A characterization of some weighted inequalities for the vector-valued weighted maximal function
The author obtains necessary and sufficient conditions of weighted weak and strong type norm inequalities for vector-valued weighted maximal functions. More precisely, let \(f=\{f_k\}_1^\infty\) be a sequence of locally integrable functions on \(\mathbb R^n\), \(|f(x)|_r= (\sum_1^\infty| f(x)|^r)\) and \(M\omega f(x)= \sup_{x\in Q}\int_Q|f(y)|\omega(y)\,dy\). The author proves that {\parindent=6,5mm \begin{itemize}\item[(1)] There is constant \(C_{r,p}\) such that \[ u\big(\{x\in\mathbb R^n:| M\omega f(x)|_r>\lambda\}\big)\leq C_{r,p} \lambda^{-p} \int_{\mathbb R^n}|f(y)|_r^p u(y)\,dy \] if and only if \(u\in A_{p}(\omega)\). \item[(2)] There is a constant \(C_{r,p}\) such that \[ \int_{\mathbb R^n}|M_\omega f(y)|_r^p u(y)\,dy\leq C_{r,p} \int_{\mathbb R^n}|f(y)|_r^p u(y)\,dy \] if and only if \(u\in A_{p}(\omega)\). \end{itemize}} The case \(\omega(x)=1\) was proved by \textit{K. F. Andersen} and \textit{R. T. John} [Stud. Math. 69, 19--31 (1980; Zbl 0448.42016)].
scientific article; zbMATH DE number 5859392
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