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

Abstract

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 $||b_Q|{|}_{L^p(\mu )}^p\lesssim \mu (Q)$, the norm inequality $\Vert g_{\lambda }^{\ast }(f){\Vert }_{L^p(\mu )}\lesssim \Vert f{\Vert }_{L^p(\mu )}$ 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 $g_{\lambda }^{\ast }$-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 $p\in (1,2]$.

arXiv comment26 pages