Weighted inequalities for a vector-valued strong maximal function
Let \(M_ sf(x)=\sup_{R\ni x}1/| R| \int_{R}| f(y)| dy\) be the strong maximal function, the supremum being taken over all rectangles R in \(R^ n\) with edges parallel to coordinate axes. For any \(q\in (1,\infty)\) let \(f(x)=\{f_ 1(x),...,f_ k(x),...\}\) be \(\ell^ q\)-valued vector-functions with \(| f(x)|_ q=\| f(x)\|_{\ell^ q}\), and let \(M_ sf=(M_ sf_ 1,...,M_ sf_ k,...)\). The authors show that \[ \int_{| M_ sf(x)| >\lambda}w(x)dx\leq c\int_{R^ n}\frac{| f(x)|_ q}{\lambda}(1+\log +\frac{| f(x)|_ q}{\lambda})^{n-1}w(x)dx \] if \(w(x)\in A_ 1({\mathcal R})\) and \[ (*)\quad \int_{R^ n}| (M_ sf)(x)|^ p_ qw(x)dx\leq c\int_{R^ n}| f(x)|^ p_ qw(x)dx \] if and only if \(w(x)\in A_ p({\mathcal R})\), \(1<p<\infty.\) Reviewer's remark. The strong type estimate (*) for vector-valued functions with the characterization of the weight function w(x) was earlier obtained by \textit{V. M. Kokilashvili} [Dokl. Akad. Nauk SSSR 239, 42-45 (1978; Zbl 0396.46034); Maximal functions and singular integrals in weighted functional spaces, Tr. Tbilis. Mat. Instituta im. A. M. Razmadze, Tom. 80 (1985; Zbl 0609.42015)] in the much more general setting including fractional maximal functions, anisotropic case and more general behaviour of f(x) than \(\ell^ q\) behaviour.
scientific article; zbMATH DE number 4107398