Local atomic decompositions for multidimensional Hardy spaces

TL;DR

This paper develops atomic characterizations of Hardy spaces associated with a class of operators on multidimensional spaces, including classical and local atoms, and applies these results to various multidimensional operators like Bessel, Laguerre, and Schrödinger operators.

Contribution

It introduces simple conditions for atomic characterizations of Hardy spaces linked to self-adjoint operators, extending to sums of operators and multidimensional cases.

Findings

Atomic characterizations include classical and local atoms.

Sum of operators satisfying assumptions also satisfies the same atomic characterization.

Characterization of Hardy spaces via maximal operators of subordinate semigroups.

Abstract

We consider a nonnegative self-adjoint operator $L$ on $L^2(X)$, where $X\subseteq \mathbb{R}^d$. Under certain assumptions, we prove atomic characterizations of the Hardy space $$H^1(L) = \l \{f\in L^1(X) \ : \ \ {\|}\sup_{t>0} \ |\exp(-tL)f \ | \ {\|}_{L^1(X)}<\infty\ \}.$$ We state simple conditions, such that $H^1(L)$ is characterized by atoms being either the classical atoms on $X\subseteq \mathbb{R}^d$ or local atoms of the form $|Q|^{-1}\chi_Q$, where $Q\subseteq X$ is a cube (or cuboid). One of our main motivation is to study multidimensional operators related to orthogonal expansions. We prove that if two operators $L_1, L_2$ satisfy the assumptions of our theorem, then the sum $L_1 + L_2$ also does. As a consequence, we give atomic characterizations for multidimensional Bessel, Laguerre, and Schr\"odinger operators. As a by-product, under the same assumptions, we characterize $H^1(L)$ also by the maximal operator related to the subordinate semigroup $\exp(-tL^\nu)$, where $\nu\in(0,1)$.