On generalized Littlewood–Paley functions

Abstract

We study the Lp boundedness of certain classes of generalized Littlewood–Paley functions S(f). We obtain Lp estimates of S(f) with sharp range of p and under optimal conditions on ?. By using these estimates along with an extrapolation argument we obtain some new and improved results on generalized Littlewood–Paley functions. The approach in proving our results is mainly based on proving vector-valued inequalities and in turn the proof of our results (in the case ?= 2) provides us with alternative proofs of the results obtained by Duoandikoetxea as his approach is based on proving certain weighted norm inequalities.