On multilinear square function and its applications to multilinear Littlewood-Paley operators with non-convolution type kernels

摘要

Let m >= 2, n >= 1 and x epsilon R-n define the multilinear square function T by T((f) over right arrow) (x) = (integral(infinity)(0)| integral(R-n)(m) K-t(x,y1,...,y(m)) Pi(m)(j) =1 f(j) (y(j) ) dy(1)...dy(m)|2dt/t)(1/2), where the kernel K satisfies a class of integral smooth conditions which is much weaker than the standard Calderon-Zygmund kernel conditions. In this paper, we first established the L-P1(w(1)) x...x L-pm (Wm) -> L-P(v (w) over right arrow) estimate of T when each p(i) > 1 and weak type L-P1 (w(1)) x...x L-pm (Wm) -> L-P,L-infinity(v (w) over right arrow) estimate of T when there is a p(i) = 1, where v (w) over right arrow = Pi(m)(j) =1w(i)(p/pi) and each w(i) is a nonnegative function on R-n. As applications of the above results, we obtained the boundedness of multilinear Littlewood Paley operators with non-convolution type kernels, including multilinear g-function, Marcinkiewicz integral and g(lambda)*-function. (C) 2014 Elsevier Inc. All rights reserved.