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

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.