A boundedness criterion for general maximal operators. (English) Zbl 1183.42024
Summary: We consider maximal operators \(M_{\mathcal B}\) with respect to a basis \({\mathcal B}\). In the case when \(M_{\mathcal B}\) satisfies a reversed weak type inequality, we obtain a boundedness criterion for \(M_{\mathcal B}\) on an arbitrary quasi-Banach function space \(X\). Being applied to specific \({\mathcal B}\) and \(X\) this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function \(M^{+}\) on \(L^p_w\).
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
References:
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