Localized Hardy Spaces $H^1$ Related to Admissible Functions on RD-Spaces and Applications to Schrödinger Operators - Mathematics > Classical Analysis and ODEsReport as inadecuate




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Abstract: Let ${\mathcal X}$ be an RD-space, which means that ${\mathcal X}$ is a spaceof homogenous type in the sense of Coifman and Weiss with the additionalproperty that a reverse doubling property holds in ${\mathcal X}$. In thispaper, the authors first introduce the notion of admissible functions $ ho$and then develop a theory of localized Hardy spaces $H^1 ho {\mathcal X}$associated with $ ho$, which includes several maximal functioncharacterizations of $H^1 ho {\mathcal X}$, the relations between $H^1 ho{\mathcal X}$ and the classical Hardy space $H^1{\mathcal X}$ viaconstructing a kernel function related to $ ho$, the atomic decompositioncharacterization of $H^1 ho {\mathcal X}$, and the boundedness of certainlocalized singular integrals in $H^1 ho{\mathcal X}$ via a finite atomicdecomposition characterization of some dense subspace of $H^1 ho {\mathcalX}$. This theory has a wide range of applications. Even when this theory isapplied, respectively, to the Schr\-odinger operator or the degenerateSchr\-odinger operator on $ n$, or the sub-Laplace Schr\-odinger operator onHeisenberg groups or connected and simply connected nilpotent Lie groups, somenew results are also obtained. The Schr\-odinger operators considered here areassociated with nonnegative potentials satisfying the reverse H\-olderinequality.



Author: Dachun Yang, Yuan Zhou

Source: https://arxiv.org/



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