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 $A_{p}^{lip_{\left( \alpha ,1\right) }^{\beta }}\left(\mathbb{R}^{d}\right) $ of functions $h$ belonging to Lipschitz space under fractional convolution whose fractional Fourier transforms $\mathcal{F}{_{\beta }}h$ belongs to {Lebesgue spaces}. We show that this space\ becomes a Banach algebra with the sum norm ${\left\Vert h\right\Vert }_{lip_{\left( \alpha,1\right) }^{\beta },p}={\left\Vert h\right\Vert _{\left( \alpha ,1\right)_{\beta }}} +{\left\Vert \mathcal{F}{{_{\beta }}h}\right\Vert _{p}}$ {and } $\Theta $ (fractional convolution) convolution operation. Also we indicate that this space becomes an essential Banach module over ${{L^{1}}\left(\mathbb{R}^{d}\right)}$ with $\Theta$ convolution.