Characteristic functions as bounded multipliers on anisotropic spaces

TL;DR

This paper proves that characteristic functions of certain domains act as bounded multipliers on specific anisotropic Banach spaces, under particular geometric and parameter conditions, advancing the understanding of functional analysis in anisotropic settings.

Contribution

It establishes the boundedness of characteristic functions as multipliers on the $U^{t,s}_p$ anisotropic spaces for domains with boundaries transversal to stable cones, under new parameter conditions.

Findings

Characteristic functions are bounded multipliers on $U^{t,s}_p$ spaces.

Results depend on boundary transversality and parameter constraints.

Advances the theory of multipliers in anisotropic Banach spaces.

Abstract

We show that characteristic functions of domains with boundaries transversal to stable cones are bounded multipliers on a recently introduced scale $U^{t,s}_p$ of anisotropic Banach spaces, under the conditions -1+1/p<s<-t<0 and -(r-1)+t<s, with 1<p<infty. (Amended after comments from the referee and M. J\'ez\'equel, January 10, 2018)

Equations

$${\mathcal{L}}_h\phi =(h\cdot \phi )\circ T^{-1}.$$

$${\mathcal{L}}_{1/|\det DT|}\phi =\sum _i\frac{(1_{{\Lambda }_i}\cdot \phi )}{|\det DT|}\circ T^{-1}.$$

$$\mathbb{F}(\phi )(\xi )$$

$${\mathbb{F}}^{-1}(\psi )(x)$$

$$a^{Op}(\phi )={\mathbb{F}}^{-1}(a(\cdot )\cdot \mathbb{F}(\phi ))=({\mathbb{F}}^{-1}a)\ast \phi .$$

$${\psi }_n^{(D)}(\xi )=\chi (2^{-n}\Vert \xi \Vert )-\chi (2^{-n+1}\Vert \xi \Vert ),n\ge 1.$$

$${\mathbb{F}}^{-1}{\psi }_n^{(D)}=2^{D(n-1)}{\mathbb{F}}^{-1}{\psi }_1^{(D)}(2^{n-1}x)\text{mbox}and(\sum _{k\le n}{\mathbb{F}}^{-1}{\psi }_k^{(D)})(x)=2^{Dn}{\mathbb{F}}^{-1}\chi (2^{n}x),$$

$$\underset{n}{\sup }\Vert {\mathbb{F}}^{-1}{\psi }_n^{(D)}{\Vert }_{L_1({\mathbb{R}}^D)}<\infty ,\underset{n}{\sup }\Vert \sum _{k\le n}{\mathbb{F}}^{-1}{\psi }_k^{(D)}{\Vert }_{L_1({\mathbb{R}}^D)}<\infty ,$$

$$\Vert {\partial }^{\beta }{\psi }_n^{(D)}{\Vert }_{L_{\infty }}\le C_{\beta }{2}^{-n|\beta |},\forall n\ge 0.$$

$$({\psi }_n^{(D)}{)}^{Op}(\phi )(x)$$

$$({\psi }_n^{(D)}{)}^{Op}\circ ({\psi }_m^{(D)}{)}^{Op}\equiv 0\text{mbox}if|n-m|\ge 2.$$

$${\psi }_{{\ell }_s}^{Op(\Gamma )}(\phi )(w)$$

$$\Vert \phi {\Vert }_{p,\Gamma }^s=\underset{{\ell }_s\in {\mathbb{Z}}_{+}}{\sup }{2}^{{\ell }_{s}s}\Vert {\psi }_{{\ell }_s}^{Op(\Gamma )}(\phi ){\Vert }_{L_p({\mu }_{\Gamma })},$$

$$\Vert \phi {\Vert }_{p,\Gamma }^s\sim \Vert \phi \circ {\pi }_{\Gamma }^{-1}{\Vert }_{B_{p,\infty }^s({\mathbb{R}}^{d_s})}.$$

$$\Vert \phi {\Vert }_{{\mathcal{U}}_p^{{\mathbf{C}}_{+},t,s}}=\underset{\Gamma \in \mathcal{F}({\mathbf{C}}_{+},C_{\mathcal{F}},r)}{\sup }\underset{\ell \in {\mathbb{Z}}_{+}}{\sup }{2}^{\ell t}\Vert {\psi }_{\ell }^{Op}(\phi ){\Vert }_{p,\Gamma }^s.$$

$$F=F_{{\omega }^{\prime }\omega }={\kappa }_{\omega }^{-1}\circ T^{-1}\circ {\kappa }_{{\omega }^{\prime }}:{\kappa }_{{\omega }^{\prime }}^{-1}(V_{{\omega }^{\prime }\omega })\to U_{\omega },$$

$$D{F}_x^{tr}({\mathbb{R}}^d\setminus {\mathbf{C}}_{\omega ,+})⋐{\mathbf{C}}_{{\omega }^{\prime },-},\forall x\in {\mathbb{R}}^d.$$

$$({\mathbb{L}}_{xy}{)}^{tr}({\mathbb{R}}^d\setminus {\mathbf{C}}_{\omega ,+})⋐{\mathbf{C}}_{{\omega }^{\prime },-}\text{mbox}and{\mathbb{L}}_{xy}(x-y)=F(x)-F(y).$$

$$\Vert 1_{\tilde{\Lambda }}\phi {\Vert }_{{\mathcal{U}}_p^{\mathbf{C},t,s}}\le C_{\tilde{\Lambda },\tilde{\mathbf{C}}}\Vert \phi {\Vert }_{{\mathcal{U}}_p^{\tilde{\mathbf{C}},t,s}},\forall \phi .$$

$$1<p<\infty \text{mbox}and\max \{t-(r-1),-1+\frac{1}{p},\}<s<-t<0,$$

$$\Vert 1_{\tilde{\Lambda }}\phi {\Vert }_{{\mathcal{U}}_p^{{\mathbf{C}}_{+},t,s}}\le C\Vert \phi {\Vert }_{{\mathcal{U}}_p^{{\mathbf{C}}_{+},t,s}}.$$

$$\Vert f{\Vert }_{L_p({\mathbb{R}}^D)}\le C{M}^{D(1/p_1-1/p)}\Vert f{\Vert }_{L_{p_1}({\mathbb{R}}^D)}.$$

$$\Vert \tilde{\psi }\ast \phi {\Vert }_{p,\Gamma }^s\le \Vert \tilde{\psi }{\Vert }_{L_1({\mathbb{R}}^d)}\underset{x\in {\mathbb{R}}^d}{\sup }\Vert \phi {\Vert }_{p,\Gamma +x}^s\le \Vert \tilde{\psi }{\Vert }_{L_1}\underset{\tilde{\Gamma }\in \mathcal{F}}{\sup }\Vert \phi {\Vert }_{p,\tilde{\Gamma }}^s.$$

$$\phi \cdot \upsilon$$

$$=\sum _{k=2}^{\infty }\sum _{j=0}^{k-2}{S}_j\phi \cdot S_k\upsilon +\sum _{k=0}^{\infty }\sum _{j=k-1}^{k+1}{S}_j\phi \cdot S_k\upsilon +\sum _{j=2}^{\infty }\sum _{k=0}^{j-2}{S}_j\phi \cdot S_k\upsilon$$

$$={\Pi }_1(\phi ,\upsilon )+{\Pi }_2(\phi ,\upsilon )+{\Pi }_3(\phi ,\upsilon ),$$

$${\Pi }_1(\phi ,\upsilon )$$

$$\text{mbox}and{\Pi }_3(\phi ,\upsilon )=\sum _{j=2}^{\infty }{S}_j\phi \cdot S^{j-2}\upsilon ={\Pi }_1(\upsilon ,\phi ).$$

$$\mathrm{supp}\mathbb{F}(S^{k-2}\phi \cdot S_k\upsilon )\subset \{2^{k-3}\le \Vert \xi \Vert \le 2^{k+1}\},\forall k\ge 2,$$

$$\mathrm{supp}\mathbb{F}(\sum _{j=k-1}^{k+1}{S}_j\phi \cdot S_k\upsilon )\subset \{\Vert \xi \Vert \le 5\cdot 2^k\},\forall k\ge 0.$$

$$\Vert 1_{\tilde{\Lambda }}{\Vert }_{B_{p,q}^t({\mathbb{R}}^d)}<\infty ,\text{mbox}if0<t<1/p\text{mbox}and0<q<\infty \text{mbox}ort=1/p\text{mbox}andq=\infty .$$

$$S_k\upsilon (x):=({\mathbb{F}}^{-1}{\psi }_k)\ast \upsilon (x)=({\mathbb{F}}^{-1}{\psi }_k^{(1)})\ast \upsilon (x_1).$$

Full text

I thank D. Terhesiu for many useful comments and encouragements, in particular during her three-months stay in Paris in 2016. I am very grateful to the anonymous referee for thoughtful remarks, including the observation that the norms only depend on the “unstable” cones $\mathbf{C}_{+}$. I thank Malo Jézéquel for very sharp questions which helped me to improve the text.

1. Introduction

A (not necessarily smooth) function $g:M\to\mathbb{C}$ is called a bounded multiplier on a Banach space $\mathcal{B}$ of distributions on a $d$-dimensional Riemann manifold $M$ if there exists $C_{g}<\infty$ so that for all $\varphi\in\mathcal{B}$ the product $g\varphi$ is a well-defined element of $\mathcal{B}$ and, in addition, $\|g\cdot\varphi\|\leq C_{g}\|\varphi\|$, where $\|\cdot\|$ is the norm of $\mathcal{B}$. One interesting special case is when $g$ is the characteristic function $1_{\Lambda}$ of an open domain $\Lambda\subset M$: Half a century ago, Strichartz [16] proved that for any $d\geq 1$, if $M=\mathbb{R}^{d}$ and $\mathcal{B}$ is the Sobolev111Recall that $\|\varphi\|_{H^{t}_{p}}=\|(\mathrm{id}+\Delta)^{t/2}\varphi\|_{L_{p}}=\|\mathbb{F}^{-1}(1+|\xi|^{2})^{t/2}\mathbb{F}\varphi\|_{L_{p}}$, with $\Delta$ the Laplacian and $\mathbb{F}$ the Fourier transform. space $H^{t}_{p}(\mathbb{R}^{d})$ for $p\in(1,\infty)$ and $t\in\mathbb{R}$, then the characteristic function $1_{\Lambda}$ of a half-space is a bounded multiplier on $H^{t}_{p}(\mathbb{R}^{d})$ if and only if $-1+1/p<t<1/p$.

In the present work, we consider a newly introduced scale $\mathcal{U}^{\mathbf{C},t,s}_{p}$ of spaces of anisotropic distributions $\mathcal{B}$ on a manifold $M$, adapted to smooth hyperbolic dynamics, and we prove the bounded multiplier property for characteristic functions of suitable subsets $\Lambda\subset M$.

Fix $r>1$, and suppose from now on that $M$ is connected and compact. The simplest hyperbolic maps on $M$ are transitive $C^{r}$ Anosov diffeomorphisms $T$. The Ruelle transfer operator associated to such a map $T$ and to a $C^{r-1}$ function $h$ on $M$ (for example, $h=1/|\det DT|$) is defined on $C^{r-1}$ functions $\varphi$ by

[TABLE]

Blank–Keller–Liverani [7] were the first to study the spectrum of such transfer operators on a suitable Banach space $\mathcal{B}$ of anisotropic distributions and to exploit this spectrum to get information on the Sinai–Ruelle–Bowen (physical) measure: The spectral radius of $\mathcal{L}_{1/|\det DT|}$ is equal to $1$, there is a simple positive maximal eigenvalue, whose eigenvector is in fact a Radon measure $\mu$, which is just the physical measure of $T$. Finally, the rest of the spectrum lies in a disc of radius strictly smaller than $1$, which implies exponential decay of correlations $\int\varphi(\psi\circ T^{n})\mathrm{d}\mu-\int\varphi\mathrm{d}\mu\int\psi\mathrm{d}\mu$ for Hölder observables $\psi$ and $\varphi$ as $n\to\infty$. (The first step in this analysis is to show the bound $\rho_{ess}<1$ for the essential spectral radius of $\mathcal{L}_{1/|\det DT|}$ on $\mathcal{B}$.)

Some natural dynamical systems originating from physics (such as Sinai billiards) enjoy uniform hyperbolicity, but are only piecewise smooth. Letting $M=\cup_{i}\Lambda_{i}$ be a (finite or countable) partition of $M$ into domains where the dynamics is smooth, one can often reduce to the smooth hyperbolic case via the decomposition

[TABLE]

This motivates studying bounded multiplier properties of characteristic functions.

In the 15 years since the publication of [7], dynamicists and semi-classical analysts have created a rich jungle of spaces of anisotropic distributions for hyperbolic dynamics (here, $d=d_{s}+d_{u}$ with $d_{s}\geq 1$ and $d_{u}\geq 1$). These spaces are usually scaled by two real numbers $v<0$ and $t>0$. Leaving aside the classical foliated anisotropic spaces of Triebel [17] (which are limited to “bunched” cases [4], and seem to fail for Sinai billiards), they come in two groups:

In the first, “geometric” group [7, 13], a class of $d_{s}$-dimensional “admissible” leaves $\Gamma$ (having tangent vectors in stable cones for $T$) is introduced, and the norm of $\varphi$ is obtained by fixing an integer $t\geq 1$ and taking a supremum, over all admissible leaves $\Gamma$, of the partial derivatives of $\varphi$ of total order at most $t$, integrated against $C^{|v|}$ test functions on $\Gamma$. Modifications of this space, for suitable noninteger $0<t<1$ and $|v|<1$, were introduced to work with piecewise smooth systems [8, 9] (only in dimension two). A version of these spaces for piecewise smooth hyperbolic flows in dimension three recently allowed to prove exponential mixing for Sinai billiard flows [3].

In the222This group could also be called pseudodifferential, or semi-classical, or Sobolev. second, “microlocal,” group [5], a third parameter $p\in[1,\infty)$ is present, and the norm (in charts) of $\varphi$ is the $L_{p}$ average of $\Delta^{t,v}(\varphi)$, where the operator $\Delta^{t,v}$ interpolates smoothly between $(\mathrm{id}+\Delta)^{v/2}$ in stable cones in the cotangent space, and $(\mathrm{id}+\Delta)^{t/2}$ in unstable cones in the cotangent space. Powerful tools are available for this microlocal approach, allowing in particular to study the dynamical determinants and zeta functions333The “kneading determinants” of by Milnor and Thurston from the 70’s are revisited as “nuclear decompositions” in [1]. much more efficiently than for the geometric spaces. Variants of these microlocal spaces (usually in the Hilbert setting $p=2$) have also been studied by the semi-classical community, starting from [10]. However, S. Gouëzel pointed out over ten years ago that characteristic functions cannot be bounded multipliers on spaces defined by conical wave front sets as in [5] or [10] (Gouëzel’s counterexamples are presented in [2, App. 1]). The microlocal spaces of the type defined in [5, 6] or [10] thus appear unsuitable to study piecewise smooth dynamics.

In order to overcome this limitation of the microlocal approach, we recently introduced [2] a new scale $\mathcal{U}^{\mathbf{C},t,s}_{p}$ of microlocal anisotropic spaces, obtained by mimicking the construction of the geometric spaces of Gouëzel–Liverani [13] (with, morally, $s=v+t$). We showed in [2] the expected bound on the essential spectral radius of the transfer operator of a $C^{r}$ Anosov diffeomorphism acting on $\mathcal{U}^{\mathbf{C},t,s}_{p}$ (if $t-(r-1)<s<-t<0$), and we conjectured that characteristic functions of domains with piecewise smooth boundaries everywhere transversal to the stable cones should be bounded multipliers on $\mathcal{U}^{\mathbf{C},t,s}_{p}$, if $s$ and $t$ satisfy additional constraints depending on $p\in(0,1)$. The main result444See Remark 2.5. of the present paper, Theorem 3.1, implies this bounded multiplier property if $\max\{t-(r-1),-1+1/p\}<s<-t<0$.

This result opens the door to the spectral study, not only of hyperbolic maps with discontinuities in arbitrary dimensions, but also (using nuclear power decompositions [1, 2]) of the hitherto unexplored topic of the dynamical zeta functions of piecewise expanding and piecewise hyperbolic maps in any dimensions. This should include billiards maps [9] and their dynamical zeta functions in arbitrary dimensions. We also hope that the spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$ will allow to extend the scope of the renewal methods introduced in [14] to dynamical systems with infinite invariant measures. (The induction procedure used there introduces discontinuities in the dynamics.) Finally, it goes without saying that suitable version of the spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$ will be useful to study flows.

F. Faure and M. Tsujii [11] recently introduced new microlocal anisotropic spaces, for which the wave front set is more narrowly constrained than for previous microlocal spaces used for hyperbolic dynamics. It would be interesting to check whether characteristic functions are bounded multipliers on these new spaces. (Note however that, contrary to the spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$ or the spaces of [10, 5, 13, 9], spaces of [11] do not appear suitable for perturbations of hyperbolic maps or flows.)

2. $\mathcal{U}^{\mathbf{C},t,s}_{p}$: A Fourier version of the Demers–Gouëzel–Liverani spaces

We recall the “microlocal” spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$, for real numbers $s$ and $t$ (in the application, $s<-t<0$) and $1\leq p\leq\infty$, introduced in [2].

2.1. Basic notation

Suppose that $d=d_{s}+d_{u}$ with $d_{u}\geq 1$ and $d_{s}\geq 1$. For $\ell\geq 1$ and $x\in\mathbb{R}^{\ell}$, $\xi\in\mathbb{R}^{\ell}$, we write $x\xi$ for the scalar product of $x$ and $\xi$. The Fourier transform $\mathbb{F}$ and its inverse $\mathbb{F}^{-1}$ are defined on rapidly decreasing functions $\varphi,\psi$ by

[TABLE]

and extended to the space of temperate distributions $\varphi,\psi$ as usual [15]. For suitable functions $a:\mathbb{R}^{d}\to\mathbb{R}$ (called “symbols”, note that, in this paper, $a$ depends only on $\xi$, while more general symbols may depend on $x$ and $\xi$), we define an operator $a^{Op}$ acting on suitable $\varphi:\mathbb{R}^{d}\to\mathbb{C}$, by

[TABLE]

Note that $\|a^{Op}\varphi\|_{L_{p}}\leq\|\mathbb{F}^{-1}a\|_{1}\|\varphi\|_{L_{p}}$ for each $1\leq p\leq\infty$, by Young’s inequality in $L_{p}$.

Fix a $C^{\infty}$ function $\chi:\mathbb{R}_{+}\to[0,1]$ with $\chi(x)=1$ for $x\leq 1$, and $\chi(x)=0$ for $x\geq 2$. For $D\geq 1$, define $\psi^{(D)}_{n}:\mathbb{R}^{D}\to[0,1]$ for $n\in\mathbb{Z}_{+}$, by $\psi^{(D)}_{0}(\xi)=\chi(\|\xi\|)$, and

[TABLE]

We set $\psi_{n}=\psi^{(d)}_{n}$. Note that

[TABLE]

so that, for any $D$,

[TABLE]

and for every multi-index $\beta$, there exists a constant $C_{\beta}$ such that

[TABLE]

We shall work with the following operators $(\psi^{(D)}_{n})^{Op}$ (putting $\psi_{n}^{Op}=(\psi^{(d)}_{n})^{Op}$):

[TABLE]

Note finally the following almost orthogonality property

[TABLE]

2.2. The local anisotropic spaces $\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$ for compact $K\subset\mathbb{R}^{d}$

Recall that a cone is a subset of $\mathbb{R}^{d}$ invariant under scalar multiplication. For two cones $\mathbf{C}$ and $\mathbf{C}^{\prime}$ in $\mathbb{R}^{d}$, we write $\mathbf{C}\Subset\mathbf{C}^{\prime}$ if $\overline{\mathbf{C}}\subset\mbox{ interior}\,(\mathbf{C}^{\prime})\cup\{0\}$. We say that a cone $\mathbf{C}$ is $d^{\prime}$-dimensional if $d^{\prime}\geq 1$ is the maximal dimension of a linear subset of $\mathbf{C}$.

Definition 2.1**.**

An unstable cone is a closed cone $\mathbf{C}_{+}$ with nonempty interior of dimension $d_{u}$ in $\mathbb{R}^{d}$ so that $\mathbb{R}^{d_{s}}\times\{0\}$ is included in555In Definitions 3.2 and 3.3, and 7 lines above Definition 3.2 of [2], the condition “$\mathbb{R}^{d_{s}}\times\{0\}$ is included in $\mathbf{C}_{-}$” can be replaced by this condition. $(\mathbb{R}^{d}\setminus\mathbf{C}_{+})\cup\{0\}$.

Recall that $r>1$. The next key ingredient is adapted from [6]:

Definition 2.2** (Admissible (or fake) stable leaves).**

Let $\mathbf{C}_{+}$ be an unstable cone, and let $C_{\mathcal{F}}>1$. Then $\mathcal{F}(\mathbf{C}_{+},C_{\mathcal{F}},r)$ (or just $\mathcal{F}$) is the set of all $C^{r}$ (embedded) submanifolds $\Gamma\subset\mathbb{R}^{d}$, of dimension $d_{s}$, with $C^{r}$ norms of submanifold charts $\leq C_{\mathcal{F}}$, and so that the straight line connecting any two distinct points in $\Gamma$ is normal to a $d_{u}$-dimensional subspace contained in $\mathbf{C}_{+}$. Denote by $\pi_{-}$ the orthogonal projection from $\mathbb{R}^{d}$ to the quotient $\mathbb{R}^{d_{s}}$ and by $\pi_{\Gamma}$ its restriction to $\Gamma$. Our assumption implies that $\pi_{\Gamma}:\Gamma\to\mathbb{R}^{d_{s}}$ is a $C^{r}$ diffeomorphism onto its image with a $C^{r}$ inverse, whose $C^{r}$ norm is bounded by a universal scalar multiple of $\mathcal{C}_{F}$. In the sequel, we replace $\mathcal{C}_{F}$ by this larger constant and we restrict to those $\Gamma$ so that $\pi_{\Gamma}$ is surjective.

Definition 2.3** (Isotropic norm on stable leaves).**

Fix an unstable cone $\mathbf{C}_{+}$. Let $\Gamma\in\mathcal{F}(\mathbf{C}_{+},C_{\mathcal{F}},r)$ and let $\varphi\in C^{0}(\Gamma)$. For $w\in\Gamma\subset\mathbb{R}^{d}$, we set

[TABLE]

where $\psi_{k}^{(d_{s})}:\mathbb{R}^{d_{s}}\to[0,1]$ is defined in (6). For all real numbers $1\leq p\leq\infty$, and $-(r-1)<s<r-1$, define an auxiliary isotropic norm on $C^{0}(\Gamma)$ as

[TABLE]

where $\mu_{\Gamma}$ is the Riemann volume on $\Gamma$ induced by the standard metric on $\mathbb{R}^{d}$.

Note that (11) is equivalent, uniformly in $\Gamma\in\mathcal{F}$, to the ([15, §2.1, Def. 2]) classical $d_{s}$-dimensional Besov norm $B^{s}_{p,\infty}$ of $\varphi$ in the chart given by $\pi_{\Gamma}^{-1}$:

[TABLE]

We next revisit the local space given in [2]:

Definition 2.4** (The local space $\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$).**

Let $r>1$, let $K\subset\mathbb{R}^{d}$ be a non-empty compact set. For an unstable cone $\mathbf{C}_{+}$, a constant $C_{\mathcal{F}}\geq 1$, and real numbers $1\leq p\leq\infty$, and $t-(r-1)<s<-t<0$, define for $\varphi\in L_{\infty}$ supported in $K$,

[TABLE]

Set $\mathcal{U}^{t,s}_{p}(K)=\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$ to be the completion of $\{\varphi\in L_{\infty}(K)\mid\|\varphi\|_{\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}}<\infty\}$ for the norm $\|\cdot\|_{\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}}$. (Note that $\mathcal{U}^{t,s}_{p}(K)$ also depends on $r$ and $C_{\mathcal{F}}$.)

Remark 2.5*.*

Beware that, in [2, Definition 3.3], the space $\mathcal{U}^{t,s}_{p}(K)$ was defined by completing $C^{\infty}(K)$ (or, equivalently, by [2, Lemma 3.4] and mollification, $C^{r-1}(K)$). We do not claim that $C^{\infty}(K)$ is dense in the space $\mathcal{U}^{t,s}_{p}(K)$ from Definition 2.4. (See however [9, Lemmas 3.7, 3.8].) But, since all results in [2] hold (except the heuristic remark after [2, Definition B.1]), with the same666In particular, [2, Lemma C.1] holds replacing $C^{\infty}(K)$ by compactly supported distributions. proofs, for the completion used in Definition 2.4, we may (abusively) use here the same notation $\mathcal{U}^{t,s}_{p}(K)$. The new definition is useful to show that (13) implies that $1_{\Lambda}\mathcal{U}^{t,s}_{p}(K)\subset\mathcal{U}^{t,s}_{p}(K)$.

The following lemma was proved777Injectivity of the embedding into distributions follows from injectivity of the embedding of the closure of the (larger) set of those tempered distributions $\varphi$ so that $\|\varphi\|_{\mathcal{U}^{t,s}_{p}}<\infty$. in [2]:

Lemma 2.6** (Comparing $\mathcal{U}_{p}^{\mathbf{C}_{+},t,s}(K)$ with classical spaces).**

Assume $-(r-1)<s<-t<0$. For any $u>t$, there exists a constant $C=C(u,K)$ such that $\|\varphi\|_{\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}}\leq C\|\varphi\|_{C^{u}}$ for all $\varphi\in C^{u}(K)$. For any $u>|t+s|$, the space $\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$ is contained in the space of distributions of order $u$ supported on $K$.

2.3. The global spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$ of anisotropic distributions

We finally introduce the global spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$ of distributions on a compact manifold $M$.

Definition 2.7**.**

An admissible chart system and partition of unity is a finite system of local charts $\{(V_{\omega},\kappa_{\omega})\}_{\omega\in\Omega}$, with open subsets $V_{\omega}\subset M$, and $C^{\infty}$ diffeomorphisms $\kappa_{\omega}:U_{\omega}\to V_{\omega}$ such that $M\subset\cup_{\omega}V_{\omega}$, and $U_{\omega}\subset\mathbb{R}^{d}$ is bounded and open, together with a $C^{\infty}$ partition of unity $\{\theta_{\omega}\}_{\omega\in\Omega}$ for $M$, subordinate to the cover $\mathcal{V}=\{V_{\omega}\}$.

Definition 2.8** (Anisotropic spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$ on $M$).**

Fix $r>1$, an admissible chart system and partition of unity, $\mathcal{C}_{F}\geq 1$ and a system of cones $\mathbf{C}=\{\mathbf{C}_{\omega,+}\}_{\omega\in\Omega}$. Fix $1\leq p\leq\infty$, and real numbers $-(r-1)<s<-t<0$. The Banach space $\mathcal{U}^{\mathbf{C},t,s}_{p}=\mathcal{U}^{\mathbf{C},t,s,r,C_{\mathcal{F}}}_{p}$ is the completion (see Remark 2.5) of $\{\varphi\in L_{\infty}(M)\mid\|\varphi\|_{\mathcal{U}^{\mathbf{C},t,s}_{p}}<\infty\}$ for the norm $\|\varphi\|_{\mathcal{U}^{\mathbf{C},t,s}_{p}}:=\max_{\omega\in\Omega}\|(\theta_{\omega}\cdot\varphi)\circ\kappa_{\omega}\|_{\mathcal{U}^{\mathbf{C}_{\omega,+},t,s}_{p}}$.

Remark 2.9* (Admissible systems $\{\mathbf{C}_{\omega,\pm}\}$).*

To get a spectral gap for the transfer operator $\mathcal{L}_{1/|\det DT|}$ associated to a $C^{\tilde{r}}$ Anosov diffeomorphism $T$ for $\tilde{r}>1$, one must take $r=\tilde{r}$ and consider an admissible chart system and partition of unity, with cones $\{\mathbf{C}_{\omega,+}\}$, satisfying the following conditions [2]:

a) Let $E^{s}$ and $E^{u}$ be the stable, respectively unstable, bundles of $T$. Then if $x\in V_{\omega}$, the cone $(D\kappa_{\omega}^{-1})^{*}_{x}(\mathbf{C}_{\omega,+})$ contains the ($d_{u}$-dimensional) normal subspace of $E^{s}(x)$, and there exists a $d_{s}$-dimensional cone $\mathbf{C}_{\omega,-}$, with nonempty interior, so that $\mathbf{C}_{\omega,+}\cap\mathbf{C}_{\omega,-}=\{0\}$, and so that $(D\kappa_{\omega}^{-1})^{*}_{x}(\mathbf{C}_{\omega,-})$ contains the ($d_{s}$-dimensional) normal subspace of $E^{u}(x)$.

b) If $V_{\omega^{\prime}\omega}=T(V_{\omega})\cap V_{\omega^{\prime}}\neq\emptyset$, the $C^{r}$ map corresponding to $T^{-1}$ in charts,

[TABLE]

extends to a bilipschitz $C^{1}$ diffeomorphism of $\mathbb{R}^{d}$ so that (by definition, $\mathbf{C}_{\omega^{\prime},-}\Subset(\mathbb{R}^{d_{s}}\setminus\mathbf{C}_{\omega^{\prime},+})$)

[TABLE]

c) Furthermore, there exists, for each $x,y$, a linear transformation $\mathbb{L}_{xy}$ so that

[TABLE]

A map $F$ satisfying (b–c) is called regular cone hyperbolic from $\mathbf{C}_{\omega,\pm}$ to $\mathbf{C}_{\omega^{\prime},\pm}$.

The anisotropic spaces $\mathcal{U}^{\mathbf{C},t,s}_{1}$ (with $p=1$) are analogues of the Blank–Keller–Gouëzel–Liverani [7, 13] spaces $\mathcal{B}^{t,|s+t|}$ associated to $T$, for integer $t$ and $s<-t$. The spaces $\mathcal{U}^{\mathbf{C},t,s}_{p}$ are somewhat similar to the Demers–Liverani spaces [8] when $p>1$ and $-1+1/p<s<-t<0$. See [2].

3. Characteristic functions as bounded multipliers

3.1. Statement of the main result

Fix $r>1$, $C_{\mathcal{F}}>0$, $p\in(1,\infty)$, an admissible chart system and partition of unity on $M$ (Definition 2.7), and an associated cone system $\mathbf{C}=\{\mathbf{C}_{\omega,+}\}$. Let $\tilde{\Lambda}\subset M$ be an open set so that $\partial\tilde{\Lambda}$ is a finite union of $C^{r}$ hypersurfaces $\partial\tilde{\Lambda}_{i}$ so that the normal vector at any $x\in\partial\tilde{\Lambda}_{i}\cap V_{\omega}$ lies in $\mathbb{R}^{d}\setminus\mathbf{C}_{\omega,+}$ (a transversality condition). We claim that if $\max\{t-(r-1),-1+1/p\}<s<-t<0$ then, for any888Given two cone systems of same cardinality, $\mathbf{C}\Subset\tilde{\mathbf{C}}$ means $\mathbf{C}_{\omega,+}\Subset\tilde{\mathbf{C}}_{\omega,+}$ for all $\omega$. cone system $\tilde{\mathbf{C}}$ with999Enlarging the cones is not a problem when studying $1_{\tilde{\Lambda}}((f\varphi)\circ F)$ for a $C^{r-1}$ function $f$ and a $C^{r}$ regular cone-hyperbolic map $F$ from $\mathbf{C}$ to $\tilde{\mathbf{C}}$ with $\mathbf{C}\Subset\tilde{\mathbf{C}}$, since the Lasota–Yorke estimate [2, Lemma 4.2] gives $\|(f\varphi)\circ F\|_{\mathcal{U}^{\tilde{\mathbf{C}},t,s}_{p}}\leq C_{f,F}\|\varphi\|_{\mathcal{U}^{\mathbf{C},t,s}_{p}}$. $\mathbf{C}\Subset\tilde{\mathbf{C}}$, there exists $C_{\tilde{\Lambda},\tilde{\mathbf{C}}}<\infty$ so that

[TABLE]

Since $t-(r-1)<s<-t$, by using suitable $C^{\infty}$ partitions of unity $h_{j}$ and $C^{r}$ coordinates $F_{j}$ (arbitrarily close to the identity, and thus regular cone hyperbolic from $\tilde{\mathbf{C}}$ to $\mathbf{C}$ if $\mathbf{C}\Subset\tilde{\mathbf{C}}$), and exploiting the Lasota–Yorke estimate [2, Lemma 4.2] for the corresponding transfer operators, we reduce to:

Theorem 3.1** (Characteristic functions of half-spaces).**

Fix $r>1$, $C_{\mathcal{F}}>0$, and an unstable cone $\mathbf{C}_{+}$. Let $K\subset\mathbb{R}^{d}$ be compact, and let $\tilde{\Lambda}\subset\mathbb{R}^{d}$ be a half-space whose unit normal vector $u_{\tilde{\Lambda}}$ lies in $\mathbb{R}^{d}\setminus\mathbf{C}_{+}$. Then for any

[TABLE]

there exists $C<\infty$ so that for any $\varphi\in\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$ we have,

[TABLE]

Since $1_{\tilde{\Lambda}}\varphi\in L_{\infty}$ if $\varphi\in L_{\infty}$ and since $\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$ is the completion of a set of bounded functions, the bound (13) implies that $1_{\tilde{\Lambda}}\varphi\in\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$ if $\varphi\in\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}(K)$ (use Cauchy sequences).

The conditions in the theorem imply $t<1-1/p$. (This does not imply $t<1/p$ if $p>2$.)

Remark 3.2* (Heuristic proof via interpolation: $t<1/p$ vs. $t<\min\{|s|,r-1-|s|\}$).*

A heuristic argument for the bounded multiplier property (13) under the conditions $-1+1/p<s<0<t<1/p$ was sketched in [2, Remark 3.9], exploiting via interpolation the fact that ([15, Thm 4.6.3/1]) the characteristic function of a half-plane in $\mathbb{R}^{n}$ is a bounded multiplier on the Besov space $B^{\tau}_{p,\infty}(\mathbb{R}^{n})$ if $\frac{1}{p}-1<\tau<\frac{1}{p}$. It does not seem easy to fill in details of this argument, and we shall prove Theorem 3.1 using paraproduct decompositions instead of interpolation. The restriction $t-(r-1)<s<-t$ is in any case necessary for applications to hyperbolic dynamics, and the bound for the essential spectral radius in [2] improves as $p\to 1$.

3.2. Basic toolbox (Nikol’skij and Young bounds, paraproduct decomposition,

and a crucial trivial observation on functions of a single variable)

The proofs below use the Nikol’skij inequality (see e.g. [15, Remark 2.2.3.4, p. 32]) which says, in dimension $D\geq 1$, that for any $p>p_{1}>0$ there exists $C$ so that for any $M>1$, and any $f$ with $\mathrm{supp}\,\mathbb{F}(f)\subset\{|\xi|\leq M\}$,

[TABLE]

We shall also use the following leafwise version of Young’s inequality (which can be proved like [6, Lemma 4.2], see [2], by using that any translation $\Gamma+x$ of $\Gamma\in\mathcal{F}$ also belongs to $\mathcal{F}$):

[TABLE]

Write $S_{k}\varphi=\psi_{k}^{Op}(\varphi)$ for $k\geq 0$, set $S_{-1}\varphi\equiv 0$, and put $S^{j}\varphi=\sum_{k=0}^{j}S_{k}\varphi$ for integer $j\geq 0$. The (a priori formal) paraproduct decomposition (see [15, §4.4]) is

[TABLE]

where we put

[TABLE]

The two key facts motivating the decomposition (16) are

[TABLE]

and

[TABLE]

Finally, the proof of Theorem 3.1 hinges on the fact that the singular set of a characteristic function is co-dimension one: We shall reduce there to the case $\partial\tilde{\Lambda}=\{x_{1}=0\}$ so that $1_{\tilde{\Lambda}}$ only depends on the first coordinate $x_{1}$ of $x\in\mathbb{R}^{d}$. We shall use below the fact that for such $\tilde{\Lambda}$ (see [15, Lemma 4.6.3.2 (ii), p. 209, Lemma 2.3.1/3, p. 48]) for all $p\in(1,\infty)$

[TABLE]

We also note for further use the trivial but absolutely essential fact that if a function $\upsilon(x)$ only depends on $x_{1}$ then $S_{k}\upsilon=(\mathbb{F}^{-1}\psi_{k})*\upsilon$ also only depends on $x_{1}$ for all $k$, and, more precisely,

[TABLE]

Indeed

[TABLE]

and, since $(2\pi)^{-(d-1)}\int_{\mathbb{R}^{d-1}}e^{i(y_{2},\ldots,y_{d})(\xi_{2},\ldots,\xi_{d})}dy_{2}\ldots dy_{d}$ (the inverse Fourier transform of the constant function) is the Dirac mass at $(\xi_{2},\ldots,\xi_{d})=0$, we get,

[TABLE]

where we used that $\psi^{(d)}_{k}(\xi_{1},0)=\psi^{(1)}_{k}(\xi_{1})$.

3.3. Multipliers depending on a single coordinate

This subsection is devoted to a classical property of multipliers depending on a single coordinate, which is instrumental in the proof of Theorem 3.1. If $1\leq p\leq\infty$, let $1\leq p^{\prime}\leq\infty$ be so that

[TABLE]

Lemma 3.3**.**

Let $d_{s}\geq 1$. Let $1<p<\infty$ and let $-1+\frac{1}{p}<s<0$. Then there exists $C<\infty$ so that for all $f,g:\mathbb{R}^{d_{s}}\to\mathbb{C}$ with $g(x)=g(x_{1})$,

[TABLE]

Remark 3.4*.*

The bound (22) is a special case of a much more general result (see e.g. [15, Cor 4.6.2.1 (40)]) which also implies that if $g(x)=g(x_{1})$ then

[TABLE]

for a constant $C$, which may depend on $p$ and $t$, but not on $f$ or $g$.

For the convenience of the reader, and as a warmup in the use of paraproducts, we include a proof of Lemma 3.3.

Proof of Lemma 3.3..

The proof uses the decomposition $\tilde{\Pi}_{1}(f,g)+\tilde{\Pi}_{2}(f,g)+\tilde{\Pi}_{3}(f,g)$ obtained from (16) by replacing $S_{k}$ and $S^{k}$ by the $d_{s}$-dimensional operators

[TABLE]

The bound for the contribution of $\tilde{\Pi}_{3}(f,g)$ is easy and does not require conditions on $s$ or $g$: Indeed, (17) and the Young inequality with the first claim of (7) imply

[TABLE]

We focus on the term for $\ell=0$ (the others are similar) and get

[TABLE]

where we used the Hölder inequality and then the Young inequality, together with the second claim of (7).

For $\tilde{\Pi}_{1}(f,g)$, we do not require any condition on $g$, and the condition on $s$ is limited to $s<0$: Indeed, exploiting again (17), we get

[TABLE]

Focusing again on the terms for $\ell=0$, we find

[TABLE]

where we used the Hölder inequality and then the Young inequality, together with the first claim of (7).

The computation for $\tilde{\Pi}_{2}(f,g)$ is trickier and will use the assumption $s>-1+1/p$ together with the Nikol’skij inequality (14). For $\ell\in\{0,\pm 1\}$, by (18), we get

[TABLE]

In the sequel, we consider the terms with $\ell=0$ (the other terms are almost identical). Setting $y=(x_{2},\ldots,x_{d_{s}})$ and applying the one-dimensional Nikol’skij inequality (14) for $1<p_{1}<p$, we have, for any function $\upsilon$,

[TABLE]

where

[TABLE]

Since $s>-1+1/p$, we may choose $p_{1}\in(1,p)$ close enough to $1$ so that

[TABLE]

Then, the right-hand side of (27) can be bounded as follows, using (28),

[TABLE]

In the last line we used (18) to exploit that there exists $C<\infty$, depending on $p>1$ and $p_{1}>1$, so that, for any $\{\upsilon_{k}\}_{k\geq 0}$ so that $\mathrm{supp}\,(\mathbb{F}(\upsilon_{k}))\subset\{|\xi|\leq 5\cdot 2^{k}\}$,

[TABLE]

(The above basically follows from Young’s inequality, see [15, Thm 2.6.3, (5), p. 96], noting that $p>1$ and $p_{1}>1$, so that $\max\{0,1/p-1,1/p_{1}-1\}=0$, and noting that $f_{j}$ in the right-hand side of [15, (5), p. 96] should be replaced by $f_{j+\ell}$, see [12, Thm 2.4.1.(II) and (III)].)

Next, recalling that $g$ only depends on $x_{1}$, using (20), and applying the Hölder inequality in $dx_{1}$ for $1/p_{1}=1/p+1/q$, we find $C$ so that for all $k$

[TABLE]

Note that (20) implies $\tilde{S}_{k}g=(\psi^{(1)}_{k})^{Op}g$. Finally, putting together (27) and (31), we find, recalling (30) and (21),

[TABLE]

where we used the one-dimensional Nikol’skij inequality for $q>p^{\prime}>1$ in (32) (recalling (18)). Together, (25), (26), and (33) give (22). ∎

3.4. Proof of Theorem 3.1

To prove the theorem, we need one last lemma. The point is that if $\Gamma$ is horizontal, i.e., $\Gamma=\mathbb{R}^{d_{s}}\times\{0\}$, then (9) implies

[TABLE]

If $\Gamma$ is an arbitrary admissible stable leaf, then we must work harder. To state the bound replacing the trivial decoupling property (34), we need notation: Defining $b:\mathbb{R}^{d}\to\mathbb{R}_{+}$ by $b(x)=1$ if $\|x\|\leq 1$ and $b(x)=\|x\|^{-d-1}$ if $\|x\|>1$, we set $b_{k}(x)=2^{dk}\cdot b(2^{k}x)$ for $k\geq 0$. (Note that $\|b_{k}\|_{L_{1}(\mathbb{R}^{d})}=\|b\|_{L_{1}(\mathbb{R}^{d})}<\infty$.)

Lemma 3.5** **(Decoupled wave packets in $\mathbb{R}^{d}$ and the cotangent space of

$\Gamma$).

Fix a compact set $K\subset\mathbb{R}^{d}$. There exists $C_{0}\in[2,\infty)$ (depending on $\mathcal{C}_{\mathcal{F}}$, $K$) so that for any $k_{s}>k+C_{0}\geq C_{0}$ and any $\Gamma\in\mathcal{F}$, the kernel $V(x,y)$ defined by $\tilde{S}_{k_{s}}((S^{k}\varphi)\circ\pi_{\Gamma}^{-1})(x)=\int_{y\in\mathbb{R}^{d}}V(x,y)\varphi(y)\mathrm{d}y$ for $x\in\mathbb{R}^{d_{s}}$ and $\varphi$ supported in $\Omega$ satisfies101010The proof shows that the same bound holds for the kernel associated to $\tilde{S}_{k_{s}}((S_{k}\varphi)\circ\pi_{\Gamma}^{-1})(x)$.

[TABLE]

The lemma implies that $\int_{y\in\mathbb{R}^{d}}V(x,y)\varphi(y)\mathrm{d}y$ is bounded by a convolution with a function in $L_{1}(\mathbb{R}^{d})$, for which (15) holds.

Proof.

The kernel $V(x,y)$ is given by the formula111111Strictly speaking, we must first integrate by parts $d_{s}+1$ times in the kernel $\int e^{i(\pi_{\Gamma}^{-1}(z)-y)\eta}\sum_{j=0}^{k}\psi_{j}(\eta)\mathrm{d}\eta$ of $(S^{k}\varphi)\circ\Pi_{\Gamma}^{-1}(z)$ for $d(z,K)>\epsilon$, to get an element of $L_{1}(dz)$.

[TABLE]

As a warmup, let us prove (34) if $\Gamma$ is horizontal or, more generally, affine: Letting $\eta=(\eta_{-},\eta_{+})$ with $\eta_{-}=\pi_{-}(\eta)\in\mathbb{R}^{d_{s}}$, we have $\pi_{\Gamma}^{-1}(z)=(z,A(z)+A_{0})$ with $A_{0}\in\mathbb{R}^{d_{u}}$ and $A:\mathbb{R}^{d_{s}}\to\mathbb{R}^{d_{u}}$ linear ($A\equiv 0$ if $\Gamma$ is horizontal), so that (using like in (20) that $\mathbb{F}^{-1}(1)$ is the Dirac at [math]), $V(x,y)$ can be rewritten as

[TABLE]

since $\psi_{j}(\eta)$ and $\psi_{k_{s}}^{(d_{s})}(\eta_{-}+A^{tr}\eta_{+})$ have disjoint supports if $k_{s}>k+C_{0}$, where $C_{0}\geq 2$ depends on $\|A\|\leq\mathcal{C}_{\mathcal{F}}$.

More generally, $\Gamma\in\mathcal{F}$ is the graph of a $C^{r}$ map $\gamma$ (with $\|\gamma\|_{C^{r}}\leq\mathcal{C}_{\mathcal{F}}$), i.e., $\pi_{\Gamma}^{-1}(z)=(z,\gamma(z))$ for $z\in\mathbb{R}^{d_{s}}$. The lemma is thus obtained integrating by parts $r$ times (in the sense of [2, App. C] if $r$ is not an integer) with respect to $z$ in the kernel $V(x,y)$, using (8), and proceeding as in the end of the proof of [1, Lemma 2.34], mutatis mutandis (using that $\|y-\pi_{\Gamma}^{-1}(x)\|>2^{-k}$ implies that either $\|y-\pi_{\Gamma}^{-1}(z)\|>2^{-k+1}$ or $\|\pi_{\Gamma}^{-1}(z)-\pi_{\Gamma}^{-1}(x)\|>2^{-k+1}$, choosing $C_{0}$ depending on $\mathcal{C}_{\mathcal{F}}$, so that $\|\pi_{\Gamma}^{-1}(z)-\pi_{\Gamma}^{-1}(x)\|>2^{-k+1}$ implies $\|z-x\|\geq 2^{-k+1}/C_{0}$). ∎

Proof of Theorem 3.1.

If $G$ is a rotation about $0\in\mathbb{R}^{d}$ then, since $\psi_{n}\circ G^{-1}=\psi_{n}$, we have $\psi_{n}^{Op}(\tilde{\varphi}\circ G)=((\psi_{n}\circ G^{tr})^{Op}\tilde{\varphi})\circ G=(\psi_{n}^{Op}\tilde{\varphi})\circ G$ (use $G^{tr}=G^{-1}$), and thus $\|\tilde{\varphi}\circ G\|_{\mathcal{U}_{p}^{\mathbf{C}_{+},t,s}}=\|\tilde{\varphi}\|_{\mathcal{U}_{p}^{G(\mathbf{C}_{+}),t,s}}$ for all $\varphi$ (use $G\circ\pi_{\Gamma}^{-1}=\pi_{G(\Gamma)}^{-1}$). It thus suffices to show (13) for $\Lambda=\{x\in\mathbb{R}^{d}\mid x_{1}>0\}$. Indeed, the assumption on $u_{\tilde{\Lambda}}$ implies that the rotation $G$ satisfying $1_{\tilde{\Lambda}}\varphi=(1_{\Lambda}(\varphi\circ G^{-1}))\circ G$ is such that $G(\mathbf{C}_{+})$ is still an unstable cone, i.e., $\mathbb{R}^{d_{s}}\times\{0\}$ is included in $(\mathbb{R}^{d}\setminus G(\mathbf{C}_{+}))\cup\{0\}$ (note that $G(u_{\tilde{\Lambda}})=(1,0,\ldots,0)$, and consider the limiting case $u_{\tilde{\Lambda}}\to\partial\mathbf{C}_{+}$).

Next, since $\varphi$ is supported in $K$, we can replace the half-space $\Lambda$ by a strip $0<x_{1}<B$, still denoted $\Lambda$, and whose characteristic function $1_{\Lambda}(x)$ still only depends on $x_{1}\in\mathbb{R}$. Without loss of generality, we may assume that $B=1$.

Our starting point is then the decomposition (16) applied to $\upsilon=1_{\Lambda}$. We consider first the term $\Pi_{3}(\varphi,1_{\Lambda})$. We will bootstrap from Lemma 3.3. Set

[TABLE]

Then $1_{\Lambda}^{k-2,\Gamma}(x_{-})$ is a function of $x_{1}$ alone (recalling (20)), and the leafwise Young inequality (15), together with the second claim of (7) and the fact that $\|1_{\Lambda}\|_{B^{1/t}_{t,\infty}(\mathbb{R})}<\infty$ (for any $1<t<\infty$, see e.g. [15, Lemma 2.3.1/3(ii), Lemma 2.3.5]), give that both $\|1_{\Lambda}^{k-2,\Gamma}\|_{B^{1/p^{\prime}}_{p^{\prime},\infty}(\mathbb{R})}$ and $\|1_{\Lambda}^{k-2,\Gamma}\|_{L_{\infty}(\mathbb{R})}$ are finite, uniformly in $\Gamma$ and $k$. Next, by (17), (15), and (22), there exists a constant $C$ so that for any $\ell\geq 0$, since $-1+1/p<s<0$,

[TABLE]

where we used (22) from Lemma 3.3 for $f(x_{-})=S_{k}\varphi(x_{-},\gamma(x_{-}))$ with $\gamma=\gamma(\Gamma)$ from the proof of Sublemma 3.5, and $g(x_{-})=1_{\Lambda}^{k-2,\Gamma}(x_{-})$. This concludes the bound for $\Pi_{3}(\varphi,1_{\Lambda})$, and we move to $\Pi_{2}(\varphi,1_{\Lambda})$. Setting

[TABLE]

we have that $1_{\Lambda,k}^{\Gamma}(x_{-})=1_{\Lambda,k}^{\Gamma}(x_{1})$, and also, recalling (19), the leafwise Young inequality (15), together with the first claim of (7), we find

[TABLE]

Thus, using (18), and applying (22) from Lemma 3.3 again, we find, since $t>0$,

[TABLE]

It remains to bound the contribution of $\Pi_{1}(\varphi,1_{\Lambda})$. This is the trickiest estimate. It will use Lemma 3.5 and our assumption $t-(r-1)<s<-t<0$. For any $\ell\geq 0$, we have, using again (15), (17), and (7),

[TABLE]

We may focus on the term $k=\ell$, as the others are almost identical. We will use the paraproduct decomposition $\tilde{\Pi}_{1}+\tilde{\Pi}_{2}+\tilde{\Pi}_{3}$ and the operators $\tilde{S}_{j}$ and $\tilde{S}^{j}$ (see (24)). Put $(S^{k-2}\varphi)^{\Gamma}=(S^{k-2}\varphi)\circ\pi_{\Gamma}^{-1}$. By (20) and (17), we have

[TABLE]

taking $C_{0}\geq 2$ from Lemma 3.5, using (9), and setting

[TABLE]

Lemma 3.3 and the Young inequality (thrice) give $C$ so that for all $j,k,m,$ and $\Gamma$

[TABLE]

where we applied (38) in the second inequality. Thus, Lemma 3.5 and the leafwise121212See §4 of Corrections and complements to [2] for the factor $2^{k(-s+\delta)}$. Young inequality (15) applied to $k_{s}=j\geq k+2+C_{0}$ gives $k_{0}\geq C_{0}$ so that for any $\delta\in(0,1)$ (recalling $0<t-s<r-1<r-\delta$)

[TABLE]

Using again (42), the finite double sum in (41) is bounded by $(C_{0}+4)C\|\varphi\|_{\mathcal{U}^{\mathbf{C}_{+},t,s}_{p}}$.

For the contribution of $\tilde{\Pi}_{1}$ in (40), using again (20) and (17), we find

[TABLE]

Setting $(S_{j}\varphi)^{\Gamma}=(S_{j}\varphi)\circ\pi_{\Gamma}^{-1}$, we bound the term for $n=k$ above131313The other terms are similar. by the sum of

[TABLE]

(which can be handled as in (43), by Lemma 3.5), and,

[TABLE]

using that $s<0$. Now, since $s+t<0$, we get, using the Young inequality,

[TABLE]

Finally, using (20) once more, we bound the contribution of $\tilde{\Pi}_{2}$ in (40):

[TABLE]

where

[TABLE]

can be bounded similarly as (43), using Lemma 3.5. For the remaining finite double sum in (44), we focus on the contributions with $\ell=0$ and $\tilde{\ell}=2$, the others being similar. Then, applying Lemma 3.3, we find

[TABLE]

using (38) once more. This ends the proof of Theorem 3.1. ∎