$L^{p}$ BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES. (17th August 2015)

• Record Type: Journal Article
• Authors: LIU, FENG; MAO, SUZHEN
• Is Part Of: Journal of the Australian Mathematical Society. Volume 99:Number 3(2015:Dec.)
• Page Start: 380
• Page End: 398
• Publication Date: 2015-08-17
• Subjects: primary 42B20, -- secondary 42B25, -- 42B99; nonisotropic dilations, -- Marcinkiewicz integrals, -- rough kernels, -- extrapolation; Mathematics -- Periodicals; Statistics -- Periodicals; Mathematical statistics -- Periodicals; 510.5
• DOI: 10.1017/S1446788715000191
• Languages: English
• ISSNs: 1446-7887

Abstract

Abstract : In an extrapolation argument, we prove certain $L^{p}\, (1<p<\infty )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions ${\rm\Omega}\in L(\log ^{+}L)^{{\it\alpha}}({\rm\Sigma})$ and the radial function $h\in {\mathcal{N}}_{{\it\beta}}(\mathbb{R}^{+})$ . As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley $g_{{\it\lambda}}^{\ast }$ -functions are given.