BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures

Yang, Dachun ; Yang, Dongyong

Canadian journal of mathematics, Tome 62 (2010) no. 6, pp. 1419-1434

Résumé

Let $\mu $ be a nonnegative Radon measure on ${{\mathbb{R}}^{d}}$ that satisfies the growth condition that there exist constants ${{C}_{0}}\,>\,0$ and $n\,\in \,(0,\,d]$ such that for all $x\,\in \,{{\mathbb{R}}^{d}}$ and $r\,>\,0$ , $\mu \left( B\left( x,\,r \right) \right)\,\le \,{{C}_{0}}{{r}^{n}}$ , where $B(x,\,r)$ is the open ball centered at $x$ and having radius $r$ . In this paper, the authors prove that if $f$ belongs to the $\text{BMO}$ -type space $\text{RBMO(}\mu \text{)}$ of Tolsa, then the homogeneous maximal function ${{\dot{\mathcal{M}}}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is not an initial cube) and the inhomogeneous maximal function ${{\overset{{}}{\mathop{\mathcal{M}}}\,}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is an initial cube) associated with a given approximation of the identity $S $ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, ${{\dot{\mathcal{M}}}_{s}}$ and ${{\mathcal{M}}_{s}}$ are bounded from $\text{RBMO(}\mu \text{)}$ to the $\text{BLO}$ -type space $\text{RBMO(}\mu \text{)}$ . The authors also prove that the inhomogeneous maximal operator ${{\mathcal{M}}_{s}}$ is bounded from the local $\text{BMO}$ -type space $\text{rbmo(}\mu \text{)}$ to the local $\text{BLO}$ -type space $\text{rblo(}\mu \text{)}$ .

Détail

DOI : 10.4153/CJM-2010-065-7

Mots-clés : 42B25, 42B30, 47A30, 43A99, Non-doubling measure, maximal operator, approximation of the identity, RBMO(μ), RBLO(μ), rbmo(μ), rblo(μ)