Interpolation inequalities between Sobolev and Morrey-Campanato spaces: A common gateway to concentration-compactness and Gagliardo-Nirenberg interpolation inequalities

TL;DR

This paper establishes new interpolation inequalities connecting Sobolev, Morrey-Campanato, and BMO spaces, providing tools for concentration-compactness and Gagliardo-Nirenberg inequalities using integral representations and maximal functions.

Contribution

It introduces novel interpolation estimates between various function spaces, including fractional Sobolev spaces, expanding the theoretical framework for analysis in PDEs and functional analysis.

Findings

Derived concentration-compactness inequalities

Established interpolation estimates involving BMO

Extended results to fractional Sobolev spaces

Abstract

We prove interpolation estimates between Morrey-Campanato spaces and Sobolev spaces. These estimates give in particular concentration-compactness inequalities in the translation-invariant and in the translation- and dilation-invariant case. They also give in particular interpolation estimates between Sobolev spaces and functions of bounded mean oscillation. The proofs rely on Sobolev integral representation formulae and maximal function theory. Fractional Sobolev spaces are also covered.