Hausdorff Besov-type and Triebel–Lizorkin-type spaces and their applications

Abstract

Let s∈R,p∈(1,∞) and τ∈(0,1/p′], where p′ denotes the conjugate index of p. In this article, the authors first introduce the Hausdorff Besov-type space BH˙p,qs,τ(Rn) with q∈[1,∞) and the Hausdorff Triebel–Lizorkin-type space FH˙p,qs,τ(Rn) with q∈(1,∞) via a class of weights defined on the upper plane, and then establish some equivalent characterizations of BH˙p,qs,τ(Rn) and FH˙p,qs,τ(Rn) via some classes of weights defined on Rn. The authors then prove that their dual spaces are Besov–Morrey and Triebel–Lizorkin–Morrey spaces, respectively. The relations between these spaces and the known Besov–Triebel–Lizorkin–Hausdorff spaces are also studied. As an application, for p∈(1,∞) and λ∈(0,n), the authors obtain the coincidence between the space FH˙p,20,(n−λ)/(np′)(Rn) and the predual space, Hp,λ(Rn), of the Morrey space Lp,λ(Rn). Moreover, characterizations of BH˙p,qs,τ(Rn) and FH˙p,qs,τ(Rn) via local means and Peetre maximal functions, as well as the boundedness of Riesz potential operators and some singular integrals on BH˙p,qs,τ(Rn) and FH˙p,qs,τ(Rn) are also obtained.