Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators

Details Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators

2009-03-26

arxiv.org/abs/0903.4576v2

Mathematics

Classical Analysis and ODEs

Nagoya Math. J. (to appear)

Scientific paper

Let ${\mathcal X}$ be a space of homogeneous type in the sense of Coifman and Weiss and ${\mathcal D}$ a collection of balls in $\cx$. The authors introduce the localized atomic Hardy space $H^{p, q}_{\mathcal D}({\mathcal X})$ with $p\in (0,1]$ and $q\in[1,\infty]\cap(p,\infty]$, the localized Morrey-Campanato space ${\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$ and the localized Morrey-Campanato-BLO space $\widetilde{\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$ with $\az\in{\mathbb R}$ and $p\in(0, \infty)$ and establish their basic properties including $H^{p, q}_{\mathcal D}({\mathcal X})=H^{p, \infty}_{\mathcal D}({\mathcal X})$ and several equivalent characterizations for ${\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$ and $\wz{\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$. Especially, the authors prove that when $p\in(0,1]$, the dual space of $H^{p, \infty}_{\mathcal D}({\mathcal X})$ is ${\mathcal E}^{1/p-1, 1}_{\mathcal D}({\mathcal X})$. Let $\rho$ be an admissible function modeled on the known auxiliary function determined by the Schr\"odinger operator. Denote the spaces ${\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$ and $\widetilde{\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$, respectively, by ${\mathcal E}^{\alpha, p}_{\rho}({\mathcal X})$ and $\widetilde{\mathcal E}^{\alpha, p}_{\rho}({\mathcal X})$, when ${\mathcal D}$ is determined by $\rho$. The authors then obtain the boundedness from ${\mathcal E}^{\alpha, p}_{\rho}({\mathcal X})$ to $\widetilde{\mathcal E}^{\alpha, p}_{\rho}({\mathcal X})$ of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley $g$-function which are defined via kernels modeled on the semigroup generated by the Schr\"odinger operator.

Affiliated with

Also associated with

No associations

Rating

Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-64753