A non-homogeneous local $Tb$ theorem for Littlewood-Paley $g_λ^{*}$-function with $L^p$-testing condition

In this paper, we present a local $Tb$ theorem for the non-homogeneous Littlewood-Paley $g_λ^{*}$-function with non-convolution type kernels and upper power bound measure $μ$. We show that, under the assumptions $\supp b_Q \subset Q$, $|\int_Q b_Q dμ| \gtrsim μ(Q)$ and $||b_Q||^p_{L^p(μ)} \lesssim μ(Q)$, the norm inequality $\big\| g_λ^{*}(f) \big\|_{L^p(μ)} \lesssim \big\| f \big\|_{L^p(μ)}$ holds if and only if the following testing condition holds : $$\sup_{Q : cubes \ in \ \Rn} \frac{1}{μ(Q)}\int_Q \bigg(\int_{0}^{\ell(Q)} \int_{\Rn} \Big(\frac{t}{t+|x-y|}\Big)^{mλ}|θ_t(b_Q)(y,t)|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{p/2} dμ(x) < \infty.$$ This is the first time to investigate $g_λ^*$-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 $p \in (1,2]$.

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