{"title":"与Schrödinger算子相关的Riesz变换的一些估计","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":"{\"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}","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}
Some Estimates for Riesz Transforms Associated with Schrödinger Operators
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
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
