Endpoint Estimates for Certain Singular Integrals with Non-smooth Kernels

Let $L$ be a closed, densely defined operator of type $\omega$ on $L^2({\mathbb{R}}^n)$ with $0\le \omega <\pi \mathrm{/}2$. We assume that $L$ possesses a bounded $H_{\mathrm{\infty }}$-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 $L^{p_0,\mathrm{\infty }}({\mathbb{R}}^n)$ 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.