We study the Triebel-Lizorkin-Morrey spaces  𝑠 𝑢,𝑝,𝑞and prove that under certain sufficient conditions on the parameters these spaces can be characterised in terms of higher-order differences. Moreover we show that some of the mentioned conditions are also necessary.

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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.

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We will prove that under certain conditions on the parameters the operators T + f = max(f, 0) and T f = |f | are bounded mappings on the Triebel-Lizorkin-Morrey and Besov-Morrey spaces. Moreover we will show that some of the conditions we mentioned before are also necessary. Furthermore we prove that for p < u in many cases the Triebel-Lizorkin-Morrey spaces do not have the Fubini property.