$A_{\infty}({{{\mathbb R}^n}})$ WEIGHTS AND THE LOCAL MAXIMAL OPERATOR

Abstract

Let $s \in (0,\,1/2)$, $M_{0,\,s}$ be the local maximal operator of John and Strömberg, and ${\cal M}_{0,\,s}$ the multi(sub)linear local maximal operator. In this paper, the authors give some characterizations of the weights $w_1,\,...,\,w_{\ell}$ for which the operator ${\cal M}_{0,\,s}$ is bounded from $L^{p_1}(\mathbb{R}^n,\,w_1) \times ... \times L^{p_{\ell}}(\mathbb{R}^n,\,w_{\ell})$ to $L^{p}(\mathbb{R}^n,\,\nu_{\vec{w}})$ with $\nu_{\vec{w}} = \prod_{k=1}^{\ell} w_k^{p/p_k}$, $p_1,\,...,\,p_{\ell} \in (0,\,\infty)$ and $1/p = \sum_{1 \leq k \leq \ell} 1/p_k$. A new characterization of $A_{\infty}(\mathbb{R}^n)$ weights and a characterization of weights $w$ which satisfies $w^{\theta} \in A_{\infty}(\mathbb{R}^n)$ for some $\theta \in (0,\,\infty)$, are also obtained.