{"title":"Some Estimates for Riesz Transforms Associated with Schrödinger Operators","authors":"Y. H. Wang","doi":"10.54503/0002-3043-2022.57.6-81-94","DOIUrl":null,"url":null,"abstract":"Abstract Let $$\\mathcal{L}=-\\Delta+V$$ be the Schrödinger operator on $$\\mathbb{R}^{n},$$ where $$n\\geq 3,$$ and nonnegative potential $$V$$ belongs to the reverse Hölder class $$RH_{q}$$ with $$n/2\\leq q<n.$$ Let $$H^{p}_{\\mathcal{L}}(\\mathbb{R}^{n})$$ denote the Hardy space related to $$\\mathcal{L}$$ and $$BMO_{\\mathcal{L}}(\\mathbb{R}^{n})$$ denote the dual space of $$H^{1}_{\\mathcal{L}}(\\mathbb{R}^{n}).$$ In this paper, we show that $$T_{\\alpha,\\beta}=V^{\\alpha}\\nabla\\mathcal{L}^{-\\beta}$$ is bounded from $$H^{p_{1}}_{\\mathcal{L}}(\\mathbb{R}^{n})$$ into $$L^{p_{2}}(\\mathbb{R}^{n})$$ for $$\\dfrac{n}{n+\\delta^{\\prime}}<p_{1}\\leq 1$$ and $$\\dfrac{1}{p_{2}}=\\dfrac{1}{p_{1}}-\\dfrac{2(\\beta-\\alpha)}{n},$$ where $$\\delta^{\\prime}=\\min\\{1,2-n/q_{0}\\}$$ and $$q_{0}$$ is the reverse Hölder index of $$V.$$ Moreover, we prove $$T^{*}_{\\alpha,\\beta}$$ is bounded on $$BMO_{\\mathcal{L}}(\\mathbb{R}^{n})$$ when $$\\beta-\\alpha=1/2.$$","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":"26 1","pages":"380-394"},"PeriodicalIF":0.3000,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.54503/0002-3043-2022.57.6-81-94","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract Let $$\mathcal{L}=-\Delta+V$$ be the Schrödinger operator on $$\mathbb{R}^{n},$$ where $$n\geq 3,$$ and nonnegative potential $$V$$ belongs to the reverse Hölder class $$RH_{q}$$ with $$n/2\leq q
期刊介绍：
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) is an outlet for research stemming from the widely acclaimed Armenian school of theory of functions, this journal today continues the traditions of that school in the area of general analysis. A very prolific group of mathematicians in Yerevan contribute to this leading mathematics journal in the following fields: real and complex analysis; approximations; boundary value problems; integral and stochastic geometry; differential equations; probability; integral equations; algebra.
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