Composition operators on Lizorkin–Triebel spaces

We deal with decay and boundedness properties of elements of radial subspaces of homogeneous Besov and Triebel-Lizorkin spaces. For the region of parameters which are of interest for us these homogeneous spaces are larger than the inhomogeneous counterparts. By switching from the inhomogeneous spaces to the homogeneous classes the properties of the radial elements change. Our investigations are based on the atomic decompositions for radial subspaces in the sense of Epperson and Frazier (J. Fourier Anal Appl. 1:311-353, 1995). Finally, we apply these results for deriving some assertions on compact embeddings on unbounded domains.

show abstract

“…A fractional order version is given by the following definition. [5]. Both references cover the case of Banach spaces only.…”

Section: Remark 14mentioning

confidence: 99%

“…Both references cover the case of Banach spaces only. However, the methods from [5] extend to the quasi-Banach space case.…”

Section: Remark 14mentioning

confidence: 99%

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

Sickel

Skrzypczak

2011

J Fourier Anal Appl

Self Cite

View full text Add to dashboard Cite

show abstract

“…This result is a variant of the Nikol'skij representation method, see 10, Proposition 2.3.2(1), p. 59], 12, 4, Proposition 4], 7, Proposition 2].…”

Section: Homogeneous Besov and Lizorkin‐triebel Spacesmentioning

confidence: 92%

Hit and Lead Profiling: Identification and Optimization of Drug‐like Molecules. Edited by B. Faller and L. Urban.

Wegscheid-Gerlach

2010

ChemMedChem

View full text Add to dashboard Cite

“…For a brief overview on homogeneous Triebel-Lizorkin spaces, we refer to [25,Chapter 5], [3] and also [21, Chapter 2].…”

Section: Preliminariesmentioning

confidence: 99%

Approximation in higher-order Sobolev spaces and Hodge systems

Bousquet

Russ

Wang

et al. 2019

Journal of Functional Analysis

View full text Add to dashboard Cite

Let d ≥ 2 be an integer, 1 ≤ l ≤ d − 1 and ϕ be a differential l-form on R d withẆ 1,d coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on R d with coefficients in L ∞ ∩Ẇ 1,d such that dϕ = dψ. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order spaceẆ 2,d/2 or more generally in the fractional Sobolev spacesẆ s,p with sp = d. We give a positive answer to this question, provided that d − κ ≤ l ≤ d − 1, where κ is the largest positive integer such that κ < min(p, d). The proof relies on an approximation result (interesting in its own right) for functions inẆ s,p by functions inẆ s,p ∩L ∞ , even thoughẆ s,p does not embed into L ∞ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.

show abstract

Composition operators on Lizorkin–Triebel spaces

Cited by 32 publications

References 31 publications

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

Hit and Lead Profiling: Identification and Optimization of Drug‐like Molecules. Edited by B. Faller and L. Urban.

Approximation in higher-order Sobolev spaces and Hodge systems

Contact Info

Product

Resources

About