A family of Dirichlet–Morrey spaces

Abstract

To each weighted Dirichlet space D p , 0 < p < 1, we associate a family of Morrey-type spaces D λ p , 0 < λ < 1, constructed by imposing growth conditions on the norm of hyperbolic translates of functions. We indicate some of the properties of these spaces, mention the characterization in terms of boundary values, and study integration and multiplication operators on them.Recall that the space BMOA consists of all functions f ∈ H 2 whose boundary values f (e iθ ) have bounded mean oscillation, that is,

Characterizations of Morrey type spaces

“…We will consider the embedding map I : D s K → T s K (µ) and find sufficient or necessary condition for the operator I to be bounded. As an extention of results of [6,11,15,19,28], we state the following theorem. Theorem 3.1.…”

“…In 1938, Morrey spaces were introduced for solutions of partial differential equations and were subsequently studied as function classes in harmonic analysis on Euclidean spaces; see [12,18,31]. The analytic Morrey spaces have attracted a lot of attention in recent years, for example, [6,9,10,23,24].…”

Higher Schwarzian derivative and Dirichlet Morrey space

“…Some lemmas are needed. The following result gives some higher derivative characterizations of D p λ (D) (see [12]).…”

“…[12] Let f be an analytic function on D and 0 < p, λ ≤ 1. Then dµ(z) = | f (z)| 2 (1 − |z| 2 ) p dm(z) is a pλ-Carleson measure if and only if dν(z)…”

Dirichlet-Morrey type spaces and Volterra integral operators

“…The Volterra integral operator T g was studied by many authors recently. For more results on operator T g , we refer to [5,7,8,9,10,11,12,13,14,15,16,17,18,19] and the references therein For any I ⊂ ∂ D, the boundary of D, let |I| be the normalized arc length of I. Let…”