Bilinear Pseudo-differential Operators with Symbols in $BS^{m}_{1,1}$ on Triebel–Lizorkin Spaces

Abstract

In this paper, bilinear pseudo-differential operators with symbols in the bilinear Hörmander symbol class $BS^{m}_{1,1}$ on Triebel–Lizorkin spaces are discussed. As a result, we can obtain the Kato–Ponce inequality in local Hardy spaces.

Appendix

In this appendix, let us consider bilinear pseudo-differential operators with symbols in $BS^{m}_{1,1}$ on Besov spaces. Let $\{\varphi _j\}_{j=0}^{\infty }$ be the partition of unity appearing in the definition of Triebel–Lizorkin spaces. For $0<p,q\le \infty$ and $s \in \mathbb {R}$, the Besov space $B^{p,q}_s(\mathbb {R}^n)$ consists of all $f \in \mathcal {S}'(\mathbb {R}^n)$ such that

$$\begin{aligned} \Vert f\Vert _{B^{p,q}_s} =\left( \sum _{j=0}^{\infty } 2^{jsq}\left\| \varphi _j(D)f\right\| _{L^p}^q\right) ^{1/q} <\infty , \end{aligned}$$

with the usual modification when $q=\infty$.

Let $\sigma \in BS^0_{1,1}$. Bényi [1] proved that

$$\begin{aligned} \Vert T_{\sigma }(f,g)\Vert _{B^{p,q}_{\min \{s_1,s_2\}}} \lesssim \Vert f\Vert _{B^{p_1,q}_{s_1}} \Vert g\Vert _{B^{p_2,q}_{s_2}} \end{aligned}$$

(3.11)

for $1 \le p_1,p_2,p \le \infty$, $1/p_1+1/p_2=1/p$, $0<q\le \infty$ and $s_1,s_2>0$. Naibo [12] considered the case $p_i \le 1$, $i=1,2$, or $p \le 1$, and proved

$$\begin{aligned} \Vert T_{\sigma }(f,g)\Vert _{B^{p,q}_s} \lesssim \Vert f\Vert _{B^{p_1,q}_s} \Vert g\Vert _{B^{p_2,\min \{p_2,1\}}_0} +\Vert f\Vert _{B^{p_1,\min \{p_1,1\}}_0} \Vert g\Vert _{B^{p_2,q}_s} \end{aligned}$$

(3.12)

for $0<p_1,p_2, p \le \infty$, $1/p_1+1/p_2=1/p$, $0<q\le \infty$ and $s>\tau _p$, where

$$\begin{aligned} \tau _p=n\left( \frac{1}{\min \{p,1\}}-1\right) . \end{aligned}$$

Since $B^{p_i,q}_{\min \{s_1,s_2\}} \hookrightarrow B^{p_i,\min \{p_i,1\}}_0$, $i=1,2$, we can obtain (3.11) from (3.12). By the same argument as in the proof of Theorem 1.1, we can prove the following.

Theorem 3.1

Let $m \in \mathbb {R}$, $0 < p_1,p_2,\widetilde{p_1}, \widetilde{p_2}, p \le \infty$, $1/p_1+1/p_2=1/\widetilde{p_1}+1/\widetilde{p_2}=1/p$, $0<q \le \infty$ and $s>\tau _p$. Then there exists a positive integer N such that

$$\begin{aligned} \Vert T_{\sigma }(f,g)\Vert _{B^{p,q}_s} \lesssim \Vert \sigma ; BS^{m}_{1,1,N}\Vert \left( \Vert f\Vert _{B^{p_1,q}_{m+s}}\Vert g\Vert _{h^{p_2}} +\Vert f\Vert _{h^{\widetilde{p_1}}} \Vert g\Vert _{B^{\widetilde{p_2},q}_{m+s}}\right) \end{aligned}$$

for all $\sigma \in BS^{m}_{1,1}$ and $f,g \in \mathcal {S}(\mathbb {R}^n)$.

It is known that

$$\begin{aligned} B^{p,\min \{p,q\}}_s \hookrightarrow F^{p,q}_s \hookrightarrow B^{p,\max \{p,q\}}_s \end{aligned}$$

for $0<p<\infty$, $0<q \le \infty$ and $-\infty<s<\infty$ ([14, Proposition 2.3.2/2]). Then, since $B^{p_i,\min \{p_i,1\}}_0 \hookrightarrow F^{p_i,2}_0=h^{p_i}$, $i=1,2$, for $p_i<\infty$ and obviously $B^{\infty ,1}_0 \hookrightarrow L^{\infty }=h^{\infty }$, Theorem 3.1 is an improvement of (3.12).

We end this paper by giving a sketch of the proof of Theorem 3.1. We use the same notation as in Sect. 3, and let $\sigma$ be a symbol as in (3.2). It follows from (3.10) that the r-th power of $\Vert T_{\sigma }(f,g)\Vert _{B^{p,q}_s}$ is estimated by

$$\begin{aligned} \sum _{j,k=0}^{\infty } \left( \sum _{\ell =0}^{\infty } 2^{\ell s q} \left\| \varphi _{\ell }(D) [m_{j_{(k,\ell )},k}\psi _{j_{(k,\ell )}}(D)f \chi _{j_{(k,\ell )}}(D)g]\right\| _{L^p}^q \right) ^{r/q}. \end{aligned}$$

We next use the following multiplier theorem instead of Lemma 2.1. If $\gamma >\tau _p+n/2$, then

$$\begin{aligned} \Vert m(D)h\Vert _{L^p} \lesssim \Vert m(B\cdot )\Vert _{L^2_{\gamma }} \Vert h\Vert _{L^p} \end{aligned}$$

for all h satisfying $\mathrm {supp}\, \widehat{h} \subset \{|\xi | \lesssim B\}$, where the implicit constant is independent of B (see [14, Theorem 1.5.2] and the remark mentioned after Lemma 2.1). Then, the 1 / q-th power of the parenthesis above is estimated by

$$\begin{aligned} \left( \sup _{\ell \ge 0} \left\| \varphi _{\ell }(2^{j+\ell }\cdot ) \right\| _{L^2_{\widetilde{s}}}\right) \left( \sum _{\ell =0}^{\infty } 2^{\ell s q} \left\| m_{j_{(k,\ell )},k}\psi _{j_{(k,\ell )}}(D)f \chi _{j_{(k,\ell )}}(D)g\right\| _{L^p}^q \right) ^{1/q}, \end{aligned}$$

where $s>\widetilde{s}-n/2>\tau _p$. In the rest of the proof, the difference between Theorems 3.1 and 1.1 is that $\sup _{\ell \ge 0}\Vert \chi _{\ell }(D)g\Vert _{L^{p_2}}$ appears instead of $\Vert \sup _{\ell \ge 0}|\chi _{\ell }(D)g|\Vert _{L^{p_2}}$, but the former $L^{p_2}$-norm is clearly estimated by the latter $L^{p_2}$-norm, and consequently $\Vert g\Vert _{h^{p_2}}$.