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Interpolation with function parameter and UMD spaces.

Fernando Cobos Díaz — 1986

Extracta Mathematicae

A (real or complex) Banach space E is said to have the unconditionaly property for martingale differences (UMD-property, for short) if E-values martingale differences are unconditional in L(E;[0,1]). The main reason for the interest in this new class of spaces is that the analogues of several classical results on martingales and singular integrals are also true for a Banach space belonging to this class.

Real interpolation and compactness.

Fernando Cobos Díaz — 1989

Revista Matemática de la Universidad Complutense de Madrid

The behavior of compactness under real interpolation real is discussed. Classical results due to Krasnoselskii, Lions-Peetre, Persson, and Hayakawa are described, as well as others obtained very recently by Edmunds, Potter, Fernández, and the author.

Limiting real interpolation methods for arbitrary Banach couples

Fernando CobosAlba Segurado — 2012

Studia Mathematica

We study limiting K- and J-methods for arbitrary Banach couples. They are related by duality and they extend the methods already known in the ordered case. We investigate the behaviour of compact operators and we also discuss the representation of the methods by means of the corresponding dual functional. Finally, some examples of limiting function spaces are given.

Embeddings of Besov spaces of logarithmic smoothness

Fernando CobosÓscar Domínguez — 2014

Studia Mathematica

This paper deals with Besov spaces of logarithmic smoothness B p , r 0 , b formed by periodic functions. We study embeddings of B p , r 0 , b into Lorentz-Zygmund spaces L p , q ( l o g L ) β . Our techniques rely on the approximation structure of B p , r 0 , b , Nikol’skiĭ type inequalities, extrapolation properties of L p , q ( l o g L ) β and interpolation.

Extreme points of the complex binary trilinear ball

Fernando CobosThomas KühnJaak Peetre — 2000

Studia Mathematica

We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space 2 . This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space 2 . As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.

Compact embeddings of Brézis-Wainger type.

Fernando CobosThomas KühnTomas Schonbek — 2006

Revista Matemática Iberoamericana

Let Ω be a bounded domain in R and denote by id the restriction operator from the Besov space B (R) into the generalized Lipschitz space Lip(Ω). We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like e(id) ~ k if α > max (1 + 2/p + 1/q, 1/p). Our estimates improve previous results by Edmunds and Haroske.

Interpolation theory and measures related to operator ideals

Cobos, Fernando — 1999

Nonlinear Analysis, Function Spaces and Applications

Given any operator ideal , there are two natural functionals γ ( T ) , β ( T ) that one can use to show the deviation of the operator T to the closed surjective hull of and to the closed injective hull of , respectively. We describe the behaviour under interpolation of γ and β . The results are part of joint works with A. Martínez, A. Manzano and P. Fernández-Martínez.

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