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Consider the multidimensional Bessel operator

Consider the Bessel operator with a potential on L2false((0,∞),xα0.16emdxfalse), namely Lffalse(xfalse)=−f′′false(xfalse)−αxf′false(xfalse)+Vfalse(xfalse)ffalse(xfalse).We assume that α>0 and V∈Lloc1false((0,∞),xα0.16emdxfalse) is a nonnegative function. By definition, a function f∈L1false((0,∞),xα0.16emdxfalse) belongs to the Hardy space scriptH1false(boldLfalse) if trueprefixsupt>0e−tboldLf∈L1false((0,∞),xα0.16emdxfalse).Under certain assumptions on V we characterize the space scriptH1false(boldLfalse) in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to boldL for α∈(0,1) with no additional assumptions on the potential V.

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).