The Bilinear Pseudo-differential Operators of $S_{0,0}$-type on $L^p\times L^q$

Abstract

In this paper, we set up the boundedness of the bilinear pseudo-differential operators of $S_{0,0}$-type on $L^p\times L^q\rightarrow B^{0}_{r,1}$. In particular, we improve the recent results of Kato-Miyachi-Tomita where they proved the boundedness on $L^2\times L^2$.

Appendix

In this section, we give the proof the Lemma 2.1.

Proof

Since $\mathbb {R}^{n}=\cup _{v\in \mathbb {Z}^{n}}(2\pi v+[-\pi ,\pi ]^{n})=:\cup _{v\in \mathbb {Z}^{n}}\Omega$, we have

$$\begin{aligned}{} & {} \varphi (R^{-1}(D-u))f(x) =R^{n}\int _{\mathbb {R}^{n}}e^{iu\cdot (x-y)}\check{\varphi }(R(x-y))f(y)dy\\{} & {} \quad =R^{n}e^{iu\cdot x}\sum _{v\in \mathbb {Z}^{n}}\int _{\Omega }e^{-iu\cdot y}\check{\varphi }(R(x-y))f(y)dy\\{} & {} \quad = R^{n}e^{iu\cdot x}\int _{[-\pi ,\pi ]^{n}}e^{-iu\cdot y}\left( \sum _{v\in \mathbb {Z}^{n}}\check{\varphi }(R(x-y-2\pi v))f(y+2\pi v)\right) dy. \end{aligned}$$

Notice that $\sum _{v\in \mathbb {Z}^{n}}\check{\varphi }(R(x-y-2\pi v)) f(y+2\pi v)$ is a $(2\pi \mathbb {Z})^{n}$-periodic function of the y-variables. Combining

$$\begin{aligned}{} & {} \left( \sum _{u\in \mathbb {Z}^{n}}|\varphi (R^{-1}(D-u))f(x)|^{2}\right) ^{\frac{1}{2}}\\{} & {} \quad \approx \left( \int _{[-\pi ,\pi ]^{n}}\big |\sum _{v\in \mathbb {Z}^{n}}\check{\varphi }(R(x-y-2\pi v))f(y+2\pi v)\big |^{2}dy\right) ^{\frac{1}{2}} \end{aligned}$$

with

$$\begin{aligned} \sup _{u\in \mathbb {Z}^{n}}\varphi (R^{-1}(D-u))f(x)\lesssim \int _{[-\pi ,\pi ]^{n}}\big |\sum _{v\in \mathbb {Z}^{n}}\check{\varphi }(R(x-y-2\pi v))f(y+2\pi v)\big |dy, \end{aligned}$$

we have

$$\begin{aligned}{} & {} \left( \sum _{u\in \mathbb {Z}^{n}}|\varphi (R^{-1}(D-u))f(x)|^{p'}\right) ^{\frac{1}{p'}}\\{} & {} \quad \lesssim R^{n}\left( \int _{[-\pi ,\pi ]^{n}}\big |\sum _{v\in \mathbb {Z}^{n}}\check{\varphi }(R(x-y-2\pi v))f(y+2\pi v)\big |^{p}dy\right) ^{\frac{1}{p}} \text{ for } 1<p\le 2. \end{aligned}$$

Since

$$\begin{aligned} \sup _{z\in \mathbb {R}^{n}}\sum _{v\in \mathbb {Z}^{n}}|\check{\varphi }(R(z-2\pi v))|^{p-1}\lesssim \sup _{z\in \mathbb {R}^{n}}\sum _{v\in \mathbb {Z}^{n}}(1+R|z-2\pi v|)^{-L}\lesssim 1, \end{aligned}$$

for $1<p\le 2$ and arbitrary large $L>0$.

Furthermore, by Hölder’s inequality, we have

$$\begin{aligned}{} & {} \left( \int _{[-\pi ,\pi ]^{n}}\big |\sum _{v\in \mathbb {Z}^{n}}\check{\varphi }(R(x-y-2\pi v))f(y+2\pi v)\big |^{p}dy\right) ^{1/p}\\{} & {} \quad \lesssim \left( \int _{[-\pi ,\pi ]^{n}}\sum _{v\in \mathbb {Z}^{n}}\big |\check{\varphi }(R(x-y-2\pi v))\big |\big |f(y+2\pi v)\big |^{p}dy\right) ^{1/p}\\{} & {} \quad =\left( \int _{\mathbb {R}^{n}}\big |\check{\varphi }(R(x-y))\big |\big |f(y)\big |^{p}dy\right) ^{1/p}. \end{aligned}$$

Since $\check{\varphi }$ is a rapidly decreasing functions, it is easy to see that we obtain the desired estimates. $\square$