Uniformly local spaces and refinements of the classical Sobolev embedding theorems

Patrick J. Rabier University of Pittsburgh

Functional Analysis mathscidoc:1912.43024

Arkiv for Matematik, 56, (2), 409-440, 2018
We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lpj,σj∩LN,1uloc 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 Lp,σuloc is a “uniformly local” subspace of Lp,σloc larger than Lp,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lpj,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∈Lp∩Lq 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.
Lebesgue point, uniformly local space, Sobolev embedding, convolution
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@inproceedings{patrick2018uniformly,
  title={Uniformly local spaces and refinements of the classical Sobolev embedding theorems},
  author={Patrick J. Rabier},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204104355117448580},
  booktitle={Arkiv for Matematik},
  volume={56},
  number={2},
  pages={409-440},
  year={2018},
}
Patrick J. Rabier. Uniformly local spaces and refinements of the classical Sobolev embedding theorems. 2018. Vol. 56. In Arkiv for Matematik. pp.409-440. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204104355117448580.
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