Trace Operators in Besov and Triebel–Lizorkin Spaces

Abstract

We determine the trace of Besov spaces B s p,q (R n) and Triebel-Lizorkin spaces F s p,q (R n)characterized via atomic decompositions-on hyperplanes R m , n > m ∈ N, for parameters 0 < p, q < ∞ and s > 1 p. The limiting case s = 1 p is investigated as well. We generalize these assertions to traces on the boundary Γ = ∂Ω of bounded C k domains Ω. Our results remain valid considering the classical spaces B s p,q , F s p,q-defined via differences. Finally, we include some density assertions, which imply that the trace does not exist when s < 1 p .

Traces of Besov and Triebel‐Lizorkin spaces on domains

“…The dichotomy results from [6,Theorem 3.5] can now be generalized to smooth domains with boundary Γ = ∂Ω. P r o o f. Recalling Remark 4.3 the proof basically is the same as the proof of Theorem [6, Theorem 3.5] now using (3.7) instead of (3.3).…”

“…The idea for this paper originates from its forerunner [6], where traces on hyperplanes R m , n > m ∈ N, in these spaces were studied. The question came up whether there are corresponding results regarding traces on domains.…”

“…This paper aims at providing a rather final answer to this question. We extend the results from [6] to traces on the boundary ∂Ω of C k domains Ω, cf. Definition 2.6.…”

“…In Section 2 we define Besov and Triebel-Lizorkin spaces on domains Ω and their boundaries ∂Ω and collect auxiliary results. In Section 3 we extent the trace results from [6], to traces on the boundary of C k domains. Finally, Section 4 contains some density assertions in form of dichotomy numbers, which imply the non-existence of a trace for s < 1 p .…”

“…Hence (3.1) is always meaningful. We recall some trace results from [6] for B-and F-spaces on hyperplanes in R n . In particular, the results obtained there were rather complete and naturally extended the ones previously known for B s p,q (R n ) to small smoothness parameters when 0 < s < (n − 1) 1 p − 1 , 0 < p < 1 (in this context we also refer to [2] …”