{"title":"$L^p$$ -沿平面曲线最大函数边界的改进","authors":"Naijia Liu, Haixia Yu","doi":"10.1007/s12220-024-01783-7","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study the <span>\\(L^p({\\mathbb {R}}^2)\\)</span>-improving bounds, i.e., <span>\\(L^p({\\mathbb {R}}^2)\\rightarrow L^q({\\mathbb {R}}^2)\\)</span> estimates, of the maximal function <span>\\(M_{\\gamma }\\)</span> along a plane curve <span>\\((t,\\gamma (t))\\)</span>, where </p><span>$$\\begin{aligned} M_{\\gamma }f(x_1,x_2):=\\sup _{u\\in [1,2]}\\left| \\int _{0}^{1}f(x_1-ut,x_2-u \\gamma (t))\\,\\text {d}t\\right| , \\end{aligned}$$</span><p>and <span>\\(\\gamma \\)</span> is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain <span>\\(M_{\\gamma }: L^p({\\mathbb {R}}^2)\\rightarrow L^q({\\mathbb {R}}^2)\\)</span> if <span>\\(\\left( \\frac{1}{p},\\frac{1}{q}\\right) \\in \\Delta \\cup \\{(0,0)\\}\\)</span> and <span>\\(\\left( \\frac{1}{p},\\frac{1}{q}\\right) \\)</span> satisfying <span>\\(1+(1 +\\omega )\\left( \\frac{1}{q}-\\frac{1}{p}\\right) >0\\)</span>, where <span>\\(\\Delta :=\\left\\{ \\left( \\frac{1}{p},\\frac{1}{q}\\right) : \\frac{1}{2p}<\\frac{1}{q}\\le \\frac{1}{p}, \\frac{1}{q}>\\frac{3}{p}-1 \\right\\} \\)</span> and <span>\\(\\omega :=\\limsup _{t\\rightarrow 0^{+}}\\frac{\\ln |\\gamma (t)|}{\\ln t}\\)</span>. This result is sharp except for some borderline cases.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"39 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"$$L^p$$ -Improving Bounds of Maximal Functions Along Planar Curves\",\"authors\":\"Naijia Liu, Haixia Yu\",\"doi\":\"10.1007/s12220-024-01783-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we study the <span>\\\\(L^p({\\\\mathbb {R}}^2)\\\\)</span>-improving bounds, i.e., <span>\\\\(L^p({\\\\mathbb {R}}^2)\\\\rightarrow L^q({\\\\mathbb {R}}^2)\\\\)</span> estimates, of the maximal function <span>\\\\(M_{\\\\gamma }\\\\)</span> along a plane curve <span>\\\\((t,\\\\gamma (t))\\\\)</span>, where </p><span>$$\\\\begin{aligned} M_{\\\\gamma }f(x_1,x_2):=\\\\sup _{u\\\\in [1,2]}\\\\left| \\\\int _{0}^{1}f(x_1-ut,x_2-u \\\\gamma (t))\\\\,\\\\text {d}t\\\\right| , \\\\end{aligned}$$</span><p>and <span>\\\\(\\\\gamma \\\\)</span> is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain <span>\\\\(M_{\\\\gamma }: L^p({\\\\mathbb {R}}^2)\\\\rightarrow L^q({\\\\mathbb {R}}^2)\\\\)</span> if <span>\\\\(\\\\left( \\\\frac{1}{p},\\\\frac{1}{q}\\\\right) \\\\in \\\\Delta \\\\cup \\\\{(0,0)\\\\}\\\\)</span> and <span>\\\\(\\\\left( \\\\frac{1}{p},\\\\frac{1}{q}\\\\right) \\\\)</span> satisfying <span>\\\\(1+(1 +\\\\omega )\\\\left( \\\\frac{1}{q}-\\\\frac{1}{p}\\\\right) >0\\\\)</span>, where <span>\\\\(\\\\Delta :=\\\\left\\\\{ \\\\left( \\\\frac{1}{p},\\\\frac{1}{q}\\\\right) : \\\\frac{1}{2p}<\\\\frac{1}{q}\\\\le \\\\frac{1}{p}, \\\\frac{1}{q}>\\\\frac{3}{p}-1 \\\\right\\\\} \\\\)</span> and <span>\\\\(\\\\omega :=\\\\limsup _{t\\\\rightarrow 0^{+}}\\\\frac{\\\\ln |\\\\gamma (t)|}{\\\\ln t}\\\\)</span>. This result is sharp except for some borderline cases.</p>\",\"PeriodicalId\":501200,\"journal\":{\"name\":\"The Journal of Geometric Analysis\",\"volume\":\"39 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Journal of Geometric Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s12220-024-01783-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01783-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
$$L^p$$ -Improving Bounds of Maximal Functions Along Planar Curves
In this paper, we study the \(L^p({\mathbb {R}}^2)\)-improving bounds, i.e., \(L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2)\) estimates, of the maximal function \(M_{\gamma }\) along a plane curve \((t,\gamma (t))\), where
and \(\gamma \) is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain \(M_{\gamma }: L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2)\) if \(\left( \frac{1}{p},\frac{1}{q}\right) \in \Delta \cup \{(0,0)\}\) and \(\left( \frac{1}{p},\frac{1}{q}\right) \) satisfying \(1+(1 +\omega )\left( \frac{1}{q}-\frac{1}{p}\right) >0\), where \(\Delta :=\left\{ \left( \frac{1}{p},\frac{1}{q}\right) : \frac{1}{2p}<\frac{1}{q}\le \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \right\} \) and \(\omega :=\limsup _{t\rightarrow 0^{+}}\frac{\ln |\gamma (t)|}{\ln t}\). This result is sharp except for some borderline cases.