Littlewood–Paley characterizations of Triebel–Lizorkin–Morrey spaces via ball averages

Abstract

Let 1<p≤u<∞ and q∈(1,∞]. In this article, the authors establish equivalent characterizations of the Triebel–Lizorkin–Morrey space Eu,q,pα(Rn) with smoothness order α∈(0,2) in terms of the Lusin-area function and the gλ∗-function generated by the difference between f(x) and its ball average B2−jf(x):=1|B(x,2−j)|∫B(x,2−j)f(y)dy,∀j∈{1,2,…},∀x∈Rn, where, for any j∈{1,2,…} and x∈Rn, B(x,2−j):={y∈Rn:|y−x|<2−j}. As applications, the authors obtain several characterizations of Eu,∞,pα(Rn) via pointwise inequalities involving ball averages.