Bernstein <Emphasis Type="Italic">n</Emphasis>-widths of Besov embeddings on Lipschitz domains

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.     

Symmetrization inequalities in the fractional case and Besov embeddings

We prove new extended forms of the Pólya-Szegö symmetrization principle in the fractional case. As a consequence we determine new results for rearrangement invariant hulls of generalized Besov spaces.     

Eigenvalue problems for a non‐self‐adjoint Bessel‐type operators in limit‐point case

It is shown in the Weyl limit‐point case that system of root functions of the non‐self‐adjoint Bessel operator and its perturbation Sturm–Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the non‐self‐adjoint Bessel operators is investigated, and it is proved that system of root functions form a Bari basis in the same Hilbert space. Copyright © 2013 John Wiley & Sons, Ltd.     

Linear average and stochastic n-widths of Besov embeddings on Lipschitz domains

In this paper, we determine the asymptotic degree of the linear average and stochastic n-widths of the compact embeddings where is a Besov space defined on the bounded Lipschitz domain .     

Lipschitz曲线上Besov-Triebel-Lizorkin空间的嵌入定理

本文用离散的Calderón型再生公式。证明了Lipschitz曲线上Beasov空间与Triebel-Lizorkin空间的嵌入定理。     

Norms of embeddings of logarithmic Bessel potential spaces

Let be a subset of with finite volume, let and let be a Young function with for large . We show that the norm on the Orlicz space is equivalent to

We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in which are sharp in their dependences on provided that is large enough.

     

Sobolev type embeddings in the limiting case

We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes the result of Hansson and Brezis-Wainger for W _n ^k/k as a special case. We deal with generalized Sobolev spaces W _A ^k , where instead of requiring the functions and their derivatives to be in L_n/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to L_n/k. We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an, earlier optimality result obtained by Hansson with respect to the Riesz potential operator. In memory of Gene Fabes. Acknowledgements and Notes This research was supported by Technion V.P.R. Fund-M. and C. Papo Research Fund.     

An Osgood type regularity criterion for the 2D Oldroyd‐B model

By means of the Littlewood‐Paley decomposition and the div‐curl Theorem by Coifman‐Lions‐Meyer‐Semmes, we prove an Osgood type regularity criterion for the 2D incompressible Oldroyd‐B model, that is, where denotes the Fourier localization operator whose spectrum is supported in the shell {|ξ|≈2^j}.     

Spaces of Besov-Sobolev type and a problem on nonlinear approximation

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space ${\dot{W}}^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki? spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best n-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.     

Spectral Theory of Riesz Potentials on Quasi‐Metric Spaces

This paper deals with spectral assertions of Riesz potentials in some classes of quasi‐metric spaces. In addition we survey briefly a few related subjects: integral operators, local means and function spaces, euclidean charts of quasi‐metric spaces, relations to fractal geometry.     

Gâteaux Differentiability of Bump Functions in Banach Spaces

Upper estimates for the order of Gâteaux smoothness of bump functions in Orlicz spaces ℓ_M(Γ) and Lorentz spaces d(w, p, Γ), Γ uncountable, are obtained.     

A note on operator‐valued Fourier multipliers on Besov spaces

Let X be a Banach space. We show that each m : ? \ {0} → L (X ) satisfying the Mikhlin condition sup_{x ≠0}(‖m (x )‖ + ‖xm ′(x )‖) < ∞ defines a Fourier multiplier on B ^s _p,q (?; X ) if and only if 1 < p < ∞ and X is isomorphic to a Hilbert space; each bounded measurable function m : ? → L (X ) having a uniformly bounded variation on dyadic intervals defines a Fourier multiplier on B ^s _p,q (?; X ) if and only if 1 < p < ∞ and X is a UMD space. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Higher‐order Sobolev‐type embeddings on Carnot–Carathéodory spaces

A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.     

Boundedness and compactness of Hardy operators on Lorentz‐type spaces

Given non‐negative measurable functions on we study the high dimensional Hardy operator between Orlicz–Lorentz spaces , where f is a measurable function of and is the ball of radius t in . We give sufficient conditions of boundedness of and . We investigate also boundedness and compactness of between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.     

Triebel-Lizorkin空间和Besov空间上关于一类带Hardy核的振荡奇异积分

In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.     

Harmonic functions on the real hyperbolic ball II Hardy‐Sobolev and Lipschitz spaces

In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started in [13]. Our focus here is on the theory of Hardy‐Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman‐Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball. In doing so, we exhibit differences of behaviour of derivatives of harmonic functions depending on the parity of the dimension of the ball and on the parity of the order of derivation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bernstein numbers of embeddings of isotropic and dominating mixed Besov spaces

The purpose of the present paper is to investigate the decay of Bernstein numbers of the embedding from into the space . The asymptotic behaviour of Bernstein numbers of the identity will be also considered.     

Weighted generalized functions and hyperbolic Riesz B‐potential

The paper is devoted to the study of the fractional integral operator, which is a negative real power of the singular wave operator generated by Bessel operator using weighted generalized functions. We give conditions for this operator to be bounded in appropriate spaces, obtain formula for the Hankel transform of this operator, and get formula of connection between this operator and natural degree of singular wave operator generated by Bessel operator.     

Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces on domains

The purpose of this paper is to develop a theory of the Besov‐Morrey spaces and the Triebel‐Lizorkin‐Morrey spaces on domains in R ⁿ. We consider the pointwise multiplier operator, the trace operator, the extension operator and the diffeomorphism operator. Not only to domains in R ⁿ we extend our definition of function spaces to compact oriented Riemannian manifolds. Among the properties above, the result for the trace operator is in particular interesting, which reflects the property of the parameters p, q in the Morrey space ??^p_q (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)