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Characterizations of some function spaces in terms of Haar wavelets

Triebel H.

Commentationes Mathematicae

2013

Vol. 53, No. 2

35--53

Some spaces Asp,q(Rn) with A = {B, F}, s ϵ R, 0 < p, q ≤ ∞, covering Besov spaces, Hölder-Zygmund spaces and Sobolev spaces, admit characterizations in terms of Haar bases. It is the main aim of this paper to extend this observation to corresponding Morreyfied spaces Lr Asp,q(Rn). As a by-product we obtain Littlewood-Paley theorems for (homogeneous and inhomogeneous) Morrey spaces Lrp(Rn), Lrp(Rn) and, in particular, L°rp(Rn).

The regularity of measures and related properties of eigenfunctions and first eigenvalues of some fractal elliptic operators

Triebel H.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr spec.

257-283

The paper deals with regularity properties of potentials of Radon measures u in IR^n, expressed in terms of some fractal quantities of u (Theorem 1). Based on these assertions, first eigenvalues and eigenfunctions of some fractal elliptic operators are considered (Theorems 2 and 3). The results are illustrated by examples.