Local Growth Envelopes of Triebel-Lizorkin Spaces of Generalized Smoothness

Abstract

The concept of local growth envelope (E LG A, u) of the quasi-normed function spaceA is applied to the Triebel-Lizorkin spaces of generalized smoothness F σ,N p,q (R n ). In order to achieve this, a standardization result for these and corresponding Besov spaces is derived.

Continuity Envelopes of Spaces of Generalized Smoothness in the Critical Case

“…The case of being an admissible function is covered by [8,Proposition 3.11]. We refer to [3, Corollary 3.10 & Remark 3.11] and to [6,Proposition 4.4] for a more general situation. Theorem 2.14 Let 0 < p, q ≤ ∞ and be a slowly varying function.…”

Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces

“…We apply some results proved in those papers, namely characterisations with atomic decompositions and with differences. We also use a standardisation result proved by Caetano and Leopold [7] to reduce these spaces to corresponding spaces where usual dyadic decompositions can be taken.…”

“…Following the proof of [7,Theorem 1,p. 432] one can conclude that the equivalence constants in (3.1.1) are independent of k.…”

Sharp embeddings of Besov spaces involving only logarithmic smoothness

“…Definition 2.1 in Section 2. By the Lorentz-Zygmund space L loc p,q; we mean the set of all measurable functions on R n with the finite quasi-norm 1 0 t q/p (1 + | ln t|) q f * (t) q dt t 1/q (1) (with the usual modification when q = ∞). First, Theorem 3.1 mentioned below states that the (continuous) embedding (2) with = + 1/r + 1/ max{p, q} − 1/q (3) holds if and only if q r. Consequently, when q r, (2) holds with any satisfying…”