Generalized Fourier Multipliers via Mittag-Leffler Functions

A Fourier multiplier related to Mittag-Leffler function is introduced. We prove that our multiplier is radial on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}} <^>{n}$$\end{document}and generalizes the Bessel function. Furthermore, we study the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{2}$$\end{document} boundedness of the related Mittag-Leffler maximal function, the Littlewood-Paley g-function, and the discrete singular integral operator. We prove that the three operators are bounded on L2(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{2}({\mathbb {R}}<^>{n})$$\end{document}. In addition, our formulation of the introduced Mittag-Leffler maximal function is a solution of a diffusion equation. Our results generalize previously known results.