arXiv:2105.14546v1 [math.CV] 30 May 2021 The Mittag Leffler space & its Littlewood- Paley Identity Himanshu Singh Abstract. We present the Littlewood-Paley Identity for the Mittag Leffler space ML 2 (C; α) of entire functions. We also briefly demonstrate the connection between the Littlewood-Paley Identity and the compact- ness of the weighted composition operator on ML 2 (C; α). Keywords. Mittag Leffler, Littlewood-Paley Identity, Fock space, Fu- bini theorem, Reproducing Kernel Hilbert space, Exponential Integrals, Compact, Weighted Composition Operator, Caputo fractional deriva- tive. 1. Introduction This paper is the chief avenue for the Littlewood-Paley Identity of the Mittag- Leffler space of entire functions for α> 0, denoted by ML 2 (C; α). ML 2 (C; α) was introduced by Rosenfeld [RRD18] to facilitate a numerical method for estimating the Caputo fractional derivative of a function, see [RD17]. It can also be envisioned as one parameter parametrization of the Fock space F 2 (C). A little to nothing is known about the Littlewood-Paley Identity for ML 2 (C; α). Therefore determining the same becomes more challenging and compelling at the same time. Recently, the Littlewood-Paley type Identities were manifested for the Bergman space and the Dirichlet space in [Sin21b] and in the Polylogarithmic Hardy space as well [Sin21a]. With the advantage of the fact that F 2 (C) can be recovered from ML 2 (C; α), when α = 1, implies that we eventually get the Littlewood-Paley Identity for F 2 (C). As a matter of fact, the estimate of Littlewood-Paley Identity is already established in Theorem 19 of [CP16]. But since that is an estimate, therefore the purpose of determining the perfect equality becomes more profound. The paper is designed as follows: • Section 2 presents the introduction of ML 2 (C; α),