Estimates of some integral operators with bounded variable kernels on the Hardy and weak Hardy spaces over $\mathbb R^n$

TL;DR

This paper establishes the boundedness of various integral operators with variable kernels on Hardy and weak Hardy spaces under new smoothness conditions, extending results to Hardy--Lorentz spaces via interpolation.

Contribution

Introduces new $L^{\sigma_1}$-$( ext{log }L)^{\sigma_2}$ conditions for variable kernels and proves their boundedness on Hardy and weak Hardy spaces, including Hardy--Lorentz spaces.

Findings

Boundedness of singular integral operators with variable kernels on Hardy spaces.

Extension of results to weak Hardy and Hardy--Lorentz spaces.

New smoothness conditions for variable kernels.

Abstract

In this paper, we first introduce $L^{\sigma_1}$-$(\log L)^{\sigma_2}$ conditions satisfied by the variable kernels $\Omega(x,z)$ for $0\leq\sigma_1\leq1$ and $\sigma_2\geq0$. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators $T_{\Omega}$, fractional integrals $T_{\Omega,\alpha}$ and parametric Marcinkiewicz integrals $\mu^{\rho}_{\Omega}$ with variable kernels on the Hardy spaces $H^p(\mathbb R^n)$ and weak Hardy spaces $WH^p(\mathbb R^n)$. Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy--Lorentz spaces $H^{p,q}(\mathbb R^n)$ for all $p<q<\infty$.