We show how certain nonconvex optimization problems that arise in image processing and computer vision can be restated as convex minimization problems. This allows, in particular, the finding of global minimizers via standard convex minimization schemes.

http://mnikolova.perso.math.cnrs.fr/ChanEseNikoSiap06.pdf

Algorithms for finding global minimizers of image segmentation and denoising models

A discrete transform and decompositions of distribution spaces

Background 1. Stationary phase 2. Non-homogeneous oscillatory integral operators 3. Pseudo-differential operators 4. The half-wave operator and functions of pseudo-differential operators 5. Lp estimates of eigenfunctions 6. Fourier integral operators 7. Local smoothing of Fourier integral operators Appendix. Lagrangian subspaces of T*IRn.

Fourier Integrals in Classical Analysis

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

We define Sobolev space W1,p for 1<p≤∞ on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.

Sobolev Spaces on an Arbitrary Metric Space

Theorie classique et espaces de Hardy. Theorie de Calderon-Zygmund. Theorie a variable reelle des espaces de Hardy. Inegalites en norme ponderees. Inegalites a valeurs vectorielles. Theories de la factorisation et inegalites en norme ponderees

Weighted norm inequalities and related topics

Studies in Advanced Mathematics

The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on euclidean space and their known extensions for doubling measures. We start by analyzing the images of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined in terms of the metric, play a basic role too. For a euclidean space equipped with one of these measures, we also consider the so-called "regular"BMO space introduced by X. Tolsa. We show that it contains the image of a Lebesgue space in the appropriate limit case and also that the image of the space "regular"BMO is contained in the adequate Lipschitz space.

/pdf/boundedness-properties-of-fractional-integral-operators-2zh4bbbcul.pdf

José García-Cuerva

Papers

Weighted norm inequalities and related topics

Studies in Advanced Mathematics

Boundedness properties of fractional integral operators associated to non-doubling measures

Hardy Spaces and Beurling Algebras

A Theory of Hardy Spaces Associated to the Herz Spaces