Function Spaces Of Besov Type Research Articles

Path Regularity of the Brownian Motion and the Brownian Sheet

By the work of P. Lévy, the sample paths of the Brownian motion are known to satisfy a certain Hölder regularity condition almost surely. This was later improved by Ciesielski, who studied the regularity of these paths in Besov and Besov-Orlicz spaces. We review these results and propose new function spaces of Besov type, strictly smaller than those of Ciesielski and Lévy, in which the sample paths of the Brownian motion almost surely lie. In the same spirit, we review and extend the work of Kamont, who investigated the same question for the multivariate Brownian sheet and function spaces of dominating mixed smoothness.

Fourier–Jacobi harmonic analysis and some problems of approximation of functions on the half-axis in L2 metric: Nikol'skii–Besov type function spaces

ABSTRACTWe use the methods of Fourier–Jacobi harmonic analysis to study problems of the approximation of functions in weighted function spaces on the half-axis . We define function spaces of Nikol'skii–Besov type and describe them in terms of best approximations. We use as the approximation tool a class of functions with the bounded spectrum, that is a class of functions for which their Fourier–Jacobi transform are functions with compact support.

Almost Diagonal Matrices and Besov-Type Spaces Based on Wavelet Expansions

This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $$\Psi $$ on domains or manifolds $$\Gamma $$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ , provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces $$b^\alpha _{p,q}(\nabla )$$ of Besov type is developed.

Global well-posedness and ill-posedness for the Navier–Stokes equations with the Coriolis force in function spaces of Besov type

We consider the initial value problems for the Navier–Stokes equations in the rotational framework. We introduce function spaces B˙p,qs(R3) of Besov type, and prove the global in time existence and the uniqueness of the mild solution for small initial data in our space B˙1,2−1(R3) near BMO−1(R3). Furthermore, we also discuss the ill-posedness for the Navier–Stokes equations with the Coriolis force, which implies the optimality of our function space B˙1,2−1(R3) for the global well-posedness.

Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type

We introduce Besov type function spaces, based on the weak L p -spaces instead of the standard L p -spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of perfect incompressible fluid in $${\mathbb{R}}^n , n \geq 3$$ . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces.

A rate of convergence for the [formula omitted] regularization of Navier–Stokes equations

A rate of convergence of the solutions of the LANS α equations with periodic boundary to the solutions of the Navier–Stokes equations as α ↓ 0 is obtained in a mixed L 1 – L 2 time–space norm for small initial data in Besov-type function spaces in which global existence and uniqueness of solutions can also be established.

The dimension of a closed subset ofR n and related function spaces

Let F 6= ∅ be a closed subset of Rn with empty interior. There are several proposals what should be called the dimension of F , globally and locally. Besides the classical Hausdorff dimension there exist nearby but, in general, not identical definitions, better adapted to the needs of measure theory, see [14] and also [9] and [7]: Ch.II,1. One aim of our paper is to contribute to this field of research by introducing two types of a dimension of F , the distributional dimension and the cascade dimension. The first notion is connected with the question whether there exist nontrivial singular distributions with a support on F and belonging to some function spaces of Besov type on Rn. The second notion is related to the eentropy and ecapacity of F and its neighbourhood and is connected with atomic representations of function spaces. Our approach is intimately linked with function spaces on Rn and on F . Hence, the second aim of this paper is to introduce some function spaces of Besov type on F . In that sense this note might be considered as a direct continuation of our paper [13]. On the other hand function spaces on (closed) subsets of Rn have been studied extensively by A.Jonsson and H.Wallin, see [5], [6], [7], [8] and [15]. Our approach is closely related to this work and should also be seen in the context of the theory developed there. The paper has two sections. Section 2 deals with function spaces on Rn and on F and related problems: dimension, extension, duality. In Section 3 we introduce two notions of dimensions. Our main results are the Theorems 2.3 and 3.3. All unimportant positive constants are denoted by c, occasionally with additional subsripts within the same formula or the same step of the proof. Furthermore, (k.l/m) refers to formula (m) in subsection k.l, whereas (j) means formula (j) in the same subsection. Similarly we refer to remarks, theorems etc.

On Function Spaces Related to Finite Element Approximation Theory

Certain function spaces of Besov type based on triangular finite elements in a two-dimensional domain are introduced and investigated. These spaces allow a systematic approach to the main estimates of finite element approximation theory in Besov-Sobolev norms, to the problem of improving the rate of approximation by choosing appropriate triangulations, and to asymptotic error estimates for elliptic boundary value problems.