\(A_{\infty}({{{\mathbb R}^n}})\) weights and the local maximal operator (Q1936097)

Let \(s\in (0,1)\), let \(f\) be a measurable function in \(\mathbb R^n\), and let \(Q\) be a cube in \(\mathbb R^n\). Let us set \(m_{0,s;Q}(f)=\inf\{\lambda>0:|\{x\in Q:|f(x)|>\lambda\}|<s|Q|\}\). The multi(sub)linear local maximal operator \(M_{0,s}\) is defined by \[ M_{0,s}(f_1,\dots,f_l)(x)=\sup_{Q\ni x}\prod^l_{k=1}m_{0,s;Q}(f_k). \] For \(\vec{P}=(p_1,\dots,p_l)\), \(p_i\in (0,+\infty]\), \(r\in (0,\min_{1\leq k\leq l}p_k)=:(0,B)\), set \(1/p=\sum_{1\leq k\leq l}1/p_k\), \(\vec{P}/r=(p_1/r,\dots,p_l/r)\), \(A_{\vec{P},\infty}(\mathbb R^n)= \cup_{0<r<B}A_{\vec{P}/r}(\mathbb R^n)\), \(\nu_{\vec{w}}=\prod^l_{k=1}w_k^{p/p_k}\). Here \(\vec{w}\in A_{\vec{P}/r}(\mathbb R^n)\) means that \[ \sup_{Q}\left(|Q|^{-1}\int_Q\nu_{\vec{w}}(x)\,dx\right)^{1/p}\prod_{k=1}^l\left(|Q|^{-1}\int_Qw_k^{-1/(p_k-1)} (x)\,dx\right)^{1/p'_k}<\infty. \] The main result of the paper is the following part of Theorem 1.1. Theorem. Let \(s\in (0,1/(2l))\), let \(w_1,\dots,w_l\) be weights, and let \(p_1,\dots,p_l\in (0,\infty)\). Then the following conditions are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] the operator \(M_{0,s}\) is bounded from \(L^{p_1}(\mathbb R^n,w_1)\times\dots\times L^{p_l}(\mathbb R^n,w_l)\) to \(L^{p}(\mathbb R^n,\nu_{\vec{w}})\); \item[(v)] \(\nu_{\vec{w}}\) belongs to the Muckenhoupt class \(A_{\infty}(\mathbb R^n)\) and there is a constant \(\gamma\in (0,\infty)\) such that for each \(1\leq k\leq l\) one has \(w^\gamma_k\in A_{\infty}(\mathbb R^n)\); \item[(vi)] \(\vec{w}\in A_{\vec{P},\infty}(\mathbb R^n)\). \end{itemize}}