new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and H?lder-Zygmund spaces on the one hand and Morrey-Campanato spaces on the other. The obtained assertions about function spaces and nonlinear heat equations are used in Chapters 5 and 6 to study Navier-Stokes equations in hybrid and global spaces. This book is addressed to graduate students and mathematicians who have a working knowledge of basic elements of (global) function spaces and who are interested in applications to nonlinear PDEs with heat and Navier-Stokes equations as prototypes. The Navier-Stokes equations are invariant under a particular change of time and space scaling. More exactly, assume that, in R3 × (0, ∞), v(t, x) and p(t, x) solve the system. (129). Critical spaces are all embedded in a same function space, as stated in the following proposition. Proposition 7 (A remarkable embedding). If X is a critical space, then X is continuously embedded in the Besov space.

Hybrid Function Spaces, Heat and Navier-stokes Equations

The theory of generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which Mizuhara and Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of the atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.

Generalized Morrey spaces and trace operator

We study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e. g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the ±-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasis is given to the quasi-Banach case and to the interpolation property.

Interpolation of Morrey-Campanato and related smoothness spaces

In this article, the authors study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e.\,g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the $\pm$-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasize is given to the quasi-Banach case and to the interpolation property.

Interpolation of Morrey-Campanato and Related Smoothness Spaces

We study embeddings of spaces of Besov-Morrey type, idΩ : N s1 p1,u1,q1(Ω) ↪→ N s2 p2,u2,q2(Ω), where Ω ⊂ R d is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of idΩ. This continues our earlier studies in [HS] related to the situation on Rd. Moreover, we can also characterise embeddings into the scale of Lp spaces or into the space of bounded continuous functions.

Embeddings of Besov-Morrey spaces on bounded domains

We consider local means with bounded smoothness for Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r-regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov-Triebel-Lizorkin spaces.

Local means, wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

This paper deals with mapping properties of classical Calderon–Zygmund operators in local and global Morrey spaces.

Calderón–Zygmund operators in Morrey spaces

This survey deals with Morrey spaces, their duals and preduals in the framework of tempered distributions on Euclidean spaces. We concentrate on basic assertions (including density and non-separability), duality, embeddings (including relations to distinguished Besov spaces) and applications to Calderon–Zygmund operators.

Morrey spaces, their duals and preduals

We prove that operators satisfying the hypotheses of the extrapolation theorem for Muckenhoupt weights are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt $$A_1$$
 weighted $$L_p$$
 spaces, which enable us to define the operators on the considered Morrey spaces by restriction. In this way, we can avoid the delicate problem of the definition of the operator, often ignored by the authors. In dealing with the extension problem through the embeddings (instead of using duality), one is neither restricted in the parameter range of the p’s (in particular $$p=1$$
 is admissible and applies to weak-type inequalities) nor the operator has to be linear. Another remarkable consequence of our results is that vector-valued inequalities in Morrey spaces are automatically deduced. On the other hand, we also obtain $$A_\infty $$
 -weighted inequalities with Morrey quasinorms.

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

The bidual of the closure of smooth functions with respect to the Morrey norm coincides with the Morrey space. This assertion is generalized to some Muckenhoupt weighted Morrey spaces. We combine this fact with basic extrapolation techniques due to Rubio de Francia adapted to weighted Morrey spaces. This leads to new results on the boundedness of operators even for the unweighted case.

Marcel Rosenthal

Papers

Local means, wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

Calderón–Zygmund operators in Morrey spaces

Morrey spaces, their duals and preduals

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

The boundedness of operators in Muckenhoupt weighted Morrey spaces via extrapolation techniques and duality