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

It is well-known that the $L^p$ boundedness and weak $(1,1)$ estiamte $(λ>2)$ of the classical Littlewood-Paley $g_λ^{*}$-function was first studied by Stein, and the weak $(p,p)$ $(p>1)$ estimate was later given by Fefferman for $λ=2/p$. In this paper, we investigated the $L^p(μ)$ boundedness of the non-homogeneous Littlewood-Paley $g_{λ,μ}^{*}$-function with non-convolution type kernels and a power bounded measure $μ$: $$ g_{λ,μ}^*(f)(x) = \bigg(\iint_{{\mathbb R}^{n+1}_{+}} \Big(\frac{t}{t + |x - y|}\Big)^{m λ} |θ_t^μf(y)|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{1/2},\ x \in {\mathbb R}^n,\ λ> 1, $$ where $θ_t^μf(y) = \int_{{\mathbb R}^n} s_t(y,z) f(z) dμ(z)$, and $s_t$ is a non-convolution type kernel. Based on a big piece prior boundedness, we first gave a sufficient condition for the $L^p(μ)$ boundedness of $g_{λ,μ}^*$. 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 $L^p(μ)$ boundedness of $g_{λ,μ}^*$-function. It is worth noting that our testing conditions are weak $(1,1)$ type with respect to measures.