Sharp maximal function and \(C_ p\) condition
Let \(f^{\#}\) denote the sharp maximal function, i.e. \[ f^{\#}(x)=\sup_{x\in Q}\frac{1}{| Q|}\int_{Q}| f(y)- \frac{1}{| Q|}\int_{Q}f(z)dz| dy, \] where the supremum is taken over all cubes Q with sides parallel to the coordinate axes, and containing x. \(C_ p\) is the weight class introduced by Muckenhoupt: a weight w(x) is said to belong to \(C_ p\), if there exist positive constants C, \(\epsilon\) such that \[ \int_{E}w(x)dx\leq C(\frac{| E|}{| Q|})^{\epsilon}\int | M_{\chi_ Q}|^ p\quad w(x)dx \] whenever E is a subset of a cube \(Q\supset {\mathbb{R}}^ n\). Here Mf denotes the Hardy-Littlewood maximal function of f. We have shown the following. Theorem. (i) Let w(x) be a weight and \(1\leq p<\infty\). Suppose \(\| f\|_{L^ p(w)}\leq C\| f^{\#}\|_{L^ p(w)},\) \(f\in L_ c^{\infty}\), then it follows that \(w\in C_ p\). (ii) If \(1<p<q<\infty\) and \(w\in C_ q\), then there exists a positive constant C such that \(\| f\|_{L^ p(w)}\leq \| Mf\|_{L^ p(w)}\leq C\| f^{\#}\|_{L^ p(w)},\) \(f\in L_ c^{\infty}\). Parallel results are given by \textit{E. Sawyer}, concerning weighted norm inequalities between singular integrals and the Hardy-Littlewood maximal function [Stud. Math. 75, 253-263 (1983; Zbl 0528.44002)]. Our proof is a modification of Sawyer's one.
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