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Local atomic decompositions for multidimensional Hardy spaces
Revista Matemática Complutense ( IF 1.4 ) Pub Date : 2020-03-28 , DOI: 10.1007/s13163-020-00354-y
Edyta Kania-Strojec , Paweł Plewa , Marcin Preisner

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

$$\begin{aligned} H^1(L) = \left\{ f\in L^1(X) \ : \ \left\| \sup _{t>0} \left| \exp (-tL)f \right| \right\| _{L^1(X)}<\infty \right\} . \end{aligned}$$

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 \(\nu \in (0,1)\).



中文翻译:

多维Hardy空间的局部原子分解

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

$$ \ begin {aligned} H ^ 1(L)= \ left \ {f \ in L ^ 1(X)\:\ \ left \ | \ sup _ {t> 0} \ left | \ exp(-tL)f \ right | \ right \ | _ {L ^ 1(X)} <\ infty \ right \}。\ end {aligned} $$

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\(\ nu \ in(0,1)\)

更新日期:2020-03-28
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