{"title":"Uniformly local spaces and refinements of the classical Sobolev embedding theorems","authors":"P. Rabier","doi":"10.4310/ARKIV.2018.V56.N2.A13","DOIUrl":null,"url":null,"abstract":"We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lj ,σj ∩LN,1 uloc with 1≤pj≤N and σj=1 when pj=1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, Lp,σ is the classical Lorentz space and L uloc is a “uniformly local” subspace of L loc larger than L p,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lj ,σj ∩Lquloc with q>N whenever pj<N and that, if so, the limit of f at infinity is uniform if the pj are suitably distributed. Only a few special cases have been considered in the literature, under much more restrictive assumptions that do not involve uniformly local spaces (pj=N and f vanishing at infinity, or ∂jf∈L∩L with p<N<q). Various similar results hold under integrability conditions on the higher order derivatives of f. All of them are applicable to g∗f with g∈L1 and f as above, or under weaker assumptions on f and stronger ones on g. When g is a Bessel kernel, the results are provably optimal in some cases.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arkiv for Matematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/ARKIV.2018.V56.N2.A13","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lj ,σj ∩LN,1 uloc with 1≤pj≤N and σj=1 when pj=1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, Lp,σ is the classical Lorentz space and L uloc is a “uniformly local” subspace of L loc larger than L p,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lj ,σj ∩Lquloc with q>N whenever pj
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