Garsia-Rodemich spaces: Local Maximal Functions and Interpolation

Abstract

We characterize the Garsia-Rodemich spaces associated with a rearrangement invariant space via local maximal operators. Let $Q_0$ be a cube in $R^n$. We show that there exists $s_0\in (0,1),$ such that for all $0<s<s_0,$ and for all r.i. spaces $X(Q_0),$ we have% \[ GaRo_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\Vert f\Vert_{GaRo_{X}}\simeq\Vert M_{0,s,Q_{0}}^{\#}f\Vert_{X}<\infty\}, \] where $M_{0,s,Q_0}^{\#}$ is the Strömberg-Jawerth-Torchinsky local maximal operator. Combined with a formula for the $K-$functional of the pair $(L^1,BMO)$ obtained by Jawerth-Torchinsky, our result shows that the $GaR{o}_X$ spaces are interpolation spaces between $L^1$ and $BMO.$ Among the applications, we prove, using real interpolation, the monotonicity under rearrangements of Garsia-Rodemich type functionals. We also give an approach to Sobolev-Morrey inequalities via Garsia-Rodemich norms, and prove necessary and sufficient conditions for $GaR{o}_X(Q_0)=X(Q_0).$ Using packings, we obtain a new expression for the $K-$functional of the pair $(L^1,BMO)$.

23 pages, rewrote introduction, made some minor corrections