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The main purpose of the present paper is to extend the theory of non-smooth atomic decompositions to anisotropic function spaces of Besov and Triebel-Lizorkin type. Moreover, the detailed analysis of the anisotropic homogeneity property is carried out. We also present some results on pointwise multipliers in special anisotropic function spaces. CMUC; Junior Research Team Fractal analysis

The present paper is devoted to the study of growth envelopes of anisotropic function spaces. An affirmative answer is given to the question of [19, Conjecture 13], whether the growth envelopes are independent of anisotropy. As an application, related anisotropic Hardy inequalities are presented and we also discuss a connection to some anisotropic fractal sets. CMUC

We present a general method for extending decomposition systems of L 2(ℝd) to decomposition systems for the anisotropic Triebel-Lizorkin and Besov spaces, Fp,qα,s and Bp,qα,s, respectively, for the full range of the indexes. Our approach is based on techniques from harmonic analysis and relies on the boundedness of almost diagonal operators on appropriate sequence spaces. Typical examples of such decomposition systems are the various wavelet-type unconditional bases for L2(ℝd). © 2004 Society for Industrial and Applied Mathematics. 36 2 441 462 Cited By :6

We study multiplication as well as Nemytskij operators in anisotropic vector-valued Besov spaces $B^{s, \omega}_p$, Bessel potential spaces $H^{s, \omega}_p$, and Sobolev-Slobodeckij spaces $W^{s, \omega}_p$. Concerning multiplication we obtain optimal estimates, which constitute generalizations and improvements of known estimates in the isotropic/scalar-valued case. Concerning Nemytskij operators we consider the acting of analytic functions on supercritial anisotropic vector-valued function spaces of the above type. Moreover, we show how the given estimates may be used in order to improve results on quasilinear evolution equations as well as their proofs., Comment: 44 pages

In this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities. To a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in Sobolev and mixed derivative embeddings with fixed integrability. We apply the general results to prove maximal \(L^p\)-\(L^q\)-regularity for the linearized, fully inhomogeneous two-phase Stefan problem with Gibbs–Thomson correction.

In this paper, we study anisotropic Bessel potential and Besov spaces, where the anisotropy measures the extra amount of regularity in certain directions. Some basic properties of these spaces are established along with applications to elliptic boundary value problems.

This work deals with decompositions in anisotropic function spaces. Defining anisotropic atoms as smooth building blocks which are the counterpart of the atoms from the works of M. Frazier and B. Jawerth, it is shown that the study of anisotropic function spaces can be done with the help of some sequence spaces in a similar way as it is done in the isotropic case. It is also shown that the subatomic decomposition theorem for isotropic function spaces, recently proved by H. Triebel, can be extended to the anisotropic case if the mean smoothness parameter is sufficiently large.

In L-2((0,1)(2)) infinitely many different biorthogonal wavelet bases may be introduced by taking tenser products of one-dimensional biorthogonal wavelet bases on the interval (0, 1). Most well-known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases. In L-2((0,1)(2)) infinitely many different biorthogonal wavelet bases may be introduced by taking tenser products of one-dimensional biorthogonal wavelet bases on the interval (0, 1). Most well-known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.

In this survey we review the recently developed theory of anisotropic spaces and representations of functions based on anisotropic multilevel ellipsoid covers (dilations) of ℝ n . We also exhibit the relations of the ellipsoid cover approach to earlier concepts of anisotropic structures as well as to the framework of general spaces of homogeneous type. A number of open problems are presented and discussed.

Estimates for Tikhonov regularized solutions of an integral equation of the first kind of convolution type are presented. They are obtained by using anisotropic Nikol'skii-Besov spaces. This article extends the results, obtained by V.I. Burenkov, I.F. Dorofeev, and A.S. Pankratov for isotropic Nikol'skii-Besov spaces, to the anisotropic case.

In this paper we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equation, where uniform trace estimates on the half-line are shown., Comment: Some typos corrected. Accepted for publication in Mathematische Nachrichten