It is well-known that the $L^p$ boundedness and weak $(1,1)$ estiamte $(\lambda>2)$ of the classical Littlewood-Paley $g_{ \lambda}^{ *}$-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 $g_{ \lambda,\mu}^{ *}$-function with non-convolution type kernels and a power bounded measure $\mu$: $$ g_{ \lambda,\mu}^*(f)(x) = \bigg(\iint_{ { \mathbb R}^{ n+1}_{ +}} \Big(\frac{ t}{ t + |x - y|}\Big)^{ m \lambda} |\theta_t^\mu f(y)|^2 \frac{ d\mu(y) dt}{ t^{ m+1}}\bigg)^{ 1/2},\ x \in { \mathbb R}^n,\ \lambda > 1, $$ 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 $g_{ \lambda,\mu}^*$. 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 $g_{ \lambda,\mu}^*$-function. It is worth noting that our testing conditions are weak $(1,1)$ type with respect to measures.
Categories: math.CA
