Inhomogeneous Lipschitz spaces of variable order and their applications

Abstract

In this article, the authors first give a Littlewood–Paley characterization for inhomogeneous Lipschitz spaces of variable order with the help of inhomogeneous Calderón identity and almost-orthogonality estimates. As applications, the boundedness of inhomogeneous Calderón–Zygmund singular integral operators of order $(ϵ,\sigma )$ on these spaces has been presented. Finally, we note that a class of pseudodifferential operators $T_a\in \mathcal{O}p{S}_{1,1}^0$ are continuous on the inhomogeneous Lipschitz spaces of variable order as a corollary. We may observe that those operators are not, in general, continuous in $L^2$.