A Characterization of Two-Weight Norm Inequality for Littlewood-Paley g(lambda)*-Function

作者：Cao, Mingming; Li, Kangwei; Xue, Qingying^*

来源：Journal of Geometric Analysis, 2018, 28(2): 842-865.

DOI：10.1007/s12220-017-9844-x

• Two-weight inequality
• Littlewood-Paley g(lambda)*-function
• Pivotal condition
• Random dyadic grids

Let n >= 2 and g(lambda)* be the well-known high-dimensional Littlewood-Paley function which was defined and studied by E.M. Stein,
g(lambda)* (f) (x) = (integral integral R+n+1 (t/t + vertical bar x - y vertical bar)(n lambda) vertical bar del P-t f (y, t)vertical bar(2) dydt/t(n-1)) (1/2), lambda > 1,
where P-t f (y, t) = p(t) * f (y), p(t) (y) = t(-n) p(y/t), and p(x) = (1 + vertical bar x vertical bar(2)) (-(n+1)/2), del = (partial derivative/partial derivative y(1) , ... , partial derivative/partial derivative y(n), partial derivative/partial derivative(t)). In this paper, we give a characterization of two-weight norm inequality for g(lambda)*-function. We show parallel to g(lambda)*(f sigma)parallel to L-(omega)(2) less than or similar to parallel to f parallel to L-2 (sigma) if and only if the two-weight Muckenhoupt A(2) condition holds, and a testing condition holds:
sup(Q:cubes in Rn) 1/sigma (Q) integral(Rn) integral integral(Q) over cap (t/t + vertical bar x - y vertical bar(n lambda) vertical bar del P-t (1 Q sigma) (y, t)vertical bar(2) wdxdt/t(n-1) dy < infinity,
where <(Q)over cap> is the Carleson box over Q and (w, sigma) is a pair of weights. We actually prove this characterization for g(lambda)*-function associated with more general fractional Poisson kernel p(alpha) (x) = (1 + vertical bar x vertical bar(2)) (-(n+alpha)/2). Moreover, the corresponding results for intrinsic g(lambda)*-function are also presented.

• 出版日期2018-4
• 单位北京师范大学