A Characterization of Two-Weight Norm Inequality for Littlewood–Paley gλ∗ -Function

Let n≥ 2 and gλ∗ be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein, gλ∗(f)(x)=(∫∫R+n+1(tt+|x-y|)nλ|∇Ptf(y,t)|2dydttn-1)1/2,λ>1,where P_tf(y, t) = p_t∗ f(y) , p_t(y) = t^-np(y/ t) , and p(x) = (1 + | x| ²) ^-(n+1)/2, ∇=(∂∂y1,…,∂∂yn,∂∂t). In this paper, we give a characterization of two-weight norm inequality for gλ∗-function. We show that ‖gλ∗(fσ)‖L2(w)≲‖f‖L2(σ) if and only if the two-weight Muckenhoupt A₂ condition holds, and a testing condition holds: supQ:cubesinRn1σ(Q)∫Rn∫∫Q^(tt+|x-y|)nλ|∇Pt(1Qσ)(y,t)|2wdxdttn-1dy<∞,where Q^ is the Carleson box over Q and (w, σ) is a pair of weights. We actually prove this characterization for gλ∗-function associated with more general fractional Poisson kernel p^α(x) = (1 + | x| ²) ^-(n+α)/2. Moreover, the corresponding results for intrinsic gλ∗-function are also presented.