Littlewood–Paley characterizations of Triebel–Lizorkin–Morrey spaces via ball averages. (February 2017) - Zhang, Junwei; Zhuo, Ciqiang; Yang, Dachun; He, Ziyi
Abstract: Let 1 < p ≤ u < ∞ and q ∈ ( 1, ∞ ] . In this article, the authors establish equivalent characterizations of the Triebel–Lizorkin–Morrey space E u, q, p α ( R n ) with smoothness order α ∈ ( 0, 2 ) in terms of the Lusin-area function and the g λ ∗ -function generated by the difference between f ( x ) and its ball average B 2 − j f ( x ) : = 1 | B ( x, 2 − j ) | ∫ B ( x, 2 − j ) f ( y ) d y, ∀ j ∈ 1, 2, … , ∀ x ∈ R n, where, for any j ∈ 1, 2, … and x ∈ R n, B ( x, 2 − j ) : = y ∈ R n : | y − x | < 2 − j . As applications, the authors obtain several characterizations of E u, ∞, p α ( R n ) via pointwise inequalities involving ball averages.