$L^p$ bounds for the parabolic Littlewood-Paley operator associated to surfaces of revolution
Bull. Korean Math. Soc. 2012 Vol. 49, No. 4, 787-797
https://doi.org/10.4134/BKMS.2012.49.4.787
Published online July 1, 2012
Feixing Wang, Yanping Chen, and Wei Yu
University of Science and Technology Beijing, University of Science and Technology Beijing, University of Science and Technology Beijing
Abstract : In this paper the authors study the $L^p$ boundedness for parabolic Littlewood-Paley operator $$ \mu_{\Phi,\Omega}(f)(x)=\left(\int_0^\infty|F_{\Phi,t}(x)|^{2}\frac{dt}{t^{3}}\right)^{1/2}, $$ where $$ F_{\Phi,t}(x)=\int_{\rho(y)\leq t} {\frac{\Omega(y)}{\rho(y)^{\alpha-1}}f(x-\Phi(y))dy} $$ and $\Omega$ satisfies a condition introduced by Grafakos and Stefanov in [6]. The result in the paper extends some known results.
Keywords : parabolic Littlewood-Paley operator, rough kernel, surfaces of revolution
MSC numbers : 42B20, 42B25
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