Garsia–Rodemich spaces: Local maximal functions and interpolation
Volume 255 / 2020
Abstract
We characterize the Garsia–Rodemich spaces $ \operatorname {GaRo}_{X}$ associated with a rearrangement invariant space via local maximal operators. Let $Q_{0}$ be a cube in $\mathbb {R}^{n}$. We show that there exists $s_{0}\in (0,1)$ such that for all $0 \lt s \lt s_{0}$, and for all r.i. spaces $X(Q_{0})$, we have\[ \operatorname {GaRo}_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\| f\|_{\operatorname {GaRo}_{X}}\simeq \| M_{s,Q_{0}}^{\#}f\|_{X} \lt \infty \}, \] where $M_{s,Q_{0}}^{\#}$ is the Strömberg–Jawerth–Torchinsky local maximal operator. Combined with a formula for the $K$-functional of the pair $(L^{1},\operatorname {BMO})$ obtained by Jawerth–Torchinsky, our result shows that the $\operatorname {GaRo}_{X}$ spaces are interpolation spaces between $L^{1}$ and $\operatorname {BMO}$. 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 $\operatorname {GaRo}_{X}(Q_{0})=X(Q_{0})$. Using packings, we obtain a new expression for the $K$-functional of the pair $(L^{1}, \operatorname {BMO})$.
