Dunkl translation and uncentered maximal operator on the real line
Summary: We establish estimates of the Dunkl translation of the characteristic function \(\chi _{[ - \varepsilon ,\varepsilon]}, \varepsilon >0\), and we prove that the uncentered maximal operator associated with the Dunkl operator is of weak type \((1,1)\). As a consequence, we obtain the \(L^p\)-boundedness of this operator for \(1<p\leq +\infty \). Remark: Note that for the centered maximal function associated to any finite reflection group on \(\mathbb R^n\) the same result has been proved by \textit{S. Thangavelu} and \textit{Y. Xu} [J. Anal. Math. 97, 25--55 (2006; Zbl 1131.43006)].
scientific article; zbMATH DE number 5282512
scientific article