Local atomic decompositions for multidimensional Hardy spaces

Abstract

We consider a nonnegative self-adjoint operator L on L 2 (X), where X ⊆ R d . Under certain assumptions, we prove atomic characterizations of the Hardy spaceWe state simple conditions, such that H 1 (L) is characterized by atoms being either the classical atoms on X ⊆ R d or local atoms of the form |Q| −1 χ Q , where Q ⊆ 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 ν ), where ν ∈ (0, 1).

Hardy spaces meet harmonic weights

“…[6]. A simple criteria for the semigroup kernel that give atomic decomposition in this case can be also found in [30]. In particular, H 1 L (0,∞)D ) (X) can be described by atomic decompositions, where atoms are either classical atoms on (0, ∞) or local atoms of the type a…”