Generalized Fourier multipliers via Mittag-Leffler functions (Q6537084)

Fourier multipliers have played an important role in harmonic analysis since from the outset. They play a decisive role in studying several integral operators such as singular integral operators, oscillatory integral operators, maximal functions, and Littlewood-Paley \(g\)-functions, among others. In this paper, the authors introduce three rough integral operators in terms of the Mittag-Leffler function. The first operator is the Mittag-Leffler Spherical Maximal Function which is a generalization of the classical Spherical Maximal Function studied by \textit{E. M. Stein} [Proc. Natl. Acad. Sci. USA 73, 2174--2175 (1976; Zbl 0332.42018)]. They show that the introduced Mittag-Leffler Spherical Maximal Function is a solution of a diffusion equation. The second operator is a Littlewood-Paley \(g\)-function type operator. It generalizes the well-known classical Marcinkiewicz integral operator introduced by \textit{E. M. Stein} [Trans. Am. Math. Soc. 88, 430--466 (1961; Zbl 0105.05104)]. The third operator is the related Discrete Singular Integral Operator with rough kernels considered in [\textit{D. Fan} et al., Acta Math. Sin., New Ser. 14, No. 2, 235--244 (1998; Zbl 0906.42009); \textit{J. Duoandikoetxea} and \textit{J. L. Rubio de Francia}, Invent. Math. 84, 541--561 (1986; Zbl 0568.42012)]. They prove that the three operators are bounded on \(L^2(\mathbb{R}^n)\). However, it is still unknown whether these operators are bounded on \(L^p(\mathbb{R}^n)\) for \(p\) is not equal to 2. This problem shall be the topic of future research.