Garsia-Rodemich Spaces: Bourgain-Brezis-Mironescu space, embeddings and rearrangement invariant spaces

TL;DR

This paper extends Garsia-Rodemich spaces, introduces a new space related to Bourgain-Brezis-Mironescu, and generalizes the construction to characterize rearrangement invariant spaces, leading to new embeddings and fractional Sobolev inequalities.

Contribution

It introduces the GaRo_X spaces associated with rearrangement invariant spaces, generalizes the Garsia-Rodemich construction, and provides new proofs of embeddings and inequalities.

Findings

Bourgain-Brezis-Mironescu space B described via Garsia-Rodemich norms.

New proof of embeddings: BMO ⊂ B ⊂ L(n',∞).

Fractional Sobolev inequalities established in the generalized framework.

Abstract

We extend the construction of Garsia-Rodemich spaces in different directions. We show that the new space \textbf{B,} introduced by Bourgain-Brezis-Mironescu \cite{bbm}, can be described via a suitable scaling of the Garsia-Rodemich norms. As an application we give a new proof of the embeddings $BMO\subset$ \textbf{B }$\subset$ $L(n^{\prime},\infty).$ We then generalize the Garsia-Rodemich construction and introduce the $GaRo_{X}$ spaces associated with a rearrangement invariant space $X,$ in such a way that $GaRo_{X}=X,$ for a large class of rearrangement invariant spaces. The underlying inequality for this new characterization of rearrangement invariant spaces is an extension of the rearrangement inequalities of \cite{milbmo}. We introduce Gagliardo seminorms adapted to rearrangement invariant spaces and use our generalized Garsia-Rodemich construction to prove Fractional Sobolev inequalities in this context.