{"title":"可变勒贝格空间上分数算子有界性的必要条件","authors":"David Cruz-Uribe, Troy Roberts","doi":"arxiv-2408.12745","DOIUrl":null,"url":null,"abstract":"In this paper we prove necessary conditions for the boundedness of fractional\noperators on the variable Lebesgue spaces. More precisely, we find necessary\nconditions on an exponent function $\\pp$ for a fractional maximal operator\n$M_\\alpha$ or a non-degenerate fractional singular integral operator\n$T_\\alpha$, $0 \\leq \\alpha < n$, to satisfy weak $(\\pp,\\qq)$ inequalities or\nstrong $(\\pp,\\qq)$ inequalities, with $\\qq$ being defined pointwise almost\neverywhere by % \\[ \\frac{1}{p(x)} - \\frac{1}{q(x)} = \\frac{\\alpha}{n}. \\] % We first prove preliminary results linking fractional averaging operators and\nthe $K_0^\\alpha$ condition, a qualitative condition on $\\pp$ related to the\nnorms of characteristic functions of cubes, and show some useful implications\nof the $K_0^\\alpha$ condition. We then show that if $M_\\alpha$ satisfies weak\n$(\\pp,\\qq)$ inequalities, then $\\pp \\in K_0^\\alpha(\\R^n)$. We use this to prove\nthat if $M_\\alpha$ satisfies strong $(\\pp,\\qq)$ inequalities, then $p_->1$.\nFinally, we prove a powerful pointwise estimate for $T_\\alpha$ that relates\n$T_\\alpha$ to $M_\\alpha$ along a carefully chosen family of cubes. This allows\nus to prove necessary conditions for fractional singular integral operators\nsimilar to those for fractional maximal operators.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces\",\"authors\":\"David Cruz-Uribe, Troy Roberts\",\"doi\":\"arxiv-2408.12745\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we prove necessary conditions for the boundedness of fractional\\noperators on the variable Lebesgue spaces. More precisely, we find necessary\\nconditions on an exponent function $\\\\pp$ for a fractional maximal operator\\n$M_\\\\alpha$ or a non-degenerate fractional singular integral operator\\n$T_\\\\alpha$, $0 \\\\leq \\\\alpha < n$, to satisfy weak $(\\\\pp,\\\\qq)$ inequalities or\\nstrong $(\\\\pp,\\\\qq)$ inequalities, with $\\\\qq$ being defined pointwise almost\\neverywhere by % \\\\[ \\\\frac{1}{p(x)} - \\\\frac{1}{q(x)} = \\\\frac{\\\\alpha}{n}. \\\\] % We first prove preliminary results linking fractional averaging operators and\\nthe $K_0^\\\\alpha$ condition, a qualitative condition on $\\\\pp$ related to the\\nnorms of characteristic functions of cubes, and show some useful implications\\nof the $K_0^\\\\alpha$ condition. We then show that if $M_\\\\alpha$ satisfies weak\\n$(\\\\pp,\\\\qq)$ inequalities, then $\\\\pp \\\\in K_0^\\\\alpha(\\\\R^n)$. We use this to prove\\nthat if $M_\\\\alpha$ satisfies strong $(\\\\pp,\\\\qq)$ inequalities, then $p_->1$.\\nFinally, we prove a powerful pointwise estimate for $T_\\\\alpha$ that relates\\n$T_\\\\alpha$ to $M_\\\\alpha$ along a carefully chosen family of cubes. This allows\\nus to prove necessary conditions for fractional singular integral operators\\nsimilar to those for fractional maximal operators.\",\"PeriodicalId\":501145,\"journal\":{\"name\":\"arXiv - MATH - Classical Analysis and ODEs\",\"volume\":\"43 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Classical Analysis and ODEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.12745\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.12745","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces
In this paper we prove necessary conditions for the boundedness of fractional
operators on the variable Lebesgue spaces. More precisely, we find necessary
conditions on an exponent function $\pp$ for a fractional maximal operator
$M_\alpha$ or a non-degenerate fractional singular integral operator
$T_\alpha$, $0 \leq \alpha < n$, to satisfy weak $(\pp,\qq)$ inequalities or
strong $(\pp,\qq)$ inequalities, with $\qq$ being defined pointwise almost
everywhere by % \[ \frac{1}{p(x)} - \frac{1}{q(x)} = \frac{\alpha}{n}. \] % We first prove preliminary results linking fractional averaging operators and
the $K_0^\alpha$ condition, a qualitative condition on $\pp$ related to the
norms of characteristic functions of cubes, and show some useful implications
of the $K_0^\alpha$ condition. We then show that if $M_\alpha$ satisfies weak
$(\pp,\qq)$ inequalities, then $\pp \in K_0^\alpha(\R^n)$. We use this to prove
that if $M_\alpha$ satisfies strong $(\pp,\qq)$ inequalities, then $p_->1$.
Finally, we prove a powerful pointwise estimate for $T_\alpha$ that relates
$T_\alpha$ to $M_\alpha$ along a carefully chosen family of cubes. This allows
us to prove necessary conditions for fractional singular integral operators
similar to those for fractional maximal operators.