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(−t L ν ), where ν ∈ (0, 1).

The atomic Hardy space for a general Bessel operator

“…Moreover, we introduce universal conditions on a semigroup kernel so that the product case can be deduced from a lower-dimensional information. The methods we use have roots in [15], however in that paper the Lebesgue measure case is considered, and here we need to adapt them to our situation, which requires some effort.…”