{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.54503/0002-3043-2022.57.6-81-94","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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