Limiting weak-type behavior for rough bilinear operators

Abstract

Let ${\Omega }_1,{\Omega }_2$ be functions of homogeneous of degree $0$ and $\vec{\Omega }=({\Omega }_1,{\Omega }_2)\in L\log L({\mathbb{S}}^{n-1})\times L\log L({\mathbb{S}}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{\vec{\Omega }}$ and bilinear singular integral $T_{\vec{\Omega }}$ associated with rough kernel $\vec{\Omega }$. For all $f,g\in L^1({\mathbb{R}}^n)$, we show that $$\underset{\lambda \to 0^{+}}{\lim }\lambda |\{x\in {\mathbb{R}}^n:M_{\vec{\Omega }}(f_1,f_2)(x)>\lambda \}{|}^2=\frac{\Vert {\Omega }_1{\Omega }_2{\Vert }_{L^{1/2}({\mathbb{S}}^{n-1})}}{{\omega }_{n-1}^2}\prod _{i=1}^2\Vert f_i{\Vert }_{L^1}$$ and $$\underset{\lambda \to 0^{+}}{\lim }\lambda |\{x\in {\mathbb{R}}^n:|T_{\vec{\Omega }}(f_1,f_2)(x)|>\lambda \}{|}^2=\frac{\Vert {\Omega }_1{\Omega }_2{\Vert }_{L^{1/2}({\mathbb{S}}^{n-1})}}{n^2}\prod _{i=1}^2\Vert f_i{\Vert }_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{\vec{\Omega }}$ and $T_{\vec{\Omega }}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.

there is some gaps needed to be fixed

References in corpus (1)

• Limiting weak-type behaviors for factional maximal operators and fractional integrals with rough kernel