Garsia-Rodemich spaces: Local Maximal Functions and Interpolation

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 Str\"{o}mberg-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 $GaRo_{X}$ 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 $GaRo_{X}(Q_{0})=X(Q_{0}).$ Using packings, we obtain a new expression for the $K-$functional of the pair $(L^{1},BMO)$.