Orbits in symmetric spaces

We characterize those elements in fully symmetric spaces on the interval (0,1) or on the semi-axis (0,∞) whose orbits are the norm-closed convex hull of their extreme points.     

Abstract Hardy-Sobolev spaces and interpolation

The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities.     

Real interpolation of vector-valued spaces in non-diagonal case

It is shown that the formula

where and is correct under the restrictions and It is also true if we suppose that

and the spaces are functional Banach or quasi-Banach lattices on the same measure space

     

Infra-Mackey spaces, weak barrelledness and barrelledness

A weaker Mackey topology, infra-Mackey topology, is introduced. For an infra-Mackey space, dual local quasi-completeness, c₀-quasi-barrelledness, Ruess' property (quasi-L) and C-quasi-barrelledness are equivalent to each other. Inspired by the definition of Mazur spaces, locally convex spaces are classified according to various conditions ensuring linear functionals continuous. In the classification, every class of special locally convex spaces is characterized by some completeness of the duals. From this, some new characterizations of quasi-barrelledness and barrelledness are given.     

Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

We characterize the real interpolation space between a weighted $L^p$ space and a weighted Sobolev space in arbitrary bounded domains in ${\mathbb{R}}^n$, with weights that are positive powers of the distance to the boundary.     

Ultraproducts of real interpolation spaces between L p -spaces

Let , be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.     

Embeddings between weak Orlicz martingale spaces

Let Φ be an increasing and convex function on [0,∞) with Φ(0)=0 satisfying that for any α>0, there exists a positive constant Cα such that Φ(αt)?CαΦ(t), t>0. Let wLΦ denote the corresponding weak Orlicz space. We obtain some embeddings between vector-valued weak Orlicz martingale spaces by establishing the wLΦ-inequalities for martingale transform operators with operator-valued multiplying sequences. These embeddings are closely related to the geometric properties of the underlying Banach space. In particular, for any scalar valued martingale f=(fn)_n?1, we claim that

     

Carlson type inequalities and embeddings of interpolation spaces

We discuss the close relation between Carlson type inequalities

and interpolation, and prove embedding results for real interpolation spaces, in particular into weighted -spaces.

     

Real interpolation method,Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.     

Notes on limits of Sobolev spaces and the continuity of interpolation scales

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.

     

On the interpolation of some generalized function spaces of different anisotropy

In the paper, the interpolation properties of the spacesH _p ^s (v; ℝ _n ) of Sobolev-Liouville type and the spacesB _{p, q} ^s (μ ℝ _n ) of Nikol'skii-Besov type generated by functions of polynomial growth that are infinitely differentiable outside of the origin are studied. Interpolation formulas for the pairs {H(v _o ),H(v ₁)} and {B(μ₀),B(μ₁)} of spaces of the above types for which the anisotropies of the interpolated spaces do not depend on each other are proved. The investigated spaces, for certain specification of the generating functions, coincide with the classical (isotropic and anisotropic) Sobolev-Liouville and Nikol'skii-Besov spaces. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 666–672, November, 1997. Translated by A. I. Shtern     

Optimal recovery of interpolation operators in Hardy spaces

We continue studies begun by C.A. Micchelli and T.J. Rivlin on optimal recovery in H^p spaces. The feature operators are various interpolation operators drawn from the theory of Walsh equiconvergence, as are the information sets. The theory is of interest in that it identifies linear algorithms which might not otherwise be isolated for study or used as approximations of the feature operators. In some cases, we can identify the optimal algorithm although we cannot explicitly determine the exact order of the approximation that it achieves. For Charles Micchelli on his sixtieth birthday, with appreciation Mathematics subject classifications (2000) 41A05, 30B30. A. Sharma: Deceased.     

On subclasses of weak Asplund spaces

Assuming the consistency of the existence of a measurable cardinal, it is consistent to have two Banach spaces, , where is a weak Asplund space such that (in the weak* topology) in not in Stegall's class, whereas is in Stegall's class but is not weak* fragmentable.

     

Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations

In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and establish smoothness properties of solutions. The results are based on maximal regularity estimates in continuous interpolation spaces. An important new ingredient is that we are able to show that quasilinear parabolic evolution equations generate a smooth semiflow on the trace spaces associated with maximal regularity, which are the natural phase spaces in this framework. Received August 10, 2000; accepted September 20, 2000.     

On the interpolation constant for Orlicz spaces

In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,

where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .

     

Quasilinear evolutionary equations and continuous interpolation spaces

In this paper we analyze the abstract parabolic evolutionary equations

     

Complex Interpolation of Weighted Besov and Lizorkin–Triebel Spaces

We study complex interpolation of weighted Besov and Lizorkin–Triebel spaces.The used weights w0,w1 are local Muckenhoupt weights in the sense of Rychkov.As a first step we calculate the Calder′on products of associated sequence spaces.Finally,as a corollary of these investigations,we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin–Triebel spaces on Rd.     

On an extreme class of real interpolation spaces

We investigate the limit class of interpolation spaces that comes up by the choice θ=0 in the definition of the real method. These spaces arise naturally interpolating by the J-method associated to the unit square. Their duals coincide with the other extreme spaces obtained by the choice θ=1. We also study the behavior of compact operators under these two extreme interpolation methods. Moreover, we establish some interpolation formulae for function spaces and for spaces of operators.