$L^p$ boundedness of non-homogeneous Littlewood-Paley $g_{\lambda ,\mu }^{\ast }$-function with non-doubling measures

Abstract

It is well-known that the $L^p$ boundedness and weak $(1,1)$ estiamte $(\lambda >2)$ of the classical Littlewood-Paley $g_{\lambda }^{\ast }$-function was first studied by Stein, and the weak $(p,p)$ $(p>1)$ estimate was later given by Fefferman for $\lambda =2/p$. In this paper, we investigated the $L^p(\mu )$ boundedness of the non-homogeneous Littlewood-Paley $g_{\lambda ,\mu }^{\ast }$-function with non-convolution type kernels and a power bounded measure $\mu$: $$g_{\lambda ,\mu }^{\ast }(f)(x)=\left({\iint }_{{\mathbb{R}}_{+}^{n+1}}\right(\frac{t}{t+|x-y|}{)}^{m\lambda }|{\theta }_t^{\mu }f(y){|}^2\frac{d\mu (y)dt}{t^{m+1}}{)}^{1/2},x\in {\mathbb{R}}^n,\lambda >1,$$ where ${\theta }_t^{\mu }f(y)={\int }_{{\mathbb{R}}^n}{s}_t(y,z)f(z)d\mu (z)$, and $s_t$ is a non-convolution type kernel. Based on a big piece prior boundedness, we first gave a sufficient condition for the $L^p(\mu )$ boundedness of $g_{\lambda ,\mu }^{\ast }$. This was done by means of the non-homogeneous good lambda method. Then, using the methods of dyadic analysis, we demonstrated a big piece global $Tb$ theorem. Finally, we obtaind a sufficient and necessary condition for $L^p(\mu )$ boundedness of $g_{\lambda ,\mu }^{\ast }$-function. It is worth noting that our testing conditions are weak $(1,1)$ type with respect to measures.

32 pages