Isotropic and dominating mixed Besov spaces: A comparison

Abstract

This paper is dedicated to the memory of Björn Jawerth.Abstract. We compare Besov spaces with isotropic smoothness with Besov spaces of dominating mixed smoothness. Necessary and sufficient conditions for continuous embeddings will be given.1991 Mathematics Subject Classification. Primary 46E35; Secondary 42B35. Key words and phrases. distribution spaces, isotropic Sobolev spaces, Sobolev spaces of dominating mixed smoothness, isotropic Besov spaces, Besov spaces of dominating mixed smoothness, embeddings.

Optimal regularity in time and space for the porous medium equation

“…Proof. This follows for bounded domains from [52,Theorem 4.6]; in our case of R d the same technique applies as mentioned in [49,Proposition 1.20] or [37,Proposition 4.4].…”

Isotropic and dominating mixed Lizorkin–Triebel spaces — a comparison

“…The known duality relations of the space S t p,q B(R d ), see [13] and references given there, yields…”

“…This example is taken from [23, 2.3.9], see also [13]. Let ̺ ∈ S(R d ) be a function such that supp F̺ ⊂ {ξ : |ξ| ≤ 1}.…”

“…There is no smallest space with this respect. We may follow the arguments we have used in the context of the analogous problem for Besov spaces, see [13]. From Theorem 3.1 we have…”

Pointwise multipliers for Besov spaces of dominating mixed smoothness - II

“…We refer to Hansen [8] and [13] for most of the details. In case p = ∞ we refer to Triebel [36, 2.5.1], in particular to Remark 7 there, where the isotropic case is treated.…”