Weighted norm inequalities for geometric fractional maximal operators
For \(0\leq \alpha<\infty\), let \(T_\alpha f\) denote one of the maximal operators \[ M_{\alpha, 0}f(x)= \sup_{x\in I}| I|^\alpha \exp\Biggl(| I|^{-1} \int_I\log| f|\Biggr), \] \[ M^*_{\alpha, 0}f(x)= \lim_{r\to 0} \sup_{x\in I}| I|^\alpha\Biggl(| I|^{-1} \int_I| f|^r\Biggr)^{1/r}, \] where \(r\to 0\) decreasingly. The authors characterize the pairs of weights \((u,v)\) for which \(T_\alpha\) is a bounded operator from \(L^p(v)\) to \(L^p(u)\), \(0< p\leq q<\infty\). The results extend the norm inequalities for \(\alpha= 0\) in earlier papers [\textit{D. Cruz-Uribe} and \textit{C. J. Neugebauer}, Publ. Math., Barc. 42, No. 1, 239-263 (1998; Zbl 0919.42014)] and [\textit{X. Yin} and \textit{B. Muckenhoupt}, Proc. Am. Math. Soc. 124, No. 1, 75-81 (1986; Zbl 0853.42015)]. As an application, the authors give lower bounds for convolutions \(\Phi*f\), where \(\Phi\) is a rapidly decreasing function.
scientific article; zbMATH DE number 1275886
scientific article