Generalized Littlewood-Paley functions on product spaces

Abstract

We are interested in investigating the L p boundedness of the product of generalized Littlewood–Paley functions S (λ) Φ (f) arising from kernels satisfying only size and cancellation conditions. We obtain L p estimates of S (λ) Φ (f) for a sharp range of p and under optimal conditions on Φ. Using these estimates and an extrapolation argument, we obtain some new and improved results on generalized Littlewood–Paley functions on product spaces. As a consequence of our main results, we get two results, one of which answers a question posed by D. Fan and H. Wu and the other one answers a question raised by Y. Wu and H. Wu. In addition, one of our lemmas on Triebel–Lizorkin spaces answers a question posed by Y. Wu and H. Wu.