L^p-Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators

Abstract

In this paper, I explore a specific class of bi-parameter pseudo-differential operators characterized by symbols $\sigma (x_1,x_2,{\xi }_1,{\xi }_2)$ falling within the product-type Hörmander class ${\mathbf{S}}_{\rho ,\delta }^m$. This classification imposes constraints on the behavior of partial derivatives of $\sigma$ with respect to both spatial and frequency variables. Specifically, I demonstrate that for each multi-index $\alpha ,\beta$, the inequality $|{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }\sigma (x_1,x_2,{\xi }_1,{\xi }_2)|\le {\mathcal{C}}_{\alpha ,\beta }{(1+|\xi \left|\right)}^m{\prod }_{i=1}^2(1+|{\xi }_i{\left|\right)}^{-\rho |{\alpha }_i|+\delta |{\beta }_i|}$ is satisfied. My investigation culminates in a rigorous analysis of the $L^p$-boundedness of such pseudo-differential operators, thereby extending the seminal findings of C. Fefferman from 1973 concerning pseudo-differential operators within the Hörmander class.

1. Introduction

Consider a Schwartz function denoted by f. We define a pseudo-differential operator $T_{\sigma }$ as follows:

$$T_{\sigma }f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi \mathrm{i}x·\xi }\widehat{f}\left(\xi \right)\sigma (x,\xi )d\xi ,$$

(1)

where $\widehat{f}\left(\xi \right)$ represents the Fourier transform of f, and $\sigma (x,\xi )\in C^{\infty }({\mathbb{R}}^n\times {\mathbb{R}}^n)$ is referred to as the symbol. Of primary interest is the symbol class denoted $S_{\rho ,\delta }^m$, commonly known as the Hörmander class. A symbol $\sigma (x,\xi )$ belongs to $S_{\rho ,\delta }^m$ if it satisfies the following differential inequalities:

$$|{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }\sigma (x,\xi )|\le {\mathcal{C}}_{\alpha ,\beta }{(1+|\xi \left|\right)}^{m-\rho \left|\alpha \right|+\delta \left|\beta \right|},$$

(2)

for all multi-indices $\alpha ,\beta$.

The $L^p$-boundedness of pseudo-differential operators, defined as in (1) and (2), has been a topic of extensive investigation in recent decades. Notably, the seminal works of Calderón and Vaillancourt [1,2] established the $L^2$-boundedness of $T_{\sigma }$ for symbols $\sigma \in S_{0,0}^0$. Furthermore, Calderón and Vaillancourt showed that $T_{\sigma }$ remains bounded on $L^2$ when the symbol $\sigma$ belongs to $S_{\rho ,\rho }^0$, $0<\rho <1$, a class known as the exotic symbol class. However, the boundedness results are not universal. For instance, consider the symbol $\sigma \left(\xi \right)$ given by the Fourier transform of the Riemann singularity distribution $R\left(x\right)=e^{\frac{i}{\left|x\right|}}{\left|x\right|}^{-\frac{3}{2}}$, then $T_{\sigma }$ is not bounded on $L^p$ for $p\ne 2$. More recently, Wang [3] investigated a subclass of the exotic symbol class and demonstrated that pseudo-differential operators belonging to this subclass are bounded on $L^p$ for $0<p<1$.

The primary objective of this paper is to extend the following theorem originally established by C. Fefferman in 1973 [4].

Theorem 1 (Fefferman).

Let $\sigma (x,\xi )\in S_{1-a,\delta }^{-\beta }$ with $0\le \delta <1-a<1$ and $\beta <na/2$. Then, $T_{\sigma }$ is bounded on $L^p$ for

$$\left|\frac{1}{p}-\frac{1}{2}\right|\le \gamma =\frac{\beta }{n}\left[\frac{n/2+\lambda }{\beta +\lambda }\right],\lambda =\frac{na/2-\beta }{1-a}.$$

To be more specific, we turn to the multi-parameter setting. Let $\sigma (x_1,x_2,{\xi }_1,{\xi }_2)\in C^{\infty }({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}\times {\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$, where $n=n_1+n_2$. We say $\sigma \in {\mathbf{S}}_{\rho ,\delta }^m$ if it satisfies the estimates

$$|{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }\sigma (x,\xi )|\le {\mathcal{C}}_{\alpha ,\beta }{(1+|\xi \left|\right)}^m\prod _{i=1}^2{\left(\frac{1}{1+|{\xi }_i|}\right)}^{\rho |{\alpha }_i|-\delta |{\beta }_i|}$$

(3)

for all multi-indices $\alpha ,\beta$. Moreover, we define the bi-parameter Hörmander class $B{S}_{\rho ,\delta }^{m_1,m_2}$, we say $\sigma \in B{S}_{\rho ,\delta }^{m_1,m_2}$ if it satisfies the estimates:

$$|{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }\sigma (x,\xi )|\le {\mathcal{C}}_{\alpha ,\beta }\prod _{i=1}^2{\left(\frac{1}{1+|{\xi }_i|}\right)}^{\rho |{\alpha }_i|-\delta |{\beta }_i|-m_i}.$$

(4)

Note that if $m=m_1+m_2$ and $m_1,m_2\le 0$, then we have

$${\mathbf{S}}_{\rho ,\delta }^m\subset B{S}_{\rho ,\delta }^{m_1,m_2}.$$

The classical theory of harmonic analysis may be described as around the Hardy–Littlewood maximal operator and its relationship with certain singular integral operators which commute with the classical one-parameter family dilations $\delta :x\to \delta x=(\delta x_1,\cdots ,\delta x_d),\delta >0$. The multi-parameter theory, sometimes called product theory corresponds to a range of questions which are concerned with issues of harmonic analysis that are invariant with respect to a family of dilations $\delta :x\to \delta x=({\delta }_1{x}_1,\cdots ,{\delta }_d{x}_d),{\delta }_i>0,i=1,\cdots ,d$. Such multi-parameter symbol classes, associated with singular integral operators, pseudo-differential operators, and Fourier integral operators, have been the subject of extensive study by various authors. Notable contributions include works by Müller, Ricci, and Stein [5], Yamazaki [6], Wang [7], Chen, Ding, and Lu [8], Huang and Chen [9,10], Hong, Zhang, and Lu [11,12,13,14], Muscalu, Pipher, Tao, and Thiele [15,16], among others.

Main Results

Main Theorem:

(a)

Let $\sigma (x_1,x_2,{\xi }_1,{\xi }_2)\in {\mathbf{S}}_{1-a,\delta }^{-\beta }$ with $0\le \delta <1-a<1$, $\beta <\frac{na}{2}$, and $n_1,n_2\ge 2$. Then, $T_{\sigma }$ is bounded on $L^p$ for

$$\left|\frac{1}{p}-\frac{1}{2}\right|\le \gamma =\frac{\beta }{n}\left[\frac{n/2+\lambda }{\beta +\lambda }\right],\lambda =\frac{na/2-\beta }{1-a}.$$

(5)

(b)

Let $\sigma (x_1,x_2,{\xi }_1,{\xi }_2)\in {\mathbf{S}}_{1-a,\delta }^{-na/2}$. The critical $L^p$ space is $L^1$, while $T_{\sigma }$ is unbounded on $L^1$, it is bounded on the Hardy space $H^1$.

Remark 1.

C. Fefferman originally proved the above theorem with symbols belonging to the classical Hörmander class $S_{\rho ,\delta }^m$. Thus, the sharpness of the theorem follows from Fefferman’s theorem, as $S_{\rho ,\delta }^m\subset {\mathbf{S}}_{\rho ,\delta }^m$.

The results in [17] lead to the following propositions.

Proposition 1.

Let $\sigma (x,\xi )\in {\mathbf{S}}_{1-a,\delta }^0$ for $0\le \delta <1-a<1$. Then, $T_{\sigma }$ is a bounded operator from $L^2$ to $L^2$.

Proposition 2.

Let $\sigma (x,\xi )\in {\mathbf{S}}_{1-a,\delta }^{-na/2}$ for $0\le \delta <1-a<1$. Then, $T_{\sigma }$ is a bounded operator from $L^{\infty }$ to $BMO$.

Here, $BMO\left({\mathbb{R}}^n\right)$ denotes the class of functions of bounded mean oscillation defined by F. John and L. Nirenberg in [18]. A locally integrable function f on ${\mathbb{R}}^n$ belongs to $BMO$ if

$${\left|\right|f\left|\right|}_{BMO}=\underset{Q}{sup}\frac{1}{\left|Q\right|}{\int }_Q|f\left(x\right)-f_Q|dx<\infty ,$$

where Q is an arbitrary cube in ${\mathbb{R}}^n$ and $f_Q=\frac{1}{\left|Q\right|}{\int }_{Q}f\left(x\right)dx$.

We will prove the $L^2$-boundedness of $T_{\sigma }$ with $\sigma$ of order 0 in Section 2, and $T_{\sigma }$ is bounded from $L^{\infty }$ to $BMO$ with $\sigma$ of order $-na/2$ in Section 3. We primarily follow the proofs in [19] and [4] to establish Propositions 1 and 2, respectively. However, a single Littlewood–Paley decomposition in the $\xi$-space is insufficient; we require a further cone decomposition to fully utilize the inequalities in (4).

2. ${\mathit{L}}^{\mathbf{2}}$-Boundedness of ${\mathit{T}}_{\mathit{\sigma }}$ of Order 0

Since ${\mathbf{S}}_{1-a,\delta }^0\subset B{S}_{1-a,\delta }^{0,0}\subset B{S}_{\delta ,\delta }^{0,0}$, it suffices to prove

Lemma 1.

Suppose that $\sigma (x,\xi )\in B{S}_{\rho ,\rho }^{0,0}$, where $0\le \rho <1$. Then the operator $T_{\sigma }$ defined in (1) is bounded from $L^2\left({\mathbb{R}}^n\right)$ to iteself.

Proof. 

First we use the Cotlar–Stein lemma to show that the lemma is true in the case $\rho =0$.

By Plancherel’s theorem, we observe that it suffices to establish the $L^2$-boundedness of the operator S defined by

$$Sf\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }f\left(\xi \right)\sigma (x,\xi )d\xi .$$

Notice that, in view of the assumption of $\sigma$, the role of x and $\xi$ in the above symbol class are perfectly symmetric. We choose a smooth non-negative function ${\varphi }_i$ that is supported in the unit cube

$$Q_1^i=\{x^i:|x_j^i|\le 1,j=1,2,\cdots ,n_i\},i=1,2$$

and for which

$$\sum _{k^i\in {\mathbb{Z}}^{n_i}}{\varphi }^i(x^i-k^i)=1.$$

To construct such a ${\varphi }^i$, simply fix any smooth, non-negative ${\varphi }_0^i$ that equals 1 on the cube $Q_{1/2}^i=1/2·Q_1^i$ and is supported in $Q_1^i$. Noting that ${\sum }_{k^i\in .{\mathbb{Z}}^{n_i}}{\varphi }_0^i(x^i-k^i)$ converges and is bounded away from 0 for all $x\in {\mathbb{R}}^n$, we take

$${\varphi }^i\left(x^i\right)={\varphi }_0^i\left(x^i\right){\left[\sum _{l^i\in {\mathbb{Z}}^{n_i}}{\varphi }_0^i(x^i-k^i)\right]}^{-1}.$$

Next, let ${\overrightarrow{k}}^i=(k^i,k^{\prime i})\in {\mathbb{Z}}^{2n_i}={\mathbb{Z}}^{n_i}\times {\mathbb{Z}}^{n_i}$ denote an element of ${\mathbb{Z}}^{2n_i}$, and similarly write ${\overrightarrow{j}}^i=(j^i,j^{\prime i})$ for another element of ${\mathbb{Z}}^{2n_i}$. We set $\overrightarrow{k}=({\overrightarrow{k}}^1,{\overrightarrow{k}}^2)$ and

$${\sigma }_{\overrightarrow{k}}(x,\xi )=\left[\prod _{i=1}^2{\varphi }^i(x^i-k^i)\right]\sigma (x,\xi )\left[\prod _{i=1}^2{\varphi }^i({\xi }^i-k^{\prime i})\right]=\varphi (x-k)\sigma (x,\xi )\varphi (\xi -k^{\prime })$$

and

$$S_{\overrightarrow{k}}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }f\left(\xi \right){\sigma }_{\overrightarrow{k}}(x,\xi )d\xi .$$

Therefore, we have the decomposition

$$Sf\left(x\right)=\sum _{\overrightarrow{k}\in {\mathbb{Z}}^{2n}}{S}_{\overrightarrow{k}}f\left(x\right).$$

The main point is then to verify the almost-orthogonality estimates as follows:

$$\left|\right|S_{\overrightarrow{j}}^{*}{S}_{\overrightarrow{k}}\left|\right|\le A\prod _{i=1}^2(1+|{\overrightarrow{j}}^i-{\overrightarrow{k}}^i{\left|\right)}^{-2N_i}$$

(6)

and

$$\left|\right|S_{\overrightarrow{k}}{S}_{\overrightarrow{j}}^{*}\left|\right|\le A\prod _{i=1}^2(1+|{\overrightarrow{j}}^i-{\overrightarrow{k}}^i{\left|\right)}^{-2N_i}$$

(7)

Here, $\left|\right|·\left|\right|$ denotes the $L^2$ operator norm, $N_i$ is sufficiently large, and the bound A is independent of $\overrightarrow{k},\overrightarrow{j}$.

Now, we can write

$$S_{\overrightarrow{j}}^{*}{S}_{\overrightarrow{k}}f\left(\xi \right)={\int }_{{\mathbb{R}}^n}f\left(\eta \right)K_{\overrightarrow{j},\overrightarrow{k}}(\xi ,\eta )d\eta ,$$

where

$$K_{\overrightarrow{j},\overrightarrow{k}}(\xi ,\eta )={\int }_{{\mathbb{R}}^n}{\overline{\sigma }}_{\overrightarrow{j}}(x,\xi ){\sigma }_{\overrightarrow{k}}(x,\xi )e^{2\pi ix·(\eta -\xi )}dx.$$

In the above integral, we integrate by parts, using the identities

$$\prod _{i=1}^2{(I-{\Delta }_{x^i})}^{N_i}{e}^{2\pi ix·(\eta -\xi )}=\prod _{i=1}^2(1+4{\pi }^2|{\eta }^i-{\xi }^i{{|}^2)}^{N_i}{e}^{2\pi ix·(\eta -\xi )}.$$

We also note that ${\sigma }_{\overrightarrow{k}}(x,\xi )$ and ${\sigma }_{\overrightarrow{j}}(x,\eta )$ are given by

$${\sigma }_{\overrightarrow{k}}(x,\xi )=\varphi (x-k)\sigma (x,\xi )\varphi (\xi -k^{\prime }),{\sigma }_{\overrightarrow{j}}(x,\xi )=\varphi (x-j)\sigma (x,\eta )\varphi (\xi -j^{\prime })$$

respectively, and so have disjoint x-support unless ${\overrightarrow{j}}^i-{\overrightarrow{k}}^i\in Q_1^i$. These observations lead to the bounds

$$\begin{array}{c}{\displaystyle |K_{\overrightarrow{j},\overrightarrow{k}}(\xi ,\eta )|\le \prod _{i=1}^d\frac{A_{N_i}{\varphi }^i({\xi }^i-{\overrightarrow{j}}^{\prime i}){\varphi }^i({\eta }^i-{\overrightarrow{k}}^{\prime i})}{(1+|{\xi }_i-{\eta }_i{\left|\right)}^{2N_i}},\mathrm{if}{\overrightarrow{j}}^i-{\overrightarrow{k}}^i\in Q_1^i,i=1,2,}\hfill \\ \\ {\displaystyle |K_{\overrightarrow{j},\overrightarrow{k}}(\xi ,\eta )|=0,\mathrm{otherwise}.}\hfill \end{array}$$

Therefore, we have

$$\underset{\xi }{sup}{\int }_{{\mathbb{R}}^n}|K_{\overrightarrow{j},\overrightarrow{k}}(\xi ,\eta )|d\eta <A\prod _{i=1}^2(1+|{\overrightarrow{j}}^i-{\overrightarrow{k}}^i{\left|\right)}^{-2N_i},$$

and

$$\underset{\eta }{sup}{\int }_{{\mathbb{R}}^n}|K_{\overrightarrow{j},\overrightarrow{k}}(\xi ,\eta )|d\xi <A\prod _{i=1}^2(1+|{\overrightarrow{j}}^i-{\overrightarrow{k}}^i{\left|\right)}^{-2N_i},$$

which implies our desired estimate (6). Moreover, as we have noted, the situation is symmetric in x and $\xi$, the same proof also shows the estimate (7). Now, it is only a matter of applying the Cotlar–Stein lemma; setting $N_i$ sufficiently large, we see

$$\sum _{\overrightarrow{k}\in {\mathbb{Z}}^n}\prod _{i=1}^2(1+|{\overrightarrow{k}}^i{\left|\right)}^{-2N_i}<\infty ,$$

and as a result, $S={\sum }_{\overrightarrow{k}}{S}_{\overrightarrow{k}}$ is bounded from $L^2\left({\mathbb{R}}^n\right)$ to itself.

Now, we prove our Lemma 1.

We start by defining a $C^{\infty }$ function $\phi$ with compact support on $\mathbb{R}$, satisfying $\phi \left(t\right)=1$ for $\left|t\right|\le 1$ and $\phi \left(t\right)=0$ for $\left|t\right|\ge 2$. For each $i=1,2$, we set ${\varphi }_0\left({\xi }_i\right)=\phi \left(\right|{\xi }_i\left|\right)$ and

$${\varphi }_{j_i}\left({\xi }_i\right)=\phi (2^{-j}|{\xi }_i\left|\right)-\phi (2^{-j+1}|{\xi }_i\left|\right),j_i\in \mathbb{Z},j_i>0,i=1,2,$$

and

$${\varphi }_j\left(\xi \right)=\prod _{i=1}^2{\varphi }_{j_i}\left({\xi }_i\right),j\in {\mathbb{Z}}^2.$$

Then, we define the partial operators

$$T_{j}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }\widehat{f}\left(\xi \right){\sigma }_j(x,\xi )d\xi ,{\sigma }_j(x,\xi )=\sigma (x,\xi ){\varphi }_j\left(\xi \right).$$

Let $\widehat{S_{j}f}\left(\xi \right)={\varphi }_j\left(\xi \right)\widehat{f}\left(\xi \right)$, and we have the decomposition of T

$$T=\sum _{j\ge 0}{T}_j=\sum _{j\ge 0}T{S}_j,\sum _{j\ge 0}=\prod _{i=1}^2\sum _{j_i\ge 0}.$$

(8)

It will be convenient to break the sum (8) into two parts

$$T=\sum _{jeven}{T}_j+\sum _{jodd},\sum _{jeven}=\prod _{i=1}^2\sum _{j_i\ge 0even},$$

so that the summands in each parts have disjoint $\xi$-support; it suffices to prove the boundedness of each sum separately.

Let us consider the sum taken over the odd j. Note that

$$T_j{T}_k^{*}=T{S}_j{\left(T{S}_k\right)}^{*}=T{S}_j{S}_k^{*}T=0,j\ne k,$$

because the supports of the multipliers corresponding to $S_j$ and $S_k$ are disjoint. Next, we estimate $T_j^{*}{T}_k$, and we write

$$T_j^{*}{T}_{k}f\left(x\right)={\int }_{{\mathbb{R}}^n}K(x,y)f\left(y\right)dy,$$

with

$$K(x,y)={\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n\times {\mathbb{R}}^n}{\overline{\sigma }}_k(z,\eta ){\sigma }_j(z,\xi )e^{2\pi i[\xi ·(z-y)-\eta ·(z-x\left)\right]}dzd\eta d\xi .$$

First, one carries integration by parts with respect to z-variable by writing

$$\prod _{i=1}^2\frac{{(I-{\Delta }_{z_i})}^{N_i}}{(1+4{\pi }^2|{\xi }_i-{\eta }_i{{|}^2)}^{N_i}}{e}^{2\pi (\xi -\eta )·z}=e^{2\pi i(\xi -\eta )·z}.$$

Next, one performs a similar process on the $\eta$-variable, beginning with

$$\prod _{i=1}^2\frac{{(I-{\Delta }_{{\eta }_i})}^{N_i}}{(1+4{\pi }^2|x_i-z_i{{|}^2)}^{N_i}}{e}^{2\pi i\eta ·(x-z)}=e^{2\pi i\eta ·(x-z)}.$$

Finally, an analogous step is carried our for $\xi$-variable. If we take into account the differential inequalities for the symbols ${\sigma }_j$, and the restrictions on their supports, we see that each order of differentiation in the $z_i$-variable gives us a factor of order

$$(1+|{\xi }_i-{\eta }_i{\left|\right)}^{-1}\sim 2^{-max\{k_i,j_i\}}$$

for every factor of order

$$(1+|{\xi }_i|+|{\eta }_i{\left|\right)}^{\rho }\sim 2^{\rho max\{k-i,j_i\}}$$

that may lose. As a result, the kernel K is dominated by a constant multiple of

$$\prod _{i=1}^2{2}^{max\{k_i,j_i\}(2\rho N_i-2N_i+2n_i)}{\int }_{{\mathbb{R}}^{n_i}}{Q}_i(x_i-z_i)Q_i(z_i-y_i)d{z}_i.$$

Now, if we let $K_i(x_i,y_i)={\int }_{{\mathbb{R}}^{n_i}}{Q}_i(x_i-z_i)Q_i(z_i-y_i)d{z}_i$, then

$${\int }_{{\mathbb{R}}^{n_i}}{K}_i(x_i,y_i)d{y}_i={\int }_{{\mathbb{R}}^{n_i}}{K}_i(x_i,y_i)d{x}_i={\left({\int }_{{\mathbb{R}}^{n_i}}(1+|z_i{\left|\right)}^{-2N_i}\right)}^2<\infty ,$$

if $2N_i>n_i$. Thus, we obtain

$$\left|\right|T_j^{*}{T}_k\left|\right|\le A\prod _{i=1}^2{2}^{max\{k_i,j_i\}(2\rho N_i-2N_i+2n_i)},j\ne k,$$

which implies that

$$\left|\right|T_j^{*}{T}_k\left|\right|\le \prod _{i=1}^2{\gamma }_i\left(j_i\right){\gamma }_i\left(k_i\right),j\ne k,$$

with ${\gamma }_i\left(j_i\right)=A·2^{-ϵj_i},ϵ>0$, if we choose $N_i$ so large that $N_i>n_i(1-\rho )$.

In order to apply the Cotlar–Stein lemma, we need to show that the partial operators $T_j$ are uniformly bounded in the norm. To prove this, we set

$${\tilde{\sigma }}_j(x,\xi )={\sigma }_j(2^{-j\rho }x,2^{j\rho }\xi ),2^{-j\rho }x=(2^{j_1\rho }{x}_1,2^{j_2\rho }{x}_2),0\le \rho <1.$$

Thus, ${\tilde{\sigma }}_j(x,\xi )\in S_{\rho ,\rho }^{0,0}$ for $m_i=0,{\rho }_i=0$ for each $i=1,2,\cdots ,d$ uniformly in j. Therefore, the operator

$${\tilde{T}}_{j}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }\widehat{f}\left(\xi \right){\tilde{\sigma }}_j(x,\xi )d\xi$$

is bounded on $L^2\left({\mathbb{R}}^n\right)$. Next, define the scaling operators given by

$${\Lambda }_{j}f\left(x\right)=f\left(2^{j\rho }x\right)=f(2^{j_1\rho }{x}_1,2^{j_2\rho }{x}_2),$$

then, as is easily verified,

$$T_j={\Lambda }_j{\tilde{T}}_j{\Lambda }_j^{-1}.$$

Now, $\left|\right|{\Lambda }_j{f\left|\right|}_{L^2}={\prod }_{i=1}^2{2}^{n_i{j}_i\rho /2}{\left|\right|f\left|\right|}_{L^2}$ and $\left|\right|{\Lambda }_j^{-1}{f\left|\right|}_{L^2}={\prod }_{i=1}^d{2}^{-n_i{j}_i\rho /2}{\left|\right|f\left|\right|}_{L^2}$; so together with the $L^2$-boundedness of ${\tilde{T}}_j$, we have

$$\left|\right|T_j\left|\right|\le A,\mathrm{uniformly}\mathrm{in}j.$$

We may therefore conclude that ${\sum }_{jodd}{T}_j$ is bounded from $L^2\left({\mathbb{R}}^n\right)$ to itself, the sum ${\sum }_{jeven}$ is treated similarly, and our Lemma 1 is proved. □

3. ${\mathit{L}}^{\mathit{p}}$-Boundedness of ${\mathit{T}}_{\mathit{\sigma }}$

We make a further decomposition, let $\xi =({\xi }_1,{\xi }_2)\in {\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}$. Recalling the definition of $\phi$ in the above section, define

$${\delta }_{\ell }\left(\xi \right)=\phi \left(2^{-\ell }\frac{|{\xi }_2|}{|{\xi }_1|}\right)-\phi \left(2^{-\ell +1}\frac{|{\xi }_2|}{|{\xi }_1|}\right),\ell \in \mathbb{Z}.$$

(9)

Note that ${\delta }_{\ell }\left(\xi \right)$ has a support in the cone region,

$${\Lambda }_{\ell }=\{({\xi }_1,{\xi }_2):2^{\ell -1}\le \frac{|{\xi }_2|}{|{\xi }_1|}\le 2^{\ell +1}\}.$$

(10)

By symmetry, we can always assume ℓ is a non-negative integer. Now for fixed j, we make a cone decomposition in the frequency space, define partial operators

$$T_{\ell j}f\left(x\right)=T_{{\sigma }_{\ell j}}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }\widehat{f}\left(\xi \right){\sigma }_{\ell j}(x,\xi )d\xi ,{\sigma }_{\ell j}(x,\xi )=\sigma (x,\xi ){\varphi }_j\left(\xi \right){\delta }_{\ell }\left(\xi \right).$$

(11)

Furthermore, we define

$$T_j^{♭}f\left(x\right)=T_{{\sigma }_j^{♭}}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }\widehat{f}\left(\xi \right){\sigma }_j^{♭}(x,\xi )d\xi ,{\sigma }_j^{♭}(x,\xi )=\sum _{\ell =j}^{\infty }\sigma (x,\xi ){\varphi }_j\left(\xi \right){\delta }_{\ell }\left(\xi \right),$$

(12)

and

$$T_j^{♯}f\left(x\right)=T_{{\sigma }_j^{♯}}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }\widehat{f}\left(\xi \right){\sigma }_j^{♯}(x,\xi )d\xi ,{\sigma }_j^{♯}(x,\xi )=\sum _{\ell =0}^j\sigma (x,\xi ){\varphi }_j\left(\xi \right){\delta }_{\ell }\left(\xi \right).$$

(13)

3.1. A Key Lemma

Let a symbol $\sigma (x,\xi )\in {\mathbf{S}}_{\rho ,\delta }^m$, then we define its norm as

$${∥\sigma ∥}_{\mathbf{S}}=\underset{\left|\alpha \right|\le k,\left|\beta \right|\le N}{sup}|{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }\sigma (x,\xi )|(1+{|\xi |)}^{-m}\prod _{i=1}^2{\left(1+|{\xi }_i|\right)}^{\rho |{\alpha }_i|-\delta |{\beta }_i|},k,N>n/2.$$

(14)

Let $r>0$ be a real number, recall the definitions of ${\varphi }_j\left(\xi \right)$ and ${\delta }_{\ell }\left(\xi \right)$, define the partial operators

$$T_{\ell j}^{r}f\left(x\right)=T_{{\sigma }_{\ell j}^r}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }\widehat{f}\left(\xi \right){\sigma }_{\ell j}^r(x,\xi )d\xi ,{\sigma }_{\ell j}^r(x,\xi )=\sigma (x,\xi ){\delta }_{\ell }\left(\xi \right){\varphi }_j\left(r\xi \right),$$

(15)

and

$$T_j^{r}f\left(x\right)=T_{{\sigma }_j^r}f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\xi }\widehat{f}\left(\xi \right){\sigma }_j^r(x,\xi )d\xi ,{\sigma }_j^r(x,\xi )=\sum _{\ell \ge 0}{\sigma }_{\ell j}^r(x,\xi ).$$

(16)

Lemma 2.

Let the symbol $\sigma (x,\xi )$ be defined as (4), and $T_{\ell j}^r,T_j^r$ defined as above, then we have

$$\begin{array}{c}{\displaystyle {∥T_j^{r}f∥}_{L^{\infty }}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }},}\\ \\ {\displaystyle {∥T_{\ell j}^{r}f∥}_{L^{\infty }}\le \mathcal{C}{2}^{-n_1\ell /2}{\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}.}\end{array}$$

(17)

Moreover, let $2^k\le r^{-1}<2^{k+1}$, if ${\sigma }_0(x,\xi )={\sum }_{j\le -k}{\sigma }_j^r(x,\xi )$, we have

$${∥T_{{\sigma }_0}f∥}_{L^{\infty }}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}.$$

(18)

Proof. 

We denote $\widehat{\sigma }(x,y)={\int }_{{\mathbb{R}}^n}{e}^{-2\pi iy·\xi }\sigma (x,\xi )d\xi$ throughout this paper. Now write

$$T_j^{r}f\left(x\right)={\int }_{{\mathbb{R}}^n}f\left(y\right){\widehat{\sigma }}_j^r(x,y-x)dy.$$

(19)

We see that $|T_j^{r}f|\le ||{\widehat{\sigma }}_j^r{(x,·)\left|\right|}_{L^1}{\left|\right|f\left|\right|}_{L^{\infty }}$, where $\left|\right|{\widehat{\sigma }}_j^r{(x,·)\left|\right|}_{L^1}={\int }_{{\mathbb{R}}^n}|{\widehat{\sigma }}_j^r(x,y)|dy$. Therefore, it suffices to show that $\left|\right|{\widehat{\sigma }}_j^r{(x,·)\left|\right|}_{L^1}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}$, $\left|\right|{\widehat{\sigma }}_{\ell j}^r{(x,·)\left|\right|}_{L^1}\le \mathcal{C}{2}^{-n_1\ell /2}{\left|\right|\sigma \left|\right|}_{\mathbf{S}}$ and $\left|\right|{\widehat{\sigma }}_0{(x,·)\left|\right|}_{L^1}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}$. Let us consider $T_j^r$, let $b={\left(2^j{r}^{-1}\right)}^{a-1}$. Applying the Cauchy–Schwartz inequality and Plancherel theorem we see

$$\begin{array}{cc}{\int }_{\left|y\right|<b}\left|{\widehat{\sigma }}_j^r(x,y)\right|dy& \le \mathcal{C}{b}^{n/2}{\left({\int }_{\left|y\right|<b}{\left|{\widehat{\sigma }}_j^r(x,y)\right|}^{2}dy\right)}^{\frac{1}{2}}\le \mathcal{C}{b}^{n/2}{\left({\int }_{{\mathbb{R}}^n}{\left|{\sigma }_j^r(x,\xi )\right|}^{2}d\xi \right)}^{\frac{1}{2}}\\ & \le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}(\mathit{since}{\sigma }_{\mathit{j}}^{\mathit{r}}\mathit{lives}\mathit{in}\left|\xi \right|\sim {\mathit{2}}^{\mathit{j}}{\mathit{r}}^{-\mathit{1}}),\hfill \end{array}$$

(20)

and

$$\begin{array}{c}{\displaystyle {\int }_{\left|y\right|\ge b}\left|{\widehat{\sigma }}_j^r(x,y)\right|dy\le \mathcal{C}{b}^{n/2-k}{\left({\int }_{\left|y\right|\ge b}{\left|y\right|}^{2k}{\left|{\widehat{\sigma }}_j^r(x,y)\right|}^{2}dy\right)}^{\frac{1}{2}}}\\ \\ {\displaystyle \le \mathcal{C}{b}^{n/2-k}{\left({\int }_{{\mathbb{R}}^n}{\left|{\nabla }_{\xi }^k{\sigma }_j^r(x,\xi )\right|}^{2}d\xi \right)}^{\frac{1}{2}}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}}\\ \\ {\displaystyle \left(\mathit{since}\sigma \mathit{lives}\mathit{in}\right|\xi |\sim {\mathit{2}}^{\mathit{j}}{\mathit{r}}^{-\mathit{1}},k>n/2).}\end{array}$$

(21)

Thus, $\left|\right|{\widehat{\sigma }}_j^r{(x,·)\left|\right|}_{L^1}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}$.

Similarly, we can prove $\left|\right|{\widehat{\sigma }}_{\ell j}^r{(x,·)\left|\right|}_{L^1}\le \mathcal{C}{2}^{-n_1\ell /2}{\left|\right|\sigma \left|\right|}_{\mathbf{S}}$ and $\left|\right|{\widehat{\sigma }}_0{(x,·)\left|\right|}_{L^1}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}$ once we note that ${\sigma }_{\ell j}^r$ supported in the region $|{\xi }_1|\sim 2^{j-\ell },\left|{\xi }_2\right|\sim 2^j$ and ${\sigma }_0$ supported in the region $\left|\xi \right|\le 1$. Therefore, we have proven Lemma 2. □

3.2. Proof of the Main Theorem

Now, we can prove our proposition, fix $f\in L^{\infty }$ and $Q\subset {\mathbb{R}}^n$ having side r and center $x_0$. We have to show that

$$\frac{1}{\left|Q\right|}{\int }_{{\mathbb{R}}^n}|T_{\sigma }f\left(x\right)-{\left(T_{\sigma }f\right)}_Q{|dx\le \mathcal{C}|\left|f\right||}_{L^{\infty }}.$$

(22)

Case one $r<1$. Now let $k\ge 0$ be an integer such that

$$2^k<r^{-1}\le 2^{k+1}.$$

(23)

A direct computation shows ${\partial }_{x_i}{T}_{{\sigma }_j^r}f\left(x\right)=T_{{\sigma }_j^{\prime r}}f\left(x\right)$, where ${\sigma }_j^{\prime r}(x,\xi )={\partial }_{x_i}{\sigma }_j^r(x,\xi )+2\pi i{\xi }_i{\sigma }_j^r(x,\xi )$. Since ${\sigma }_j^{\prime r}(x,\xi )$ is supported in $\left|\xi \right|\sim 2^j{r}^{-1}$, then an elementary computation gives that $\left|\right|{\sigma }_j^{\prime r}{\left|\right|}_{\mathbf{S}}\le \mathcal{C}{r}^{-1}{2}^{-jc}$ for $-k<j\le 0$. By the Lemma 2,

$$\begin{array}{c}{\displaystyle {∥{\partial }_{x_i}\sum _{j=-k}^0{T}_{{\sigma }_j^r}f∥}_{L^{\infty }}\le \sum _{j\le 0}\left|\right|T_{{\sigma }_j^{\prime r}}f{\left|\right|}_{L^{\infty }}}\hfill \\ \\ {\displaystyle \le \mathcal{C}{r}^{-1}\sum _{j\le 0}{2}^{-jc}{\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}\le \mathbb{C}{r}^{-1}{\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}.}\hfill \end{array}$$

(24)

Therefore, $|{\sum }_{j\le 0}{T}_{{\sigma }_j^r}f\left(x\right)-a_Q|$ remains bounded in Q for some constant $a_Q$, so that

$$\frac{1}{\left|Q\right|}{\int }_Q\left|\sum _{j\le 0}{T}_{{\sigma }_j^r}f\left(x\right)-a_Q\right|dx\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}.$$

(25)

Now, let us turn to $j>0$ and let ${\sigma }^r(x,\xi )={\sum }_{j>0}{\sigma }_j^r(x,\xi )$. Fix a bump function $\lambda$ on ${\mathbb{R}}^n$, with $0\le \lambda \le 10,\lambda \ge 1$ on Q and $\widehat{\lambda }$ is supported in $\left|\xi \right|\le r^{a-1}$. Then, we write

$$\lambda \left(x\right)T_{{\sigma }^r}f\left(x\right)=T_{{\sigma }^r}\left(\lambda f\right)\left(x\right)+[\lambda ,T_{{\sigma }^r}]f\left(x\right)=I_1+I_2.$$

(26)

In order to estimate $I_1$, we write

$$T_{{\sigma }^r}\left(\lambda f\right)=(T_{{\sigma }^r}·G_{-na/4})·\left(G_{na/4}\left(\lambda f\right)\right),{\widehat{G}}_{\alpha }\left(\xi \right)={(1+|\xi |}^2{)}^{-\alpha }.$$

Observe that $T_{{\sigma }^r}·G_{-na/4}$ is a pseudo-differential operator with symbol ${\sigma }^r{(x,\xi )(1+|\xi |}^2{)}^{na/4}\in S_{1-a,\delta }^0$, so that by Hörmander’s $L^2$ result,

$$\left|\right|T_{{\sigma }^r}{\left(\lambda f\right)\left|\right|}_{L^2}^2\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}^2\left|\right|G_{na/4}{\left(\lambda f\right)\left|\right|}_{L^2}^2\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}^2{\left|\right|f\left|\right|}_{L^{\infty }}^2\left|\right|G_{na/4}\lambda {\left|\right|}_{L^2}^2,$$

(27)

now since for $\alpha >0,G_{\alpha }\in L^1$, Young’s inequality leads to $\left|\right|G_{na/4}{\lambda \left|\right|}_{L^2}^2\le \left|\right|G_{na/4}{\left|\right|}_{L^1}^2{\left|\right|\lambda \left|\right|}_{L^2}^2\le \mathcal{C}\left|Q\right|$. Therefore,

$$\begin{array}{c}{\displaystyle \frac{1}{\left|Q\right|}{\int }_Q|T_{{\sigma }^r}\left(\lambda f\right)\left(x\right)|dx\le {\left(\frac{1}{\left|Q\right|}{\int }_Q{\left|T_{{\sigma }^r}\left(\lambda f\right)\left(x\right)\right|}^{2}dx\right)}^{\frac{1}{2}}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}.}\end{array}$$

(28)

To estimate $I_2$, we set ${\theta }_{\ell j}^r(x,\xi )={\int }_{{\mathbb{R}}^n}{e}^{2\pi ix·\eta }\widehat{\lambda }\left(\eta \right)[{\sigma }_{\ell j}^r(x,\xi )-{\sigma }_{\ell j}^r(x,\xi +\eta )]d\eta$, then we can rewrite $I_2$ as $I_2={\sum }_{\ell \ge 0}{\sum }_{j>0}{T}_{{\theta }_{\ell j}^r}f\left(x\right)$. Notice that ${\theta }_{\ell j}^r$ is supported in $|{\xi }_1|\sim 2^{j-\ell }{r}^{-1},|{\xi }_2|\sim 2^j{r}^{-1}$. Hence, simple calculations show that $\left|\right|{\theta }_{\ell j}^r{\left|\right|}_{\mathbf{S}}\le \mathcal{C}{2}^{(\ell -j)(1-a)}{\left|\right|\sigma \left|\right|}_{\mathbf{S}}$. Applying the Lemma 2, we have

$$\begin{array}{c}{\displaystyle \left|\right|[\lambda ,T_{{\sigma }^r}]{f\left|\right|}_{L^{\infty }}\le \sum _{\ell \ge 0}\sum _{j>0}\left|\right|T_{{\theta }_{\ell j}^r}{f\left|\right|}_{L^{\infty }}\le \sum _{\ell \ge 0}\sum _{j>0}\mathcal{C}{2}^{-n_1\ell /2}\left|\right|{\theta }_{\ell j}^r{\left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}}\\ \\ {\displaystyle \le \mathcal{C}\sum _{\ell \ge 0}\sum _{j>0}\mathcal{C}{2}^{-n_1\ell /2}{2}^{(\ell -j)(1-a)}{\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }}\le {\mathcal{C}\left|\right|\sigma \left|\right|}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }},n_1\ge 2.}\end{array}$$

(29)

Now putting (28) and (29) into (26), we obtain

$$\frac{1}{\left|Q\right|}{\int }_Q|\lambda \left(x\right)·T_{{\sigma }^r}{f\left(x\right)|dx\le \mathcal{C}|\left|\sigma \right||}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }},$$

and since $\left|\lambda \right(x\left)\right|\ge 1$ on Q, we have

$$\frac{1}{\left|Q\right|}{\int }_Q|T_{{\sigma }^r}{f\left(x\right)|dx\le \mathcal{C}|\left|\sigma \right||}_{\mathbf{S}}{\left|\right|f\left|\right|}_{L^{\infty }},$$

and together with (25), this proves (22).

Case two $r>1$. We make the decomposition by setting $r=1$, then in the region $\left|\xi \right|<1$, applying the Lemma 2 and for $\left|\xi \right|\ge 1$, repeating the proof above, we finally proved our main theorem.

Remark 2.

The above proof shows if $a>1/2$, the main theorem holds for $n_1=1$ or $n_2=1$.

Remark 3.

We posit that our results can be further extended to encompass the general bi-parameter Hörmander class $B{S}_{\rho ,\delta }^{m_1,m_2}$. This broader class encompasses a wider range of operators and offers opportunities for deeper exploration and generalization of our findings.

4. Conclusions

We extend Fefferman’s results on pseudo-differential operators to the bi-parameter setting by introducing a novel decomposition on the phase space, known as the cone decomposition. This decomposition allows us to analyze pseudo-differential operators in a more intricate and nuanced manner, capturing the behavior of operators across multiple dimensions.

One key observation is that although there are infinitely many partial operators arising from the cone decomposition, the sum of their norms remains finite. This finiteness arises from the decay present in each partial operator norm, ensuring convergence in the overall operator analysis.

However, a challenge emerges when one subspace dimension is 1. In such cases, the decay of the partial operator norm may vanish, complicating the analysis and necessitating alternative approaches for handling these scenarios.

Overall, our paper contributes to advancing the understanding of pseudo-differential operators in bi-parameter settings, laying the groundwork for future research in this area. Through our novel decomposition approach and careful analysis, we uncover insights that extend Fefferman’s seminal work and pave the way for broader applications in harmonic analysis and operator theory.