An Endpoint Version of Uniform Sobolev Inequalities

TL;DR

This paper establishes endpoint restricted weak type Sobolev inequalities and Stein-Tomas restriction inequalities, extending classical results and characterizing the range of exponents where these inequalities hold.

Contribution

It introduces an endpoint version of uniform Sobolev inequalities using Bourgain's interpolation method and characterizes the exponents for Stein-Tomas inequalities.

Findings

Restricted weak type inequalities hold at endpoints.

Classical Sobolev inequalities are recovered via real interpolation.

Characterization of exponent ranges for Stein-Tomas inequalities.

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.