Littlewood-Paley Characterizations of Weighted Anisotropic Triebel-Lizorkin Spaces via Averages on Balls II

This article is the second part of two works of the authors on the same topic. Let Ad, be the matrix diag{2a',..., 2a.}, with a := (ai,..., an) E (0, o)n, and let w E A (Ad) be a Muckenhoupt Aoc-weight with respect to A. In this article, the authors characterize the weighted anisotropic Triebel Lizorkin space F2,',q(Ad,; w) with smoothness order a E (0, 2(_) in terms of the Lusin-area function and the Littlewood Paley gA*-function, defined via the difference between f (x) and its ball average Bb -k f (X) : 1Bp(x, b -k)1 fgp(x,b-k) f (y) dg, V X E II) Vk E {1,2,...}, where b := Idet And, a(Ad,) denotes the set of all eigenvalues of Ad, zeta E (1, minflAl : zeta E 0-(Ad)}], zeta_ := log zeta_. Further, p denotes the step homogeneous quasi -norm associated with Ad, and, for any k E {1, 2,...} and x E 71, Bp(x,b k) := {y E r:Th : p(x y) < Irk}. As applications, the authors obtain a series of characterizations for weighted anisotropic Triebel Lizorkin spaces Fp (Ad; w) via pointwise inequalities involving ball averages.