Littlewood–Paley and Multiplier Theorems on Weighted Spaces over Locally Compact Vilenkin Groups

Abstract

We extend the Littlewood–Paley theorem toLpw(G), whereGis a locally compact Vilenkin group andware weights satisfying the MuckenhouptApcondition. As an application we obtain a mixed-norm type multiplier result onLpw(G) and prove the sharpness of our result. We also obtain a sufficient condition for φ∈L∞(Γ) to be a multiplier on the power weightedLpα(G) in terms of its smoothness condition.