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

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 B H ˙ p , q s , τ ( R n ) with q ∈ [ 1 , ∞ ) and the Hausdorff Triebel–Lizorkin-type space F H ˙ p , q s , τ ( R n ) with q ∈ ( 1 , ∞ ) via a class of weights defined on the upper plane, and then establish some equivalent characterizations of B H ˙ p , q s , τ ( R n ) and F H ˙ p , q s , τ ( R n ) via some classes of weights defined on R n . 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 F H ˙ p , 2 0 , ( n − λ ) / ( n p ′ ) ( R n ) and the predual space, H p , λ ( R n ) , of the Morrey space L p , λ ( R n ) . Moreover, characterizations of B H ˙ p , q s , τ ( R n ) and F H ˙ p , q s , τ ( R n ) via local means and Peetre maximal functions, as well as the boundedness of Riesz potential operators and some singular integrals on B H ˙ p , q s , τ ( R n ) and F H ˙ p , q s , τ ( R n ) are also obtained.