Lipschitz Spaces Under Fractional Convolution

Abstract

We describe a generalization of Lipschitz spaces under fractional convolution and discuss some basic properties of these spaces. The aim of this work is to introduce and study a linear space Alip p β(α,1) (Rd ) of functions h belonging to Lipschitz space under fractional convolution whose fractional Fourier transforms F<inf>β</inf>h belongs to Lebesgue spaces. We show that this space becomes a Banach algebra with the sum norm ∥h∥<inf>lip</inf>β <inf>(</inf>α,<inf>1),p</inf> = ∥h∥<inf>(α,1)β</inf> + ∥F<inf>β</inf>h∥<inf>p</inf> and Θ (fractional convolution) convolution operation. Also we indicate that this space becomes an essential Banach module over L1 (Rd) with Θ convolution. © 2023 Tbilisi Centre for Mathematical Sciences. All rights reserved.