Weighted norm inequalities for general maximal operators (Q756067)

Let \({\mathfrak B}\) be a basis in \({\mathbb{R}}^ n\), that is a collection of open sets in \({\mathbb{R}}^ n\), and \(w: {\mathbb{R}}^ n\to [0,\infty)\) be a measurable weight such that \(w(B)=\int_{B}w(y)dy<\infty\) for each \(B\in {\mathfrak B}.\) \(M_{{\mathfrak B},w}\) is the corresponding maximal operator defined by \[ M_{{\mathfrak B},w}: f\to M_{{\mathfrak B},w}f(x)=\sup_{x\in B}\frac{1}{w(B)}\int_{B}| f(y)| w(y)dy, \] if \(x\in B\) and \(M_{{\mathfrak B},w}f(x)=0\) otherwise. If \(w\equiv 1\), we just write \(M_{{\mathfrak B}}f(x).\) The author studies weighted inequalities for this operator for various bases \({\mathfrak B}\). After introducing an analogue of Muckehoupt condition with respect to \({\mathfrak B}\) it appeared that roughly speaking, all main classical results may be obtained in this new setting. In particular criteria of boundedness of the operator \(M_{{\mathfrak B}}\) in weighted spaces and its vector-valued analogues are obtained as well as some other results of this nature.