A non-homogeneous local $Tb$ theorem for Littlewood-Paley $g_{\lambda }^{\ast }$-function with $L^p$-testing condition

In this paper, we present a local $Tb$ theorem for the non-homogeneous Littlewood-Paley $g_{\lambda }^{\ast }$-function with non-convolution type kernels and upper power bound measure $\mu$. We show that, under the assumptions $\supp b_Q \subset Q$,$|\int_Q b_Q d\mu| \gtrsim \mu(Q)$and$||b_Q||^p_{L^p(\mu)} \lesssim \mu(Q)$,thenorminequality$\big| g_{\lambda}^{*}(f) \big|{L^p(\mu)} \lesssim \big| f \big|{L^p(\mu)}$ holds if and only if the following testing condition holds : $$\underset{Q:cubesin\text{Rn}}{\sup }\frac{1}{\mu (Q)}{\int }_Q\left({\int }_0^{\mathrm{\ell }(Q)}{\int }_{\text{Rn}}\right(\frac{t}{t+\mathrm{\mid }x-y\mathrm{\mid }}{)}^{m\lambda }\mathrm{\mid }{\theta }_t(b_Q)(y,t){\mathrm{\mid }}^2\frac{d\mu (y)dt}{t^{m+1}}{)}^{p\mathrm{/}2}d\mu (x)<\mathrm{\infty }\mathrm{.}$$ This is the first time to investigate $g_{\lambda }^{\ast }$-function in the simultaneous presence of three attributes : local, non-homogeneous and $L^p$-testing condition. It is important to note that the testing condition here is $L^p$ type with $p \in (1,2]$.