On function spaces of Lorentz–Sobolev type

Abstract

We work with Triebel-Lizorkin spaces F s q L p,r (R n ) and Besov spaces B s q L p,r (R n ) with Lorentz smoothness. Using their characterizations by real interpolation we show how to transfer a number of properties of the usual Triebel-Lizorkin and Besov spaces to the spaces with Lorentz smoothness. In particular, we give results on diffeomorphisms, extension operators, multipliers and we also show sufficient conditions on parameters for F s q L p,r (R n ) and B s q L p,r (R n ) to be multiplication algebras.

Traces of some weighted function spaces and related non‐standard real interpolation of Besov spaces

“…In [33, Chapter 4] and [38, Section 2.2.1], Triebel describes four key‐problems for Triebel–Lizorkin and Besov spaces $$A_{p,q}^{s}(\mathbb {R}^{n})$$, $$A \in \lbrace B,F\rbrace$$: traces on hyperplanes, invariance with respect to diffeomorphisms of $$\mathbb {R}^{n}$$ onto itself, the existence of linear extension operators of the corresponding spaces $$A_{p,q}^{s}\big (\mathbb {R}^{n}_{+}\big )$$ on $$\mathbb {R}^{n}_{+}$$ to $$A_{p,q}^{s}(\mathbb {R}^{n})$$ and several types of pointwise multipliers. For $$F_{q}^{s}L_{p,r}(\mathbb {R}^{n})$$, three of these key‐problems can be treated satisfactorily using the interpolation formula (4.2) (see [5]). However, this is not the case for the problem of traces as we are going to see now.…”

“…Based on our results in Section 3 we can now easily describe the resulting space via its wavelet representation. Here we benefit from the very recent outcome in [5]. In the very end we present an alternative argument which works for all $$\alpha \in \mathbb {R}$$.…”

Traces of some weighted function spaces and related non-standard real interpolation of Besov spaces

“…In [31,Chapter 4] and [36, Section 2.2.1], Triebel describes four key-problems for Triebel-Lizorkin and Besov spaces A s p,q (R n ), A ∈ {B, F }: traces on hyperplanes, invariance with respect to diffeomorphisms of R n onto itself, the existence of linear extension operators of the corresponding spaces A s p,q (R n + ) on R n + to A s p,q (R n ) and several types of pointwise multipliers. For F s q L p,r (R n ), three of these key-problems can be treated satisfactorily using the interpolation formula (4.2) (see [4]). However, this is not the case for the problem of traces as we are going to see now.…”

“…Based on our results in Section 3 we can now easily describe the resulting space via its wavelet representation. Here we benefit from the very recent outcome in [4]. In the very end we present an alternative argument which works for all α ∈ R.…”

Wavelets and Real Interpolation of Besov Spaces

“…For the real interpolation of Besov spaces, we can refer to [9][10][11][12][13][14][15][16]. When the index p is fixed, it has been shown that (B s 0 ,q 0 p , B s 1 ,q 1 p ) θ,r are still Besov spaces, see [4,9,16].…”