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.
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