Endpoint Estimates for Certain Singular Integrals with Non-smooth Kernels

TL;DR

This paper proves endpoint boundedness of certain singular integrals related to operators with bounded $H_$-calculus and heat kernel bounds, including applications to Hardy and Kolmogorov operators.

Contribution

It establishes new endpoint Lorentz space estimates for singular integrals associated with a broad class of operators with non-smooth kernels.

Findings

Boundedness from $L^{p_0,1}$ to $L^{p_0, abla}$ for singular integrals.

Endpoint estimates for Hardy and Kolmogorov operators.

Application of heat kernel bounds to functional calculus.

Abstract

Let $L$ be a closed, densely defined operator of type $ \omega $ on $ L^2(\mathbb{R}^n)$ with $0 \leq \omega < \pi/2 $. We assume that $ L $ possesses a bounded $ H_\infty $-functional calculus and that its heat kernel satisfies suitable upper bounds. In this paper, we establish the boundedness from Lorentz spaces $ L^{p_0,1}(\mathbb{R}^n) $ to $ L^{p_0,\infty}(\mathbb{R}^n)$ for some singular integrals associated with $ L $, including the vertical square function and the functional calculus of Laplace transform type, where $p_0$ is determined by the upper bound of the heat kernel. As concrete applications, we obtain the endpoint estimates for the above singular integrals associated with both the Hardy operator and the Kolmogorov operator.