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

We extend the construction of Garsia-Rodemich spaces in different directions. We show that the new space $\mathbf{\text{B,}}$ 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 $BMO\subset$ $\mathbf{\text{B}}$$\subset$ $L(n^{\mathrm{\prime }},\mathrm{\infty })\mathrm{.}$ We then generalize the Garsia-Rodemich construction and introduce the $GaR{o}_X$ spaces associated with a rearrangement invariant space $X,$ in such a way that $GaR{o}_X=X,$ 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.