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

We extend the Littlewood]Paley theorem to L p G., where G is a locally w compact Vilenkin group and w are weights satisfying the Muckenhoupt Ap condition. As an application we obtain a mixed-norm type multiplier result on L p G. and prove the sharpness of our result. We also obtain a sufficient condition w for f gL` G. to be a multiplier on the power weighted L p G. in terms of its a smoothness condition.