Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains

Abstract

In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.

Compact embeddings in Besov-type and Triebel–Lizorkin-type spaces on bounded domains

“…Remark 3.7. As mentioned above essentially the same proof for Triebel-Lizorkin-type spaces can be found in [52] with additional restrictions p, q ≥ 1. Note that, in view of the coincidence (2.16), also the Triebel-Lizorkin-Morrey spaces E s u,p,q are covered by our theorem.…”

“…Another not universal construction for the smooth domains was given by Moura, Neves and Schneider in [20]. The Rychkov universal extension operator for the Triebel-Lizorkin type spaces defined on Lipschitz domains was recently constructed by Zhou, Hovemann and Sickel in [52] with additional assumption p, q ∈ [1, ∞). Here we considered all admissible parameters p and q.…”

“…We follow Rychkov's approach from [23] and construct such an operator, for all possible values of p, q ∈ (0, ∞]. We learned only recently that in [51,52] the authors followed a similar approach to construct such an extension operator adapted to their purposes, that is, for p, q ≥ 1. For the convenience of the reader we keep our argument for the full range of parameters here.…”

Quarklet Characterizations for Triebel-Lizorkin Spaces

“…The reason for this is the more difficult behavior of complex interpolation in the context of the spaces E s u,r,q (R). More details concerning this topic can be found in [48], see in particular Proposition 1.6 in [48]. Now we prove the corresponding lower estimate.…”

“…Here the infimum is taken over all representations of the form (48). To prove this we proceed in a similar way as it is described in the proofs of the Propositions 5, 6 and 7.…”