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 Rⁿand generalizes the Bessel function. Furthermore, we study the L² 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). In addition, our formulation of the introduced Mittag-Leffler maximal function is a solution of a diffusion equation. Our results generalize previously known results.