$L^{p}$ BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES. (17th August 2015) - LIU, FENG; MAO, SUZHEN
Abstract : In an extrapolation argument, we prove certain $L^{p}\, (1<p<∞ )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions ${\rmΩ}\in L(\log ^{+}L)^{{\itα}}({\rmΣ})$ and the radial function $h\in {\mathcal{N}}_{{\itβ}}(\mathbb{R}^{+})$ . As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley $g_{{\itλ}}^{* }$ -functions are given.