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arXiv · 1612.01056

An Endpoint Version of Uniform Sobolev Inequalities

Tianyi Ren·Yakun Xi·Cheng Zhang
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Abstract

We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. 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 [2]. 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.

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Tianyi Ren, Yakun Xi, Cheng Zhang. 2016-12-04. An Endpoint Version of Uniform Sobolev Inequalities. https://arxiv.org/abs/1612.01056

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