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We study nuclear embeddings for spaces of Morrey type, both in its sequencespace version and as smoothness spaces of functions defined on a bounded domain$\Omega \subset {\mathbb R}^d$. This covers, in particular, the meanwhilewell-known and completely answered situation for spaces of Besov andTriebel-Lizorkin type defined on bounded domains which has been considered fora long time. The complete result was obtained only recently. Compact embeddingsfor function spaces of Morrey type have already been studied in detail, alsoconcerning their entropy and approximation numbers. We now prove the first andcomplete nuclearity result in this context. The concept of nuclearity hasalready been introduced by Grothendieck in 1955. Again we rely on suitablewavelet decomposition techniques and the famous Tong result (1969) whichcharacterises nuclear diagonal operators acting between sequence spaces of$\ell_r$ type, $1 \leq r \leq\infty$.

Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces