Oscillations and differences in Triebel-Lizorkin-Morrey spaces

06/27/2023

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by   Marc Hovemann, et al.

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In this paper we are concerned with Triebel-Lizorkin-Morrey spaces ℰ^s_u,p,q(Ω) of positive smoothness s defined on (special or bounded) Lipschitz domains Ω⊂ℝ^d as well as on ℝ^d. For those spaces we prove new equivalent characterizations in terms of local oscillations which hold as long as some standard conditions on the parameters are fulfilled. As a byproduct, we also obtain novel characterizations of ℰ^s_u,p,q(Ω) using differences of higher order. Special cases include standard Triebel-Lizorkin spaces F^s_p,q (Ω) and hence classical L_p-Sobolev spaces H^s_p(Ω). Key words: Triebel-Lizorkin-Morrey space, Morrey space, Lipschitz domain, oscillations, higher order differences