Generalized Littlewood-Paley functions on product spaces

Authors: HUSSAIN ALQASSEM, LESLIE CHENG, YIBIAO PAN

Abstract: We are interested in investigating the $L^{p}$ boundedness of the product of generalized Littlewood-Paley functions $S_{\Phi }^{(\lambda )}(f)$ arising from kernels satisfying only size and cancellation conditions. We obtain $L^{p}$ estimates of $S_{\Phi }^{(\lambda )}(f)$ for a sharp range of $p$ and under optimal conditions on $\Phi $. 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.

Keywords: Littlewood-Paley functions, Triebel-Lizorkin spaces, Orlicz spaces, block spaces, extrapolation, $L^{p}$

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