unique solution $u\in S_{t_{0}}$, and this solution depends analytically on the initial data. Moreover, in Theorem 8.4, for all $(u_{0},u_{1})\in X$ there exists a $t_{0}>0$ such that there is a unique solution $u\in S_{t_{0}}$ to (8.4).

Proof.

The proof is almost completely analogous to that of [23, Theorem 1.4], although we rely on Theorems 8.1 and 5.1 instead of results about adapted Besov spaces. We briefly sketch the idea (see also [20, Section 8]).

Write (8.4) as

[TABLE]

with

[TABLE]

for $u_{1},u_{2},u_{3}\in S_{T}$ and $t\in[0,T]$. We then say that (8.4) is quantitatively well posed if

[TABLE]

for some $C\geq 0$ independent of $f,g\in X$ and $u_{1},u_{2},u_{3}\in S_{T}$.

Now, to prove (8.9) one can rely on Theorems 8.1 and 5.1 for initial data in $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{2})$, and homogeneous Strichartz estimates (cf. [18]) for initial data in $W^{s,2}(\mathbb{R}^{2})$. On the other hand, the proof of (8.10) relies on Hölder’s inequality and inhomogeneous Strichartz estimates. We refer to [23] for details. ∎

One can equally well extend other results in [23] to initial data in $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$. In particular, if $\varepsilon>1/2$ in Theorem 8.4, then one obtains global existence for the defocusing equation with the initial data space in (8.5) and the solution space in (8.6).

Remark 8.5**.**

The main difference between Theorem 8.4 and the results in [23] concerns the second inclusion in (8.6) and (8.8). Although standard embeddings between Besov and Sobolev spaces, combined with the $\varepsilon$ loss in (8.5) and (8.7), imply that the spaces of initial data in (8.5) and (8.7) are not fundamentally larger than those in [23], or vice versa, $\mathcal{H}_{\mathrm{dec}}^{p,q;s}(\mathbb{R}^{n})$ satisfies sharp embeddings into the $L^{p}$-based Sobolev scale, cf. Theorem 6.2. By contrast, the adapted Besov spaces from [23] embed in a sharp manner into the standard Besov scale. Hence Theorems 8.4 and 6.2 show that $L^{p}$ regularity of the initial data is pointwise preserved in an optimal sense, whereas [23, Theorem 1.4] yields the corresponding statement for Besov regularity.

Acknowledgments

Yung would like to thank Andreas Seeger for a helpful discussion that inspired the proof of Theorem 6.2.