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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 30, Issue 5

Issues

An endpoint version of uniform Sobolev inequalities

Tianyi RenORCID iD: http://orcid.org/0000-0002-3938-9784 / Yakun Xi / Cheng Zhang
Published Online: 2018-06-20 | DOI: https://doi.org/10.1515/forum-2018-0042

Abstract

We prove an endpoint version of the uniform Sobolev inequalities in [C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 1987, 329–347]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [J. Bourgain, Esitmations de certaines functions maximales, C. R. Acad. Sci. Paris 310 1985, 499–502]. We also prove restricted weak type Stein–Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.

Keywords: Uniform Sobolev inequalities; Stein–Tomas inequality; Bourgain’s interpolation

MSC 2010: 42B35; 42B20

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About the article


Received: 2018-02-13

Published Online: 2018-06-20

Published in Print: 2018-09-01


Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1279–1289, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2018-0042.

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