Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through \(L^{p{\infty}}\)
Through a suite of theorems, propositions, lemmas, corollaries and remarks, the author proves a set of inequalities. Let us consider, as an example, the fractional maximal operator \(M_ \alpha\) defined by \[ M_ \alpha f(x)= \sup_ Q| Q|^{\alpha/(n-1)} \int_ Q | f| dx,\quad x\in \mathbb{R}^ n, \] where \(f\in L^ p\), \(Q\) is a cube in \(\mathbb{R}^ n\) containing \(x\) with the sides parallel to the coordinate axes, \(| Q|\) is the finite Borel measure of \(Q\) and \(0<\alpha< n/p\). In these circumstances the inequality \[ \int (M_ \alpha f)^ q d\omega\leq C\bigl(\int | f|^ p dx\bigr)^{q/p} \] holds iff \[ \sup_{x\in Q}\bigl(| Q|_ \omega/| Q|^{1-\alpha p/n}\bigr)\in L^{q/(p-q)}(\omega), \] with \(0< q< p<\infty\), \(\omega\) being an arbitrary measure.
scientific article; zbMATH DE number 39670
scientific article