Embedding estimates and fractional smoothness

TL;DR

This paper provides an intrinsic proof of the Bourgain-Brezis-Mironescu theorem, extends it to higher-order gradients, and introduces new inequalities for fractional embeddings, highlighting the role of functional geometry and Fourier analysis.

Contribution

It offers a novel intrinsic proof of a key embedding theorem, extends it to higher-order cases, and develops new inequalities relevant to plasma physics.

Findings

New proof of Bourgain-Brezis-Mironescu theorem

Extension to higher-order gradient forms

Introduction of bilinear fractional embedding inequalities

Abstract

A short intrinsic proof is given for the Bourgain-Brezis-Mironescu theorem with an extension for higher-order gradient forms. This argument illustrates the role of functional geometry and Fourier analysis for obtaining embedding estimates. New Hausdorff-Young inequalities are obtained for fractional embedding as an extension of the classical Aronszajn-Smith formula. These results include bilinear fractional embedding as suggested by the Landau collision operator in plasma dynamics.