Fourier multipliers on a vector-valued function space

TL;DR

This paper establishes sharp multiplier theorems for vector-valued function spaces, extending classical results and providing conditions under which certain Fourier multipliers are bounded, with implications for Triebel-Lizorkin spaces.

Contribution

The paper generalizes and improves existing multiplier theorems for vector-valued functions, including sharp conditions on Sobolev spaces for boundedness.

Findings

Derived new boundedness conditions for Fourier multipliers on vector-valued spaces.

Extended results to Triebel-Lizorkin spaces for p=∞.

Proved the sharpness of the Sobolev space conditions in the multiplier theorem.

Abstract

We study multiplier theorems on a vector-valued function space, which is a generalization of the results of Calder\'on-Torchinsky and Grafakos-He-Honz\'ik-Nguyen, and an improvement of the result of Triebel. For $0<p<\infty$ and $0<q\leq \infty$ we obtain that if $r>\frac{d}{s-(d/\min{(1,p,q)}-d)}$, then $$\big\Vert \big\{\big( m_k \hat{f_k}\big)^{\vee}\big\}_{k\in\mathbb{N}}\big\Vert_{L^p(l^q)}\lesssim_{p,q} \sup_{l\in\mathbb{N}}{\big\Vert m_l(2^l\cdot)\big\Vert_{L_s^r(\mathbb{R}^d)}} \big\Vert \big\{f_k\big\}_{k\in\mathbb{N}}\big\Vert_{L^p(l^q)}, ~~f_k\in\mathcal{E}(A2^k),$$ under the condition $\max{(|d/p-d/2|,|d/q-d/2|)}<s<d/\min{(1,p,q)}$. An extension to $p=\infty$ will be additionally considered in the scale of Triebel-Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by a smaller Sobolev space $L_s^r$ with $r\leq \frac{d}{s-(d/\min{(1,p,q)}-d)}$.

Equations

$$\big{\|}\big{\{}\big{(}m_{k}\widehat{f_{k}}\big{)}^{\vee}\big{\}}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(\ell^{q})}\lesssim_{p,q}\sup_{l\in\mathbb{Z}}{\big{\|}m_{l}(2^{l}\cdot)\big{\|}_{L_{s}^{r}(\mathbb{R}^{d})}}\big{\|}\big{\{}f_{k}\big{\}}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(\ell^{q})},~{}~{}f_{k}\in\mathcal{E}(A2^{k}),$$

$$\displaystyle\big{|}\partial_{\xi}^{\beta}m(\xi)\big{|}\lesssim_{\beta}|\xi|^{-|\beta|}$$

$$\displaystyle\sup_{l\in\mathbb{Z}}{\big{\|}m(2^{l}\cdot)\widehat{\phi}\big{\|}_{L^{2}_{s}(\mathbb{R}^{d})}}<\infty$$

$$(I-\Delta)^{s/2}f:=\big{(}(1+4\pi^{2}|\cdot|^{2})^{s/2}\widehat{f}\big{)}^{\vee}$$

$$\|f\|_{L^{r}_{s}(\mathbb{R}^{d})}:=\big{\|}(I-\Delta)^{s/2}f\big{\|}_{L^{r}(\mathbb{R}^{d})}$$

$$\sup_{l\in\mathbb{Z}}{\big{\|}m(2^{l}\cdot)\widehat{\phi}\big{\|}_{L_{s}^{r}(\mathbb{R}^{d})}}<\infty\qquad\text{ for }~{}r>d/s.$$

$$L_{A}^{p}(\ell^{q}):=\big{\{}\{f_{k}\}_{k\in\mathbb{Z}}\subset S^{\prime}:f_{k}\in\mathcal{E}(A2^{k}),\big{\|}\big{\{}f_{k}\big{\}}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(\ell^{q})}<\infty\big{\}}.$$

$$f_k=f_k\ast {\Pi }_k\text{ in the sense of tempered distribution}$$

$$\Vert \{f_k{\}}_{k\in \mathbb{Z}}{\Vert }_{L^p({\ell }^q)}=\Vert \{f_k\ast {\Pi }_k{\}}_{k\in \mathbb{Z}}{\Vert }_{L^p({\ell }^q)}.$$

$$\sup_{l\in\mathbb{N}}{\big{\|}m_{l}(2^{l}\cdot)\big{\|}_{L_{s}^{2}(\mathbb{R}^{d})}}<\infty$$

$$s>\Big{\{}\begin{array}[]{ll}d/\min{(1,p,q)}-d/2&\text{if}\quad q<\infty\\ d/p+d/2&\text{if}\quad q=\infty\end{array}.$$

$$\big{\|}\big{\{}\big{(}m_{k}\widehat{f_{k}}\big{)}^{\vee}\big{\}}_{k\in\mathbb{N}}\big{\|}_{L^{p}(\ell^{q})}\lesssim_{p,q}\sup_{l\in\mathbb{N}}{\big{\|}m_{l}(2^{l}\cdot)\big{\|}_{L_{s}^{2}}}\big{\|}\big{\{}f_{k}\big{\}}_{k\in\mathbb{N}}\big{\|}_{L^{p}(\ell^{q})}.$$

$${\tau }^{(s,p)}:=\frac{d}{s-(d/\min (1,p)-d)},{\tau }^{(s,p,q)}:=\frac{d}{s-(d/\min (1,p,q)-d)}.$$

$$\mathcal{L}_{s}^{r}[\mathbf{m}]:=\sup_{l\in\mathbb{Z}}{\big{\|}m_{l}(2^{l}\cdot)\big{\|}_{L_{s}^{r}(\mathbb{R}^{d})}}.$$

$$\max{\big{(}\big{|}d/p-d/2\big{|},\big{|}d/q-d/2\big{|}\big{)}}<s<d/\min{(1,p,q)}.$$

$${\mathcal{L}}_s^r[\mathbf{m}]<\infty \text{ for }r>{\tau }^{(s,p,q)}.$$

$$\big{\|}\big{\{}\big{(}m_{k}\widehat{f_{k}}\big{)}^{\vee}\big{\}}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(\ell^{q})}\lesssim_{p,q}\mathcal{L}_{s}^{r}[\mathbf{m}]\big{\|}\big{\{}f_{k}\big{\}}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(\ell^{q})}.$$

$$\big{|}d/q-d/2\big{|}<s<d/\min{(1,q)}.$$

$${\mathcal{L}}_s^r[\mathbf{m}]<\infty ,\text{ for }r>{\tau }^{(s,q)}.$$

$$\displaystyle\sup_{P\in\mathcal{D},\ell(P)\leq 2^{-\mu}}{\Big{(}\frac{1}{|P|}\int_{P}{\sum_{k=-\log_{2}{\ell(P)}}^{\infty}{\big{|}\big{(}m_{k}\widehat{f_{k}}\big{)}^{\vee}(x)\big{|}^{q}}}dx\Big{)}^{1/q}}$$

$$\displaystyle\lesssim_{q}\mathcal{L}_{s}^{r}[\mathbf{m}]\sup_{P\in\mathcal{D},\ell(P)\leq 2^{-\mu}}{\Big{(}\frac{1}{|P|}\int_{P}{\sum_{k=-\log_{2}{\ell(P)}}^{\infty}{|f_{k}(x)|^{q}}}dx\Big{)}^{1/q}}$$

$$\max{\big{(}\big{|}d/p-d/2\big{|},\big{|}d/q-d/2\big{|}\big{)}}<s<d/\min{(1,p,q)}$$

$$\sup_{l\in\mathbb{Z}}{\big{\|}m(2^{l}\cdot)\widehat{\phi}\big{\|}_{L^{r}_{s}(\mathbb{R}^{d})}}<\infty\quad\text{ for }~{}r>\tau^{(s,p,q)}.$$

$$\|T_{m}f\|_{\dot{F}_{p}^{\alpha,q}(\mathbb{R}^{d})}\lesssim\sup_{l\in\mathbb{Z}}{\big{\|}m(2^{l}\cdot)\widehat{\phi}\big{\|}_{L^{r}_{s}(\mathbb{R}^{d})}}\|f\|_{\dot{F}_{p}^{\alpha,q}(\mathbb{R}^{d})}.$$

$$\big{\|}\big{(}m_{k}\widehat{\Psi_{k}}\big{)}(2^{k}\cdot)\big{\|}_{L_{s}^{r}(\mathbb{R}^{d})}=\big{\|}m_{k}(2^{k}\cdot)\widehat{\Psi_{0}}\big{\|}_{L_{s}^{r}(\mathbb{R}^{d})}\lesssim\|m_{k}(2^{k}\cdot)\|_{L_{s}^{r}(\mathbb{R}^{d})}$$

$$Supp(m_{k})\subset\big{\{}\xi\in\mathbb{R}^{d}:|\xi|\leq 2^{k}\big{\}}$$

$$\|f\|_{\dot{F}_{p}^{\alpha,q}(\mathbb{R}^{d})}:=\big{\|}\big{\{}2^{\alpha k}\phi_{k}\ast f\big{\}}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(\ell^{q})}<\infty,\quad 0<p<\infty~{}\text{ or }~{}p=q=\infty,$$

$$\|f\|_{\dot{F}_{\infty}^{\alpha,q}(\mathbb{R}^{d})}:=\sup_{P\in\mathcal{D}}{\Big{(}\frac{1}{|P|}\int_{P}{\sum_{k=-\log_{2}{\ell(P)}}^{\infty}{2^{\alpha kq}\big{|}\phi_{k}\ast f(x)\big{|}^{q}}}dx\Big{)}^{1/q}},\quad 0<q<\infty$$

$${\dot{F}}_p^{0,2}({\mathbb{R}}^d)=H^p({\mathbb{R}}^d)$$

$${\dot{F}}_p^{\alpha ,2}({\mathbb{R}}^d)=H_{\alpha }^p({\mathbb{R}}^d)$$

$$BMO$$

Full text

[TABLE]

under the condition $\max{(|d/p-d/2|,|d/q-d/2|)}<s<d/\min{(1,p,q)}$. An extension to $p=\infty$ will be additionally considered in the scale of Triebel-Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by Sobolev spaces $L_{s}^{r}$ with $r\leq\frac{d}{s-(d/\min{(1,p,q)}-d)}$.

The author is supported in part by NRF grant 2019R1F1A1044075 and by a KIAS Individual Grant MG070001 at Korea Institute for Advanced Study.

1. Introduction and main results

Let $S(\mathbb{R}^{d})$ denote the Schwartz space and $S^{\prime}(\mathbb{R}^{d})$ the space of tempered distributions. For the Fourier transform of $f\in S(\mathbb{R}^{d})$ we use the definition $\widehat{f}(\xi):=\int_{\mathbb{R}^{d}}{f(x)e^{-2\pi i\langle x,\xi\rangle}}dx$ and denote by $f^{\vee}(\xi):=\widehat{f}(-\xi)$ the inverse Fourier transform of $f$. We also extend these transforms to the space of tempered distributions.

For $m\in L^{\infty}(\mathbb{R}^{d})$ the multiplier operator $T_{m}$ is defined by T_{m}f(x):=\big{(}m\widehat{f}\big{)}^{\vee}(x) for $f\in S(\mathbb{R}^{d})$. The classical Mikhlin multiplier theorem [16] states that if a function $m$ satisfies

[TABLE]

for all multi-indices $\beta$ with |\beta|\leq\big{[}d/2\big{]}+1, then the operator $T_{m}$ is bounded in $L^{p}(\mathbb{R}^{d})$ for $1<p<\infty$. In [14] Hörmander sharpened the result of Mikhlin, using the weaker condition

[TABLE]

for $s>d/2$, where $L^{2}_{s}(\mathbb{R}^{d})$ denotes the standard fractional Sobolev space on $\mathbb{R}^{d}$ and $\phi$ is a Schwartz function on $\mathbb{R}^{d}$, which generates a Littlewood-Paley partition of unity via a dyadic dilation, defined in Section 2. Calderón and Torchinsky [3] proved that if (1.1) holds for $s>d/p-d/2$, then $m$ is a Fourier multiplier of Hardy space $H^{p}(\mathbb{R}^{d})$ for $0<p\leq 1$. A different proof was given by Taibleson and Weiss [23]. It turns out that the condition $s>d/\min{(1,p)}-d/2$ is optimal for the boundedness to hold and it is natural to ask whether (1.1) can be weakened by replacing $L^{2}_{s}(\mathbb{R}^{d})$ by other function spaces. Baernstein and Sawyer [1] obtained endpoint $H^{p}(\mathbb{R}^{d})$ estimates by using Herz space conditions for \big{(}m(2^{j}\cdot)\widehat{\phi}\big{)}^{\vee} and these estimates were improved and extended to Triebel-Lizorkin spaces by Seeger [21] and Park [18]. On the other hand, for $1<p<\infty$, using an interpolation method, Calderón and Torchinsky [3] replaced $L_{s}^{2}(\mathbb{R}^{d})$ in (1.1) by $L_{s}^{r}(\mathbb{R}^{d})$ for the $L^{p}$-boundedness to hold and the assumption in their result was replaced by a weaker one by Grafakos, He, Honzík, and Nguyen [11]. Let $(I-\Delta)^{s/2}$ be the inhomogeneous fractional Laplacian operator, explicitly given by

[TABLE]

and let $L_{s}^{r}(\mathbb{R}^{d})$ be the space containing tempered distributions $f$, defined on $\mathbb{R}^{d}$, for which the norm

[TABLE]

is finite.

Theorem A**.**

Let $1<p<\infty$ and $|d/p-d/2|<s<d$. Suppose that

[TABLE]

Then $T_{m}$ is bounded in $L^{p}(\mathbb{R}^{d})$.

We also refer to [12, 13] for further improvement of the multiplier theorem by using Lorentz space conditions.

A vector-valued version of Hörmander’s multiplier theorem was studied by Triebel [24], [26, 2.4.9]. For $r>0$ let $\mathcal{E}(r)$ denote the space of all distributions whose Fourier transform is supported in \big{\{}\xi\in\mathbb{R}^{d}:|\xi|\leq 2r\big{\}}. Let $A>0$. For $0<p<\infty$ and $0<q\leq\infty$ or for $p=q=\infty$ we define

[TABLE]

To give a rigorous definition of the space, we recall that for each $f_{k}\in\mathcal{E}(A2^{k})$

[TABLE]

where $\Pi_{k}$ is a Schwartz function whose Fourier transform is equal to $1$ on the ball of radius $A2^{k+1}$, centered at [math] and is supported in a larger ball. Since convolution between a tempered distribution and a Schwartz function is a smooth function, $f_{k}\ast\Pi_{k}$ is actually a smooth function and thus, the norm $\|\{f_{k}\}_{k\in\mathbb{Z}}\|_{L^{p}(\ell^{q})}$ can be interpreted as

[TABLE]

In the rest of this paper, we think of $f_{k}\in\mathcal{E}(A2^{k})$ as a smooth function $f_{k}\ast\Pi_{k}$.

Then $L_{A}^{p}(\ell^{q})$ is a quasi-Banach space (Banach space if $p,q\geq 1$) with a (quasi-)norm $\|\cdot\|_{L^{p}(\ell^{q})}$ ( see [26] for more details ).

Theorem B**.**

Let $0<p<\infty$, $0<q\leq\infty$, and $A>0$. Suppose $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{N}$, and $\{m_{k}\}_{k\in\mathbb{N}}$ satisfies

[TABLE]

for

[TABLE]

Then

[TABLE]

It was first proved that if (1.2) holds for $s>d/2$, then (1.3) works for $1<p,q<\infty$, by using Hörmander’s multiplier theorem. For the case $0<p<\infty$ and $0<q\leq\infty$, it is easy to obtain that (1.3) is true under the assumption (1.2) with $s>d/2+d/\min{(p,q)}$. Then a complex interpolation method is applied to derive $s>d/\min{(1,p,q)}-d/2$ for general $0<p,q<\infty$. However, the method cannot be applied to the endpoint case $q=\infty$ and thus the assumption $s>d/p+d/2$ is required when $q=\infty$, which is stronger than seemingly “natural” condition $s>d/\min{(1,p)}-d/2$.

The aim of this paper is to provide an improvement of Theorem B, which would be actually a vector-valued extension of Theorem A in the full range $0<p\leq\infty$. Let

[TABLE]

For $\mathbf{m}:=\{m_{k}\}_{k\in\mathbb{Z}}$, throughout this work we will use the notation:

[TABLE]

Theorem 1.1**.**

Let $0<p<\infty$ and $0<q\leq\infty$, $A>0$, and

[TABLE]

Suppose $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$ and $\mathbf{m}:=\{m_{k}\}_{k\in\mathbb{Z}}$ satisfies

[TABLE]

Then

[TABLE]

Moreover, the inequality also holds for $p=q=\infty$.

Theorem 1.1 can be extended to the case $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space. To describe this, let $\mathcal{D}$ denote the collection of all dyadic cubes in $\mathbb{R}^{d}$ and for each $P\in\mathcal{D}$ let $\ell(P)$ be the side length of $P$.

Theorem 1.2**.**

Let $0<q<\infty$, $A>0$, $\mu\in\mathbb{Z}$, and

[TABLE]

Suppose $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$ and $\mathbf{m}:=\{m_{k}\}_{k\in\mathbb{Z}}$ satisfies

[TABLE]

Then

[TABLE]

uniformly in $\mu$.

As a corollary of the two theorems, we can prove the $\dot{F}_{p}^{\alpha,q}$-boundedness of the operator $T_{m}$, which is a generalization of Theorem A and an improvement of the result in [25].

Corollary 1.3**.**

Let $0<p,q\leq\infty$ and $\alpha\in\mathbb{R}$. Suppose

[TABLE]

and $m\in L^{\infty}(\mathbb{R}^{d})$ satisfies

[TABLE]

Then

[TABLE]

This follows from setting $m_{k}=m\widehat{\phi_{k}}$ and $f_{k}=2^{\alpha k}\widetilde{\phi_{k}}\ast f$ where $\widetilde{\phi_{k}}:=\phi_{k-1}+\phi_{k}+\phi_{k+1}$. The detailed proof is omitted as standard arguments are applicable. We refer the reader to Section 2 for the definition of Triebel-Lizorkin spaces $\dot{F}_{p}^{\alpha,q}(\mathbb{R}^{d})$. As the space $\dot{F}_{p}^{\alpha,q}$ is a generalization of many function spaces such as Lebesgue space, Hardy space and $BMO$, Corollary 1.3 also implies the boundedness of $T_{m}$ on such function spaces.

It turns out that the condition $s>|d/p-d/2|$ is optimal for the $L^{p}$-boundedness to hold in Theorem A and the proof can be found in Slavíková [22]. Moreover, Grafakos and Park [12] recently proved that the condition $r>d/s$ should be also necessary in the theorem, using properties of Bessel potentials, which will be described in (9.2) later. We now consider the sharpness of the condition $r>\tau^{(s,p,q)}$ in Theorem 1.1. Our claim is that (1.4) fails for $r=\tau^{(s,p,q)}$.

Theorem 1.4**.**

Let $0<p<\infty$, $0<q\leq\infty$, and $d/\min{(1,p,q)}-d<s<d/\min{(1,p,q)}.$ Then there exists $\mathbf{m}:=\{m_{k}\}_{k\in\mathbb{Z}}$ such that $\mathcal{L}_{s}^{\tau^{(s,p,q)}}[\mathbf{m}]<\infty$, but (1.4) does not hold.

Remark that the assumption $d/\min{(1,p,q)}-d<s<d/\min{(1,p,q)}$ is clearly weaker than \max{\big{(}\big{|}d/p-d/2\big{|},\big{|}d/q-d/2\big{|}\big{)}}<s<d/\min{(1,p,q)} in Theorem 1.1.

We first study Theorem 1.2, using a proper separation of $f_{k}$ and $F_{\infty}$-variants of Peetre’s maximal inequality, introduced by the author [17]. For the proof of Theorem 1.1, the case $0<p=q\leq\infty$ can be handled in a easy way via the $L^{p}$-boundedness of $T_{m_{k}}$, which is stated in Lemma 6.1, and thus our interest will be given to the case $p\not=q$. For the case $0<p\leq 1$ and $p<q\leq\infty$ we will establish a discrete characterization of $L_{A}^{p}(\ell^{q})$ by using the $\varphi$-transform of Frazier and Jawerth [7, 8, 9, 10] and apply atomic decomposition of discrete function space $\dot{f}_{p}^{0,q}$ in [10], which is analogous to the atomic decomposition of $H^{p}(\mathbb{R}^{d})$. When $0<q\leq 1$ and $q<p<\infty$, the proof relies on a characterization of $L_{A}^{p}(\ell^{q})$ by a dyadic version of the Fefferman-Stein sharp maximal function [6]. The remaining case $1<p<\infty$ and $1<q\leq\infty$ follows from a combination of complex interpolation techniques in Proposition 5.1 and duality arguments in Lemma 4.1. The central idea to prove Theorem 1.4 is a necessary condition for a vector-valued inequality of convolution operator in the paper of Christ and Seeger [4] and a behavior of variants of Bessel potentials in the paper of Grafakos and Park [12]. See (9.3) and (9.2) below.

Basic setting : The constant $A$ plays a minor role in the results and in fact, it affects the results only up to a constant. Hence, we fix $A=2^{-2}$ in the proof to avoid unnecessary complications. Moreover, if $f_{k}\in\mathcal{E}(2^{k-2})$, then \big{(}m_{k}\widehat{f_{k}}\big{)}^{\vee}=\big{(}(m_{k}\widehat{\Psi_{k}})\widehat{f_{k}}\big{)}^{\vee} where $\Psi_{k}\in S(\mathbb{R}^{d})$ is a Schwartz function having the properties that $Supp(\widehat{\Psi_{k}})\subset\{\xi\in\mathbb{R}^{d}:|\xi|\leq 2^{k}\}$ and $\widehat{\Psi_{k}}(\xi)=1$ for $|\xi|\leq 2^{k-1}$. This function will be officially defined in Section 3.1, using dyadic dilation $\Psi_{k}(x)=2^{kd}\Psi_{0}(2^{k}x)$. Then the Kato-Ponce inequality [15] yields that for $1<r<\infty$ and $s\geq 0$,

[TABLE]

and this enables us to assume that

[TABLE]

in the proof. With this assumption, we can write \big{(}m_{k}\widehat{f_{k}}\big{)}^{\vee}(x)=m_{k}^{\vee}\ast f_{k}(x).

This paper is organized as follows. Section 2 is dedicated to preliminaries, introducing definitions and general properties which will be used in our proofs. Two characterizations of $L^{p}_{A}(\ell^{q})$ will be given in Section 3.1, and by using one of them we dualize the function space $L^{p}_{A}(\ell^{q})$ for $1<p<\infty$ and $1\leq q<\infty$ in Section 4. In Section 5 we present a complex interpolation theorem for multipliers on $L^{p}_{A}(\ell^{q})$, based on the idea of Triebel [26, 2.4.9]. Section 6 contains a lemma which will play a fundamental role in the proof of both Theorem 1.1 and 1.2. The proof of Theorem 1.1, 1.2, and 1.4 will be provided in the last three sections.

Notations : We use standard notations. Let $\mathbb{N}$ be the collection of all natural numbers and $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$. Denote by $\mathbb{Z}$ and $\mathbb{R}$ the set of all integers and the set of all real numbers, respectively. Let $\mathcal{D}$ stand for the set of all dyadic cubes in $\mathbb{R}^{d}$ as above and for each $k\in\mathbb{Z}$, let $\mathcal{D}_{k}$ be the subset of $\mathcal{D}$ consisting of the cubes with side length $2^{-k}$. We use the symbol $X\lesssim Y$ to indicate that $X\leq CY$ for some constant $C>0$, possibly different at each occurrence, and $X\approx Y$ if $A\lesssim B$ and $B\lesssim A$ simultaneously.

2. Preliminaries

2.1. Function spaces

Let $\Phi_{0}$ be a Schwartz function so that Supp(\widehat{\Phi_{0}})\subset\big{\{}\xi\in\mathbb{R}^{d}:|\xi|\leq 2\big{\}} and $\widehat{\Phi_{0}}(\xi)=1$ for $|\xi|\leq 1$ and define $\phi:=\Phi_{0}-2^{-d}\Phi_{0}(2^{-1}\cdot)$ and $\phi_{k}:=2^{kd}\phi(2^{k}\cdot)$. Then $\{\phi_{k}\}_{k\in\mathbb{Z}}$ forms a (homogeneous) Littlewood-Paley partition of unity. That is, Supp(\widehat{\phi_{k}})\subset\big{\{}\xi\in\mathbb{R}^{d}:2^{k-1}\leq|\xi|\leq 2^{k+1}\big{\}} and $\sum_{k\in\mathbb{Z}}{\widehat{\phi_{k}}(\xi)}=1$ for $\xi\not=0$.

For $0<p,q\leq\infty$ and $\alpha\in\mathbb{R}$, the (homogeneous) Triebel-Lizorkin space $\dot{F}_{p}^{\alpha,q}(\mathbb{R}^{d})$ is defined by the collection of all $f\in S^{\prime}/\mathcal{P}$ (tempered distribution modulo polynomials) such that

[TABLE]

[TABLE]

where the supremum is taken over all dyadic cubes in $\mathbb{R}^{d}$. Then these spaces provide a general framework that unifies classical function spaces:

[TABLE]

Note that $H^{p}(\mathbb{R}^{d})=L^{p}(\mathbb{R}^{d})$ if $1<p<\infty$.

2.2. Maximal inequalities

A crucial tool in theory of function spaces is the maximal inequalities of Fefferman and Stein [5] and Peetre [20].

Let $\mathcal{M}$ be the Hardy-Littlewood maximal operator, defined by

[TABLE]

where the supremum is taken over all cubes containing $x$, and for $0<t<\infty$ let \mathcal{M}_{t}f:=\big{(}\mathcal{M}(|f|^{t})\big{)}^{1/t}. Then the Fefferman-Stein vector-valued maximal inequality [5] states that for $0<r<p,q<\infty$,

[TABLE]

The inequality (2.1) also holds for $0<p\leq\infty$ and $q=\infty$.

For $k\in\mathbb{Z}$ and $\sigma>0$ we now define the Peetre maximal operator $\mathfrak{M}_{\sigma,2^{k}}$ by the formula

[TABLE]

It is known in [20] that for $f\in\mathcal{E}(A2^{k})$,

[TABLE]

Then (2.1) and (2.2) yield the following maximal inequality: Suppose $f_{k}\in\mathcal{E}(A2^{k})$ for some $A>0$. Then for $0<p<\infty$ or $p=q=\infty$, we have

[TABLE]

if $\sigma>d/\min{(p,q)}$.

Furthermore, a $\dot{F}_{\infty}$-version of (2.3) is recently given by the author [17] : Suppose $f_{k}\in\mathcal{E}(A2^{k})$ for some $A>0$. Then for $0<q<\infty$ and $\mu\in\mathbb{Z}$, we have

[TABLE]

uniformly in $\mu$ if $\sigma>d/q$. We remark that (2.4) does not hold when $\mathfrak{M}_{\sigma,2^{k}}f_{k}$ is replaced by $\mathcal{M}_{r}f_{k}$ for all $0<r<\infty$.

As an application of (2.4), we have

[TABLE]

See [17] for more details.

2.3. $\varphi$-transform in $\dot{F}_{p}^{0,q}$

For a sequence of complex numbers $\mathbf{b}:=\{b_{Q}\}_{Q\in\mathcal{D}}$ we define

[TABLE]

[TABLE]

where

[TABLE]

Then the Triebel-Lizorkin space $\dot{F}_{p}^{0,q}(\mathbb{R}^{d})$ can be characterized by the discrete function space $\dot{f}_{p}^{0,q}$: For $Q\in\mathcal{D}$ let $x_{Q}$ be the lower left corner of $Q$. Every $f\in\dot{F}_{p}^{0,q}(\mathbb{R}^{d})$ can be written as

[TABLE]

where $\varphi_{k}$ and $\widetilde{\varphi_{k}}$ are Schwartz functions with localized frequency, involving Littlewood-Paley decomposition, $\varphi^{Q}(x):=|Q|^{1/2}\varphi_{k}(x-x_{Q})$, $\widetilde{\varphi}^{Q}(x):=|Q|^{1/2}\widetilde{\varphi_{k}}(x-x_{Q})$ for each $Q\in\mathcal{D}_{k}$, and $b_{Q}:=\langle f,\widetilde{\varphi}^{Q}\rangle$. To be specific, since $\sum_{k\in\mathbb{Z}}\widehat{\varphi_{k}}(\xi)\widehat{\widetilde{\varphi_{k}}}(\xi)=1$ for $\xi\not=0$, we have $f=\sum_{k\in\mathbb{Z}}\varphi_{k}\ast\widetilde{\varphi_{k}}\ast f$ in $S^{\prime}/\mathcal{P}$ and for each $k\in\mathbb{Z}$

[TABLE]

Moreover, in the case, we have

[TABLE]

The converse estimate is also true. For any sequence $\mathbf{b}:=\{b_{Q}\}_{Q\in\mathcal{D}}$ of complex numbers satisfying $\|\mathbf{b}\|_{\dot{f}_{p}^{0,q}}<\infty$,

[TABLE]

belongs to $\dot{F}_{p}^{0,q}$ and indeed,

[TABLE]

See [7, 8] for more details.

2.4. Atomic decomposition of $\dot{f}_{p}^{0,q}$

Let $0<p\leq 1$ and $p\leq q\leq\infty$. A sequence of complex numbers $\mathbf{r}:=\{r_{Q}\}_{Q\in\mathcal{D}}$ is called an $\infty$-atom for $\dot{f}_{p}^{0,q}$ if there exists $Q_{0}\in\mathcal{D}$ such that

[TABLE]

and

[TABLE]

Then the following atomic decomposition of $\dot{f}_{p}^{0,q}$ holds:

Lemma 2.1**.**

[9, 10]** Suppose $0<p\leq 1$, $p\leq q\leq\infty$, and $\mathbf{b}:=\{b_{Q}\}_{Q\in\mathcal{D}}\in\dot{f}_{p}^{0,q}$. Then there exist $C_{p,q}>0$, a sequence of scalars $\{\lambda_{j}\}$, and a sequence of $\infty$-atoms $\mathbf{r}_{j}=\{r_{j,Q}\}_{{Q\in\mathcal{D}}}$ for $\dot{f}_{p}^{0,q}$ so that

[TABLE]

and

[TABLE]

Moreoever, it follows that

[TABLE]

3. Characterizations of $L^{p}_{A}(\ell^{q})$

As mentioned in Section 1, we assume $A=2^{-2}$.

3.1. Characterization of $L^{p}_{A}(\ell^{q})$ by using a method of $\varphi$-transform

We will study properties of $\{f_{k}\}_{k\in\mathbb{Z}}\in L^{p}_{A}(\ell^{q})$, which are analogous to (2.6), (2.7), and (2.8).

Suppose that $\Psi_{0}\in S(\mathbb{R}^{d})$ satisfies

[TABLE]

For each $k\in\mathbb{Z}$ and $Q\in\mathcal{D}_{k}$ let $\Psi_{k}:=2^{kd}\Psi_{0}(2^{k}\cdot)$ and

[TABLE]

where $x_{Q}$ denotes the lower left corner of the cube $Q$ as before.

Lemma 3.1**.**

Let $0<p<\infty$ or $p=q=\infty$.

1. (1)

Assume $f_{k}\in\mathcal{E}(2^{k-2})$ for each $k\in\mathbb{Z}$. Then there exists a sequence of complex numbers $\mathbf{b}:=\{b_{Q}\}_{Q\in\mathcal{D}}$ such that

[TABLE] 2. (2)

For any sequence $\mathbf{b}=\{b_{Q}\}_{Q\in\mathcal{D}}$ of complex numbers satisfying $\|\mathbf{b}\|_{\dot{f}_{p}^{0,q}}<\infty$,

[TABLE]

satisfies

[TABLE]

For the case $p=\infty$ and $0<q<\infty$ we introduce

[TABLE]

for $\mu\in\mathbb{Z}$.

Lemma 3.2**.**

Let $0<q<\infty$ and $\mu\in\mathbb{Z}$.

1. (1)

Assume $f_{k}\in\mathcal{E}(2^{k-2})$ for each $k\geq\mu$. Then there exists a sequence of complex numbers $\mathbf{b}:=\{b_{Q}\}_{Q\in\mathcal{D},\ell(Q)\leq 2^{-\mu}}$ such that

[TABLE]

and

[TABLE] 2. (2)

For any sequence $\mathbf{b}:=\{b_{Q}\}_{Q\in\mathcal{D},\ell(Q)\leq 2^{-\mu}}$ of complex numbers satisfying $\|\mathbf{b}\|_{\dot{f}_{\infty}^{0,q}(\mu)}<\infty$,

[TABLE]

satisfies

[TABLE]

Proof of Lemma 3.1.

(1) Since $Supp(\widehat{f_{k}}(2^{k}\cdot))\subset\{|\xi|\leq 1/2\}$, $\widehat{f_{k}}$ admits the decomposition

[TABLE]

using a scaling argument and the Fourier series representation of $\widehat{f_{k}}(2^{k}\cdot)$. Then we have

[TABLE]

For any $Q\in\mathcal{D}_{k}$ we write

[TABLE]

where $l:=(l_{1},\dots,l_{d})\in\mathbb{Z}^{d}$. That is, $Q_{k,l}$ is the dyadic cube, contained in $\mathcal{D}_{k}$, whose lower left corner is $x_{Q_{k,l}}=2^{-k}l$. Now we use the notations

[TABLE]

[TABLE]

Then (3.2) can be expressed as

[TABLE]

In addition, for a.e. $x\in\mathbb{R}^{d}$ there exists the unique dyadic cube $Q_{0}\in\mathcal{D}_{k}$ whose interior contains $x$, and this yields that

[TABLE]

Here, the inequality holds due to the fact that

[TABLE]

which is valid even for $f_{k}$ without Fourier support condition. Then we can easily see that for $\sigma>d/\min{(p,q)}$, using (3.4) and (2.3),

[TABLE]

as desired.

(2) For a given $\mathbf{b}:=\{b_{Q}\}_{Q\in\mathcal{D}}$ and $k\in\mathbb{Z}$ let

[TABLE]

Setting

[TABLE]

[TABLE]

for each $k\in\mathbb{Z}$ and $x\in\mathbb{R}^{d}$, we can write

[TABLE]

Choose $0<\epsilon<\min{(1,p,q)}$ and $M>d/\epsilon$. Observe that $|\Psi^{Q}(x)|\lesssim_{M}2^{-jM}|Q|^{-1/2}$ on $E_{j}^{k}$ and then the embedding $\ell^{\epsilon}\hookrightarrow\ell^{1}$ shows that

[TABLE]

Finally, as a result of the maximal inequality (2.1), we obtain

[TABLE]

as required. ∎

Proof of Lemma 3.2.

(1) The proof is very similar to that of Lemma 3.1. Indeed, using (3.3), (3.4) and (2.4) with $\sigma>d/q$, it can be verified that

[TABLE]

(2) We note that

[TABLE]

Let

[TABLE]

and choose $M>d/\min{(1,q)}$. Using Hölder’s inequality if $q>1$ or the embedding $\ell^{q}\hookrightarrow\ell^{1}$ if $q\leq 1$, we obtain

[TABLE]

which further implies that

[TABLE]

For each $P\in\mathcal{D}$ and $m\in\mathbb{Z}^{d}$ let P+\ell(P)m:=\big{\{}x+\ell(P)m:x\in P\big{\}} and denote by $\mathcal{D}_{k}(P,m)$ the subfamily of $\mathcal{D}_{k}$ that contains any dyadic cubes belonging to $P+\ell(P)m$. Then in the last expression we decompose

[TABLE]

which is possible because $P$ and $Q$’s are dyadic cubes with $\ell(Q)=2^{-k}\leq\ell(P)$.

We first see that

[TABLE]

On the other hand, if $|m|>2d$ and $Q\in\mathcal{D}_{k}(P,m)$ then

[TABLE]

and therefore

[TABLE]

Now we apply the triangle inequality if $q\geq 1$ or $\ell^{q}\hookrightarrow\ell^{1}$ if $q<1$ to obtain that

[TABLE]

Since $M\min{(1,q)}>d$ and $2^{k}\ell(P)\geq 1$, the above expression is bounded by

[TABLE]

Combining these estimates, taking a supremum over ${P\in\mathcal{D},\ell(P)\leq 2^{-\mu}}$, and using (3.6), we conclude that

[TABLE]

3.2. Characterization of $L^{p}_{A}(\ell^{q})$ by using a sharp maximal function

Given a locally integrable function $f$ on $\mathbb{R}^{d}$ the Fefferman-Stein sharp maximal function $f^{\sharp}$ is defined by

[TABLE]

where $f_{P}:=\frac{1}{|P|}\int_{P}{f(z)}dz$ and the supremum is taken over all cubes $P$ containing $x$ ( not necessarily dyadic cubes ). Then a fundamental inequality of Fefferman and Stein [6] says that for $1<p<\infty$ and $1\leq p_{0}\leq p$, if $f\in L^{p_{0}}(\mathbb{R}^{d})$, then we have

[TABLE]

Using this result, it can be proved that for $0<q<p<\infty$,

[TABLE]

where the supremum in the $L^{p}$-norm is taken over all cubes containing $x$. See [19], [21, Proposition 6.1 and 6.2] for more details.

By following the proof of the estimate (3.7) in [6] we can actually replace the maximal functions by dyadic maximal ones. For locally integrable function $f$ we define the dyadic maximal function

[TABLE]

and the dyadic sharp maximal funtion

[TABLE]

where the supremums are taken over all dyadic cubes $P$ containing $x$. Then for $1<p<\infty$, $1\leq p_{0}\leq p$, and $f\in L^{p_{0}}$ we have

[TABLE]

We now provide a characterization of $L^{p}_{A}(\ell^{q})$ for $0<q<p<\infty$, which is the analogue of (3.8).

Lemma 3.3**.**

Let $0<q<p<\infty$. Suppose $f_{k}\in\mathcal{E}(2^{k-2})$ for each $k\in\mathbb{Z}$. Then

[TABLE]

where the supremum is taken over all dyadic cubes containing $x$.

The proof of the above lemma is almost same as that of [19, Lemma 2.3], and for completeness we give a brief proof here.

Proof.

The direction $"\gtrsim"$ is immediate because the right-hand side of (3.10) is bounded by \big{\|}\mathcal{M}_{q}\big{(}\big{\|}\{f_{k}\}_{k\in\mathbb{Z}}\big{\|}_{\ell^{q}}\big{)}\big{\|}_{L^{p}(\mathbb{R}^{d})} and the $L^{p}$-boundedness of $\mathcal{M}_{q}$ yields the desired estimate.

For the opposite direction, using (3.9), the left-hand side of (3.10) is smaller than a constant times

[TABLE]

and the sharp maximal function can be controlled by the sum of

[TABLE]

[TABLE]

The first term clearly gives the expected upper bound and thus it is enough to show that

[TABLE]

If $\ell(P)\leq 2^{-k-1}$ then there exists the unique dyadic cube $Q_{P}\in\mathcal{D}_{k}$ containing $P$. Then, using Taylor’s formula, we can bound \mathfrak{N}^{q}\big{(}\{f_{k}\}_{k\in\mathbb{Z}}\big{)} by

[TABLE]

for some $\psi_{k}\in S(\mathbb{R}^{d})$ with Supp(\widehat{\psi_{k}})\subset\big{\{}\xi\in\mathbb{R}^{d}:|\xi|\lesssim 2^{k}\big{\}}. Moreover, (3.5) implies that that for any $\sigma>0$

[TABLE]

and this yields that

[TABLE]

We observe that for each $Q_{P}\in\mathcal{D}_{k}$, the infimum over $w\in O_{P}$ in the preceding expression is less than

[TABLE]

since $x\in P\subset Q_{P}$. Choosing $\sigma>p,q$, the last expression can be further controlled by

[TABLE]

The proof of this estimate is contained in [19, Lemma 2.2] and we omit it here. This completes the proof of (3.11).

∎

4. Dualization of $L_{A}^{p}(\ell^{q})$ via a discrete function space $\dot{f}_{p}^{0,q}$

Suppose $1<p<\infty$ and $1\leq q<\infty$. Let $1<p^{\prime}<\infty$ and $1<q^{\prime}\leq\infty$ be the Hölder conjugates of $p$ and $q$, respectively. Then it is known in [9] that the dual of $\dot{f}_{p}^{0,q}$ is $\dot{f}_{p^{\prime}}^{0,q^{\prime}}$. Indeed, for $\{b_{Q}\}_{Q\in\mathcal{D}}\in\dot{f}_{p^{\prime}}^{0,q^{\prime}}$

[TABLE]

In this section, we dualize $L^{p}_{A}(\ell^{q})$ through the relationship between the vector-valued space $L_{A}^{p}(\ell^{q})$ and the discrete space $\dot{f}_{p}^{0,q}$ in Lemma 3.1.

For any $\{f_{k}\}_{k\in\mathbb{Z}}\in L^{p}_{A}(\ell^{q})$ and $Q\in\mathcal{D}$ we define the operator $\mathfrak{U}_{Q}$ by

[TABLE]

where we recall that $x_{Q}$ is the lower left corner of $Q\in\mathcal{D}$. Furthermore, for any $\{r_{Q}\}_{Q\in\mathcal{D}}\in\dot{f}_{p}^{0,q}$ and $k\in\mathbb{Z}$ we define the operator $\mathfrak{V}_{k}^{\Psi_{0}}$ by

[TABLE]

Then for each $k\in\mathbb{Z}$

[TABLE]

and it follows from Lemma 3.1 (2) that

[TABLE]

Lemma 4.1**.**

Let $1<p<\infty$ and $1\leq q<\infty$. Suppose $f_{k}\in\mathcal{E}(2^{k-2})$ for $k\in\mathbb{Z}$. Then

[TABLE]

Proof.

By using Lemma 3.1 and (4.1)

[TABLE]

Moreover, for each $k\in\mathbb{Z}$

[TABLE]

where $\widetilde{\Psi_{k}}:=\Psi_{k}(-\cdot)$, and this proves the lemma. ∎

5. Complex Interpolation theorem for multipliers on $L^{p}_{A}(\ell^{q})$

In this section, we obtain an interpolation theorem for multipliers on $L^{p}_{A}(l^{q})$ by using the complex method of Triebel [26, 2.4.9], which is a generalization of the well-known results of Calderón [2] and Calderón and Torchinsky [3].

Let $\Omega:=\{z\in\mathbb{C}:0<Re(z)<1\}$ be a strip in the complex plane $\mathbb{C}$ and $\overline{\Omega}$ denote its closure. We say that the mapping $z\mapsto f^{z}\in S^{\prime}(\mathbb{R}^{n})$ is a $S^{\prime}$-analytic function in $\Omega$ if the following properties are satisfied:

1. (1)

For any $\varphi\in S(\mathbb{R}^{n})$ with compact support, g(x,z):=\big{(}\varphi\widehat{f^{z}}\big{)}(x) is a uniformly continuous and bounded function in $\mathbb{R}^{n}\times\overline{\Omega}$. 2. (2)

For any $\varphi\in S(\mathbb{R}^{n})$ with compact support and any fixed $x\in\mathbb{R}^{n}$, h_{x}(z):=\big{(}\varphi\widehat{f^{z}}\big{)}^{\vee}(x) is an analytic function in $\Omega$.

Let $0<p_{0},p_{1},q_{0},q_{1}<\infty$. Then we define F\big{(}L^{p_{0}}_{A}(\ell^{q_{0}}),L^{p_{1}}_{A}(\ell^{q_{1}})\big{)} to be the collection of all systems $\mathbf{f}^{z}:=\{f_{k}^{z}\}_{k\in\mathbb{Z}}$ such that each $f_{k}^{z}$ is a $S^{\prime}$-analytic function in $\Omega$,

[TABLE]

and

[TABLE]

Moreover, for $\mathbf{f}^{z}\in F(L_{A}^{p_{0}}(\ell^{q_{0}}),L_{A}^{p_{1}}(\ell^{q_{1}}))$,

[TABLE]

For $0<\theta<1$ the intermediate space $(L_{A}^{p_{0}}(\ell^{q_{0}}),L_{A}^{p_{1}}(\ell^{q_{1}}))_{\theta}$ is defined by

[TABLE]

and the (quasi-)norm in the space is

[TABLE]

where the infimum is taken over all admissible system \mathbf{f}^{z}=\{f_{k}^{z}\}_{k\in\mathbb{Z}}\in F\big{(}L_{A}^{p_{0}}(\ell^{q_{0}}),L_{A}^{p_{1}}(\ell^{q_{1}})\big{)} such that $f_{k}=f_{k}^{\theta}$. It is known in [26, 2.4.9] that for any $0<p_{0},p_{1},q_{0},q_{1}<\infty$ and $0<\theta<1$

[TABLE]

when $1/p=(1-\theta)/p_{0}+\theta/p_{1}$ and $1/q=(1-\theta)/q_{0}+\theta/q_{1}$.

Proposition 5.1**.**

Let $0<p_{0},p_{1},q_{0},q_{1}<\infty$, $s_{0},s_{1}\geq 0$, and $1<r_{0},r_{1}<\infty$. Suppose that for any $\{g_{k}\}_{k\in\mathbb{Z}}\in L_{A}^{p_{0}}(\ell^{q_{0}})$ and $\{h_{k}\}_{k\in\mathbb{Z}}\in L_{A}^{p_{1}}(\ell^{q_{1}})$,

[TABLE]

[TABLE]

Then for any $0<\theta<1$ and $p,q,r,s$ satisfying

[TABLE]

[TABLE]

and $\{f_{k}\}_{k\in\mathbb{Z}}\in L_{A}^{p}(\ell^{q})$, we have

[TABLE]

Proof.

Suppose $p,q,r,s$ satisfy (5.4) and (5.5), and $\{f_{k}\}_{k\in\mathbb{Z}}\in L_{A}^{p}(\ell^{q})$. Then, due to (5.1), for any $\epsilon>0$ there exists \mathbf{f}^{z}=\{f^{z}_{k}\}\in\big{(}L_{A}^{p_{0}}(\ell^{q_{0}}),L_{A}^{p_{1}}(\ell^{q_{1}})\big{)}_{\theta} such that $f_{k}=f_{k}^{\theta}$ and

[TABLE]

Now let

[TABLE]

and

[TABLE]

where $Arg(\sigma_{k,s})$ means the argument of $\sigma_{k,s}$. Then we note that $\sigma_{k,s}^{\theta}=m_{k}$ and F_{k}^{z}:=\big{(}\sigma_{k,s}^{z}\big{)}^{\vee}\ast f_{k}^{z} is a $S^{\prime}(\mathbb{R}^{d})$-analytic function in $\Omega$. Moreover,

[TABLE]

From (5.2),

[TABLE]

and similarly, thanks to (5.3),

[TABLE]

Therefore, once we prove

[TABLE]

then we are done by using (5.1) and taking $\epsilon\to 0$.

Let us prove (5.6). By using Hörmander’s multiplier theorem, \big{\|}\sigma_{j,s}^{it}(2^{j}\cdot)\big{\|}_{L_{s_{0}}^{r_{0}}(\mathbb{R}^{d})} is controlled by a constant times

[TABLE]

On the other hand, \big{\|}\sigma_{j,s}^{1+it}(2^{j}\cdot)\big{\|}_{L_{s_{1}}^{r_{1}}(\mathbb{R}^{d})} is less than a constant multiple of

[TABLE]

which finishes the proof of (5.6).

∎

6. The Key Lemma

Suppose that (1.5) holds. Then for $1<r_{0}<r_{1}<\infty$ and $s\geq 0$ we have

[TABLE]

The proof of this will be given in Appendix. Now the principal ingredient in the proof of Theorem 1.1 and 1.2 is the following lemma:

Lemma 6.1**.**

Suppose $0<p\leq\infty$ and $k\in\mathbb{Z}$. Suppose $f_{k}\in\mathcal{E}(2^{k-2})$ and $\{m_{k}\}_{k\in\mathbb{Z}}$ satisfies $(\ref{assumptionm})$. Then for

[TABLE]

we have

[TABLE]

Proof.

This is trivial when $1<p<\infty$, due to Theorem A, and thus we are mainly concerned with the case $0<p\leq 1$ or $p=\infty$, assuming $d/\min{(1,p)}-d/2<s<d/\min{(1,p)}$, which implies that $1<\tau^{(s,p)}<2$. Furthermore, thanks to (6.1), we may also assume that $\tau^{(s,p)}<r<2$.

When $p=1$ or $p=\infty$, it follows immediately from Young’s inequality that

[TABLE]

On the other hand, using a dilation, Hölder’s inequality with $r>1$, and the Hausdorff-Young inequality with $1<r<2$, we obtain

[TABLE]

which ends the argument.

For $0<p<1$, Bernstein’s inequality ( see [26, 1.3.2] ) proves that

[TABLE]

and then using a dilation, Hölder’s inequality with $t:=\frac{1}{1-p+p/r}>1$, and the Hausdorff-Young inequality with $1<r<2$, we have

[TABLE]

since $r^{\prime}=pt^{\prime}$. This completes the proof. ∎

7. Proof of Theorem 1.2

Let $|d/q-d/2|<s<d/\min{(1,q)}$ and $r>\tau^{(s,q)}$. Suppose $\nu\geq\mu$ and $P\in\mathcal{D}_{\nu}$ (i.e. $\ell(P)=2^{-\nu}\leq 2^{-\mu}$). Let $P^{*}=9P$ denote the concentric dilate of $P$ by a factor of $9$. Note that $P^{*}$ is a union of some dyadic cubes near $P$. Then we decompose

[TABLE]

We observe that, due to (1.5),

[TABLE]

and then $\mathcal{U}_{P}$ is estimated by

[TABLE]

due to Lemma 6.1. We now claim that for any $\sigma>0$

[TABLE]

This follows immediately from Young’s inequality if $q\geq 1$. For $0<q<1$, we write

[TABLE]

The integral in the preceding expression can be estimated, using Hölder’s inequality with $1/q>1$, by

[TABLE]

which is clearly smaller than a constant multiple of $2^{-kd}$ for sufficiently large $M>0$. This, together with (3.5), yields that

[TABLE]

and we finally arrive at the desired estimate (7.2). Therefore we have

[TABLE]

Choosing $\sigma>d/q$ and applying the maximal inequality (2.4), we conclude that

[TABLE]

To estimate $\mathcal{V}_{P}$ we note that $r>\tau^{(s,q)}$ implies that $s-d/r>d/\min{(1,q)}-d$ and there exists $\epsilon>0$ so that $s-\epsilon-d/r>d/\min{(1,q)}-d\geq 0$. Then we see that for $x\in P$

[TABLE]

and the integral is less than a constant times

[TABLE]

by applying Hölder’s inequality and the Hausdorff-Young inequality. This proves that

[TABLE]

where the maximal inequality (2.3) and the embedding (2.5) are applied.

By taking the supremum of $\mathcal{U}_{P}$ and $\mathcal{V}_{P}$ over all dyadic cubes $P$ whose side length is less or equal to $2^{-\mu}$, the proof of Theorem 1.2 is complete.

8. Proof of Theorem 1.1

A straightforward application of Lemma 6.1 proves the special case $0<p=q\leq\infty$ and therefore we work only with the case $p\not=q$ and $0<p<\infty$.

8.1. The case $0<p\leq 1$ and $p<q\leq\infty$

Assume $d/p-d/2<s<d/p$. Then $1<\tau^{(s,p)}<2$ and we may assume $\tau^{(s,p)}<r<2$ because of (6.1). According to Lemma 3.1 and Lemma 2.1, if $Supp(\widehat{f_{k}})\subset\{\xi:|\xi|\leq 2^{k-1}\}$ for each $k\in\mathbb{Z}$, then there exist $\{b_{Q}\}_{Q\in\mathcal{D}}\in\dot{f}_{p}^{0,q}$, a sequence of scalars $\{\lambda_{j}\}$, and a sequence of $\infty$-atoms $\{r_{j,Q}\}$ for $\dot{f}_{p}^{0,q}$ such that

[TABLE]

and

[TABLE]

Then by applying $\ell^{p}\hookrightarrow\ell^{1}$ and Minkowski’s inequality with $q/p>1$, we have

[TABLE]

Therefore, it suffices to show that the supremum in the above expression is dominated by a constant times $\mathcal{L}_{s}^{r}[\mathbf{m}]$, which is equivalent to

[TABLE]

where $\{r_{Q}\}$ is an $\infty$-atom for $\dot{f}_{p}^{0,q}$ associated with $Q_{0}\in\mathcal{D}$ and

[TABLE]

Suppose $Q_{0}\in\mathcal{D}_{\nu}$ for some $\nu\in\mathbb{Z}$. Then the condition $Q\subset Q_{0}$ ensures that $A_{Q_{0},k}$ vanishes unless $\nu\leq k$, and thus our actual goal now is to prove

[TABLE]

We observe that for $x\in\mathbb{R}^{d}$

[TABLE]

and for $0<t<\infty$

[TABLE]

by using the argument in (3.1) and the estimate (2.9). Moreover,

[TABLE]

Let $Q_{0}^{*}$ and $Q_{0}^{**}$ denote the concentric dilates of $Q_{0}$ with side length $9\ell(Q_{0})$ and $81\ell(Q_{0})$, respectively. Then we write

[TABLE]

Using Hölder’s inequality and Lemma 6.1 with $\tau^{(s,q)}\leq\tau^{(s,p)}<r$ and

[TABLE]

the first one is controlled by

[TABLE]

and we see that, from (3.1) and (8.2),

[TABLE]

Now using the embedding $L_{s}^{r}(\mathbb{R}^{d})\hookrightarrow L_{s-(d/p-d/\min{(1,q)})}^{r}(\mathbb{R}^{d})$, we obtain

[TABLE]

which finishes the proof of

[TABLE]

To handle the term (8.1) we make use of the embedding $\ell^{p}\hookrightarrow\ell^{q}$ to obtain

[TABLE]

Then, writing

[TABLE]

the proof of (8.1) will be complete once we establish the estimates that for some $\delta>0$

[TABLE]

[TABLE]

It follows from the embedding $\ell^{p}\hookrightarrow\ell^{1}$ that

[TABLE]

We notice that the assumption $r>\tau^{(s,p)}$ is equivalent to $s>d/r+d/p-d$ and therefore there exists $M>d(1-p)$ such that $s>d/r+M/p>d/r+d/p-d$. Recall that $x_{Q}$ denotes the left lower corner of $Q\in\mathcal{D}$ and observe that for $Q\subset Q_{0}^{*}$

[TABLE]

where we utilized Hölder’s inequality if $0<p<1$ and the fact that $|x-x_{Q}|\lesssim|x-y|$ for $x\in(Q_{0}^{**})^{c}$ and $y\in Q\subset Q_{0}^{*}$. Moreover, Hölder’s inequality with $r>1$ and the Hausdorff-young inequality yield that

[TABLE]

Furthermore, (3.5) proves that for $\sigma>d/p$

[TABLE]

Consequently,

[TABLE]

where we applied (2.3) with $\sigma>d/p$ and (8.3) to obtain \big{\|}\mathfrak{M}_{\sigma,2^{k}}A_{Q_{0},k}\big{\|}_{L^{p}(Q_{0})}\lesssim 1. Then (8.5) follows with $\delta=M/p-(d/p-d)>0$.

To verify (8.6) we see that, similar to (7.1), under the assumption (1.5),

[TABLE]

and, it follows from Lemma 6.1 that

[TABLE]

In addition, for sufficiently large $L>0$,

[TABLE]

because $|y-x_{Q}|\gtrsim\ell(Q_{0})$ and

[TABLE]

for $y\in(Q_{0}^{*})^{c}$ and $Q\subset Q_{0}$. Due to (8.2), we have

[TABLE]

and, using Hölder’s inequality (if $p<1$), we obtain that

[TABLE]

for $N>d(1-p)$ and $L-N/p>d$.

Finally, we have

[TABLE]

and this leads to (8.6) with $\delta=L-N/p+d/p-2d>0$.

8.2. **The case $0<q\leq 1$ and $q<p<\infty$ **

Assume $s>d/\min{(1,q)}-d/2$ and $r>\tau^{(s,q)}$. As in the proof of Theorem 1.2, we select $\epsilon>0$ so that $s-\epsilon-d/r>d/\min{(1,q)}-d$.

We first consider the case $p>d/\epsilon$. In view of Lemma 3.3 we can write

[TABLE]

Now let $x\in P\in\mathcal{D}_{\nu}$ for some $\nu\in\mathbb{Z}$ and define $P^{*}=9P$ as before. Using (7.3),

[TABLE]

for $\sigma>d/q$. Then the $L^{p}$ boundedness of $\mathcal{M}_{q}$ and Peetre’s maximal inequality (2.3) yield that

[TABLE]

Furthermore, it follows from (7.4) that

[TABLE]

Then via the $L^{p}$ boundedness of $\mathcal{M}_{q}$, (2.3) with $\epsilon>d/p$, and the embedding $\ell^{q}\hookrightarrow\ell^{\infty}$ we have

[TABLE]

This proves that for $d/\epsilon<p<\infty$

[TABLE]

The general case $q<p<\infty$ follows from the interpolation method in Proposition 5.1 between (8.7) and $L^{q}(\ell^{q})$ estimate with the same values of $s$ and $r$.

8.3. The case $1<p<\infty$ and $1<q\leq\infty$

The proof is based on a suitable use of the complex interpolation method in Proposition 5.1 and the duality property in Lemma 4.1.

Step 1. We claim that for $2<p<\infty$, $d/2-d/p=d/p^{\prime}-d/2<s<d$, and $r>d/s$.

[TABLE]

Choose $\epsilon>0$ and $\widetilde{p}$ such that $s>d/r+\epsilon$ and $\max{(d/\epsilon,p)}<\widetilde{p}<\infty$. Then, by using Lemma 3.3 and the arguments used in obtaining (8.7), we can prove that

[TABLE]

Now (8.8) follows from the interpolation with the $L^{p^{\prime}}(\ell^{p^{\prime}})$ boundedness with the same values of $r$ and $s$ because $p^{\prime}<p<\widetilde{p}$.

Step 2. We prove that for $1<p<2$, $d/p-d/2=d/2-d/p^{\prime}<s<d$, and $r>d/s$,

[TABLE]

Suppose that $\{f_{k}\}_{k\in\mathbb{Z}}\in L_{A}^{p}(\ell^{p^{\prime}})$. By using Lemma 4.1, the left-hand side of (8.9) can be dualized and estimated by

[TABLE]

which can be also written as

[TABLE]

This is clearly majorized, using Hölder’s inequality, by

[TABLE]

Moreover, the result in Step 1 and (4.2) yield that the $L^{p^{\prime}}(\ell^{p})$-norm in the above expression is smaller than a constant times

[TABLE]

which proves (8.9).

Step 3. Let $1<p<\infty$ and $q$ is between $p$ and $p^{\prime}$ so that $|d/p-d/2|>|d/q-d/2|$. Suppose $|d/p-d/2|<s<d$ and $r>d/s$. We interpolate two cases $(p,p^{\prime})$ and $(p,p)$ by using Proposition 5.1 with the same values of $s$ and $r$. Then we establish the estimate

[TABLE]

Step 4. Let $1<q<\infty$ and $p$ is between $q$ and $q^{\prime}$ so that $|d/q-d/2|>|d/p-d/2|$. Suppose $|d/q-d/2|<s<d$ and $r>d/s$. We interpolate two cases $(q^{\prime},q)$ and $(q,q)$ by using Proposition 5.1 with the same values of $s$ and $r$. Then we have the estimate

[TABLE]

Step 5. Let $1<p<\infty$ and $q=\infty$. Suppose $d/2<s<d$ and $r>d/s$. An argument similar to that used in Step 2, with Lemma 4.1 and the result for $1<p<\infty$ and $q=1$, leads to the desired estimate. We skip the details to avoid unnecessary repetition.

9. Proof of Theorem 1.4

We now describe the proof of Theorem 1.4, using the ideas in [4, 12]. Suppose $0<p<\infty$ or $p=q=\infty$.

9.1. Necessary conditions for vector-valued operator inequalities

We investigate necessary conditions for the inequality that for $K\in\mathcal{E}(1)$,

[TABLE]

for some $\mathcal{A}>0$.

An immediate consequence is that

[TABLE]

which follows from setting $f_{0}=4^{d}\Psi_{0}(4\cdot)$ and $f_{k}=0$ for $k\not=0$ so that

[TABLE]

Moreover, it is known in [4] that if (9.1) holds for $0<q\leq p<\infty$, then

[TABLE]

Now we consider the case $1<p,q<\infty$. Using the dualization argument in Lemma 4.1, which was used to obtain (8.9), the $L^{p}(\ell^{q})$ boundedness also implies that

[TABLE]

Therefore it is clear from (9.2) that

[TABLE]

and if $1<p\leq q<\infty$ ( that is, $1<q^{\prime}\leq p^{\prime}<\infty$ ), then we have

[TABLE]

from the estimate (9.3).

We note that if $K\in\mathcal{E}(1)$, then Bernstein’s inequality shows that

[TABLE]

Therefore, we conclude that

Lemma 9.1**.**

Let $0<p<\infty$ and $0<q\leq\infty$. Suppose that $K\in\mathcal{E}(1)$. If (9.1) holds, then

[TABLE]

where we adhere to the standard convention that $p^{\prime}=\infty$ for $p\leq 1$ and $q^{\prime}=\infty$ for $q\leq 1$.

On the other hand, when $p\geq 1$, (9.1) implies that the convolution operator with $K$ is bounded in $L^{p}(\mathbb{R}^{d})$. Indeed, for any $f\in L^{p}(\mathbb{R}^{d})$ let

[TABLE]

Then using the identity $K=4^{d}\Psi_{0}(4\cdot)\ast K$, we have

[TABLE]

where the last inequality follows from Young’s inequality with $p\geq 1$. Hence it follows that

[TABLE]

By additionally assuming that $K\in\mathcal{E}(1)$ is a nonnegative function, we obtain that

[TABLE]

and this, together with (9.4), yields the following lemma:

Lemma 9.2**.**

Let $0<p<\infty$ and $0<q\leq\infty$. Suppose that $K\in\mathcal{E}(1)$ is a nonnegative function on $\mathbb{R}^{d}$. If (9.1) holds, then

[TABLE]

9.2. Construction of examples

Note that $s<d/\min{(1,p,q)}$ implies $\min{(1,p,q)}<\tau^{(s,p,q)}$. Choosing

[TABLE]

we define

[TABLE]

Then it is proved in [12] that

[TABLE]

where $d-t+s=s-d/\min{(1,p,q)}+d>0$.

Let $\eta\in S(\mathbb{R}^{d})$ have the properties that $\eta\geq 0$, $\eta(x)\geq c>0$ on $\{x\in\mathbb{R}^{d}:|x|\leq 1/100\}$ for some $c>0$, and $Supp(\widehat{\eta})\subset\{\xi\in\mathbb{R}^{d}:|\xi|\leq 1/10\}$. We define

[TABLE]

and

[TABLE]

We first observe that

[TABLE]

and this yields that

[TABLE]

where the Kato-Ponce inequality is applied. Then using (9.2), we obtain that

[TABLE]

and using change of variables, the second term is estimated by a constant times

[TABLE]

because $\tau^{(s,p,q)}(d-t+s)=d$ and $\tau^{(s,p,q)}\gamma/2>1$ with the choice of $t$ and $\gamma$ in (9.5). Finally, we have

[TABLE]

Now we suppose (1.4) holds with $m_{k}=m_{k}^{(t,\gamma)}$ and $A=2^{-2}$, which is equivalent to (9.1) with $K=K^{(t,\gamma)}$ and $\mathcal{A}=\mathcal{L}_{s}^{\tau^{(s,p,q)}}[\mathbf{m}]$. Then it follows from Lemma 9.2 that

[TABLE]

since $K^{(t,\gamma)}$ is a nonnegative function. However,

[TABLE]

where the inequality follows from the fact that $\mathcal{H}^{(t,\gamma)},\eta\geq 0$ and $\mathcal{H}^{(t,\gamma)}(x-y)\geq\mathcal{H}^{(t,\gamma)}(x)\mathcal{H}^{(t,\gamma)}(y)$. This yields that

[TABLE]

since $\gamma\min{(1,p,q)/2}<1$, which contradicts (9.7).

Appendix A Proof of (6.1)

(6.1) is a consequence of the following lemma:

Lemma A.1**.**

Let $1<r_{0}<r_{1}<\infty$ and $s\geq 0$. Suppose that $f\in L_{s}^{r_{1}}(\mathbb{R}^{d})$ is supported in $\{x\in\mathbb{R}^{d}:|x|\leq B\}$ for some $B>0$. Then $f\in L_{s}^{r_{0}}(\mathbb{R}^{d})$ and indeed,

[TABLE]

Proof.

Let $\Gamma\in S(\mathbb{R}^{d})$ satisfy $Supp(\Gamma)\subset\{x\in\mathbb{R}^{d}:|x|\leq 2B\}$ and $\Gamma(x)=1$ for $|x|\leq B$. Define the multiplication operator $T$ by

[TABLE]

Using Hölder’s inequality and the Kato-Ponce inequality [15], we obtain that for each $n\in\mathbb{N}_{0}$,

[TABLE]

Then we interpolate these estimates to extend to

[TABLE]

for all $s\geq 0$.

Now suppose $g\in S(\mathbb{R}^{d})$ has compact support in $\{x\in\mathbb{R}^{d}:|x|\leq B\}$ so that $g=Tg$. Then (A.1) implies that

[TABLE]

from which the desired result follows, using the density of $S(\mathbb{R}^{d})$ in the two Banach spaces $L_{s}^{r_{0}}(\mathbb{R}^{d})$ and $L_{s}^{r_{1}}(\mathbb{R}^{d})$.

∎

Acknowledgement

Part of this research was carried out during my stay at the University of Missouri-Columbia. I would like to thank Professor L. Grafakos for his invitation, hospitality, and very useful discussions during the stay. I also would like to express gratitude to the anonymous referees for the careful reading and very useful comments.