Bulletin of the
Korean Mathematical Society BKMS

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