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

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 Schrödinger 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 \(\nu \in (0,1)\).

我们考虑\（L ^ 2（X）\）上的非负自伴随算子L，其中\（X \ subseteq {{\ mathbb {R}}} ^ d \）。在某些假设下，我们证明了哈代空间的原子特征

我们陈述简单条件，使得\（H ^ 1（L）\）的特征是原子是\（X \ subseteq {\ mathbb {R} ^ d} \）上的经典原子或形式为\ （| Q | ^ {-1} \ chi _Q \），其中\（Q \ subseteq X \）是一个立方体（或长方体）。我们的主要动机之一是研究与正交展开有关的多维算子。我们证明，如果两个算子\（L_1，L_2 \）满足我们定理的假设，那么总和\（L_1 + L_2 \）也是如此。结果，我们给出了多维Bessel，Laguerre和Schrödinger算子的原子表征。作为副产品，在相同的假设下，我们表征\（H ^ 1（L）\）也由与下级半群\（\ exp（-tL ^ \ nu）\）相关的最大运算符组成，其中\（\ nu \ in（0,1）\）。