Local atomic decompositions for multidimensional Hardy spaces

Abstract

AbstractWe consider a nonnegative self-adjoint operator L on $$L^2(X)$$ L 2 ( X ) , where $$X\subseteq {{\mathbb {R}}}^d$$ X ⊆ R d . Under certain assumptions, we prove atomic characterizations of the Hardy space $$\begin{aligned} H^1(L) = \left\{ f\in L^1(X) \ : \ \left\| \sup _{t>0} \left| \exp (-tL)f \right| \right\| _{L^1(X)}<\infty \right\} . \end{aligned}$$ H 1 ( L ) = f ∈ L 1 ( X ) : sup t > 0 exp ( - t L ) f L 1 ( X ) < ∞ . We state simple conditions, such that $$H^1(L)$$ H 1 ( L ) is characterized by atoms being either the classical atoms on $$X\subseteq {\mathbb {R}^d}$$ X ⊆ R d or local atoms of the form $$|Q|^{-1}\chi _Q$$ | Q | - 1 χ Q , where $$Q\subseteq X$$ 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 $$L_1, L_2$$ L 1 , L 2 satisfy the assumptions of our theorem, then the sum $$L_1 + L_2$$ 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 $$H^1(L)$$ H 1 ( L ) also by the maximal operator related to the subordinate semigroup $$\exp (-tL^\nu )$$ exp ( - t L ν ) , where $$\nu \in (0,1)$$ ν ∈ ( 0 , 1 ) .