Decompositions with atoms and molecules for variable exponent Triebel-Lizorkin-Morrey spaces

We continue the study of the variable exponent Morreyfied Triebel-Lizorkin spaces introduced in a previous paper. Here we give characterizations by means of atoms and molecules. We also show that in some cases the number of zero moments needed for molecules, in order that an infinite linear combination of them (with coefficients in a natural sequence space) converges in the space of tempered distributions, is much smaller than what is usually required. We also establish a Sobolev type theorem for related sequence spaces, which might have independent interest.publishe

Generalized 2-microlocal Besov spaces

Wir verallgemeinern die von Bony 1984 eingeführten 2- mikrolokalen Funtionenräume auf die Besovraumskala. Wir beweisen die Charakterisierung der Räume durch lokale Mittel und folgern damit ein Theorem für punktweise Multiplikatoren und die Invarianz der Räume unter Diffeomorphismen. Im 5. Kapitel zeigen wir die Charakterisierung der Räume durch Zerlegungen mit Atomen, Molekülen und Wavelets. Mit Hilfe der Waveletcharakterisierung folgern wir anschließend Abbildungseigenschaften von Pseudodifferentialoperatoren, Verbindungen zu den Räumen variierender Glattheit und beweisen die Schärfe der Einbettungen. Das letzte Kapitel ist eine Anwendung der vorigen Theorie um zur lokalen Regularitätstheorie zu gelangen.We generalize the 2-microlocal spaces, introduced by Bony in 1984 onto the Besov spaces scale. We prove a charcaterzation by local means of these spaces and conclude a theorem on pointwise multipliers and the invariance of the spaces under diffeomorphisms. The 5 th chapter contains decompositions by atoms, molecules and wavelets. The waveletcharacterization leads us to mapping properties of pseudodifferential operators, connections to the spaces of varying smoothness and the sharpness of embeddings. The last chapter applies the developed theory to get connections to local regularity theory

Variable exponent Triebel-Lizorkin-Morrey spaces

We introduce variable exponent versions of Morreyfied Triebel-Lizorkin spaces. To that end, we prove an important convolution inequality which is a replacement for the Hardy-Littlewood maximal inequality in the fully variable setting. Using it we obtain characterizations by means of Peetre maximal functions and use them to show the independence of the introduced spaces from the admissible system used.publishe