A characterization of a multiple weights class
The author considers multisublinear fractional maximal functions \[ {\mathcal M}_{\alpha,m} \left(\vec{f}\right)(x):= \sup\limits_{x\in B} |B|^{\frac{\alpha}{n}}\prod\limits_{j=1}^{m} \frac{1}{|B|}\int\limits_B |f_j(y_j)|dy_j. \] Here the supremum is taken over all balls \(B\) in \({\mathbb R}^n\). Let \(0\leq\alpha<mn, 1<p_1,\dots,p_m<\infty\), \(\frac{1}{p}=\frac{1}{p_1}+\cdots+\frac{1}{p_m}, \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}\). Suppose that a collection of \(m\) weights satisfies the following conditions: \[ \left(w_1\cdots w_m\right)^q\in A_{1+q\left(m-\frac{1}{p}\right)}({\mathbb R}^n), \] \[ w_j^{-p_j^\prime} \in A_{1+p_j^\prime\left(\frac{1}{q}+m-\frac{1}{p}-\frac{1}{p_j^\prime}\right)}({\mathbb R}^n)\quad (j=1,\dots,m), \] where \(A_p\) is the Muckenhoupt weight class. The main result of the paper under review is the following inequality: \[ \left\|{\mathcal M}_{\alpha,m} \left(\vec{f}\right)(w_1\cdots w_m)\right\|_{L^q({\mathbb R}^n)}\leq C\prod\limits_{j=1}^m \|f_j w_j\|_{L^{p_j}({\mathbb R}^n)} \] for some constant \(C>0\) independent of the collection of positive measurable functions \(\vec{f}=(f_1,\dots,f_m)\). This result improves an inequality proved by \textit{X. Chen} and \textit{Q. Xue} [J. Math. Anal. Appl. 362, No. 2, 355--373 (2010; Zbl 1200.26023)] under an additional unnatural condition.
scientific article; zbMATH DE number 6146577
scientific article