A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp -testing condition

Abstract

Abstract In this paper, we present a local Tb theorem for the non-homogeneous Littlewood–Paley g λ * {g_{\lambda}^{*}} -function with non-convolution type kernels and upper power bound measure μ. Actually, we show that, under the assumptions supp ⁡ b Q ⊂ Q {\operatorname{supp}b_{Q}\subset Q} , | ∫ Q b Q d μ | ≳ μ ( Q ) {\lvert\int_{Q}b_{Q}\,d\mu|\gtrsim\mu(Q)} and ∥ b Q ∥ L p ⁢ ( μ ) p ≲ μ ⁢ ( Q ) {\|b_{Q}\|^{p}_{L^{p}(\mu)}\lesssim\mu(Q)} , the norm inequality ∥ g λ * ⁢ ( f ) ∥ L p ⁢ ( μ ) ≲ ∥ f ∥ L p ⁢ ( μ ) {\|g_{\lambda}^{*}(f\/)\|_{L^{p}(\mu)}\lesssim\|f\/\|_{L^{p}(\mu)}} holds if and only if the following testing condition holds: sup Q : cubes in ⁢ ℝ n 1 μ ⁢ ( Q ) ∫ Q ( ∫ 0 ℓ ⁢ ( Q ) ∫ ℝ n ( t t + | x - y | ) m ⁢ λ | θ t ( b Q ) ( y , t ) | 2 d ⁢ μ ⁢ ( y ) ⁢ d ⁢ t t m + 1 ) p / 2 d μ ( x ) < ∞ . \sup_{Q:\text{cubes in }\mathbb{R}^{n}}\frac{1}{\mu(Q)}\int_{Q}\Biggr{(}\int_{% 0}^{\ell(Q)}\int_{\mathbb{R}^{n}}\Bigl{(}\frac{t}{t+|x-y|}\Big{)}^{m\lambda}|% \theta_{t}(b_{Q})(y,t)|^{2}\frac{d\mu(y)\,dt}{t^{m+1}}\Bigg{)}^{p/2}d\mu(x)<\infty. This is the first time to investigate the g λ * {g_{\lambda}^{*}} -function in the simultaneous presence of three attributes: local, non-homogeneous and L p {L^{p}} -testing condition. It is important to note that the testing condition here is an L p {L^{p}} type with p ∈ ( 1 , 2 ] {p\in(1,2]} .