Integral Restriction for Bilinear Operators

Abstract

We consider the integral domain restriction operator $T_{\Omega}$ for certain bilinear operator $T$. We obtain that if $(s,p_1,p_2)$ satisfies $\frac{1}{p_1}+\frac{1}{p_2}\geq \frac{2}{\min\{1,s\}}$ and $\|T\|_{L^{p_1}\times L^{p_2}\rightarrow L^s}\lt\infty$, then $\|T_{\Omega}\|_{L^{p_1}\times L^{p_2}\rightarrow L^s}\lt\infty$. For some special domain $\Omega$, this property holds for triplets $(s,p_1,p_2)$ satisfying $\frac{1}{p_1}+\frac{1}{p_2}\gt\frac{1}{\min\{1,s\}}$.