Boundedness of singular integrals and their commutators with BMO functions on Hardy spaces

TL;DR

This paper provides conditions under which certain singular integrals and their commutators with BMO functions are bounded on Hardy spaces, with applications to various operators on manifolds and Euclidean spaces.

Contribution

It establishes new boundedness criteria for singular integrals and their commutators on Hardy spaces, extending to operators on manifolds and spectral multipliers.

Findings

Boundedness of singular integrals from Hardy spaces to Lebesgue spaces.

Weak-type boundedness of commutators with BMO functions on Hardy spaces.

Applicability to Riesz transforms, square functions, and spectral multipliers.

Abstract

In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0< p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded on Hardy space $H_L^1$. We then show that our sufficient conditions are applicable to the following cases: (i) $T$ is the Riesz transform or a square function associated with the Laplace-Beltrami operator on a doubling Riemannian manifold, (ii) $T$ is the Riesz transform associated with the magnetic Schr\"odinger operator on an Euclidean space, and (iii) $T = g(L) $ is a singular integral operator defined from the holomorphic functional calculus of an operator $L$ or the spectral multiplier of a non-negative self adjoint operator $L$.