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This book contains an expanded version of lectures delivered by the authors at the CRM in Spring of 2009. It contains four series of lectures. The first one is an application of harmonic analysis and the Heisenberg group to understand human vision. The second and third series of lectures cover some of the main topics on linear and multilinear harmonic analysis. The last one is a clear introduction to a deep result of De Giorgi, Moser and Nash on regularity of elliptic partial differential equations in divergence form.



Chapter 1. Models of the Visual Cortex in Lie Groups

The most classical and exhaustive theory which states and studies the phenomenological laws of visual reconstruction is Gestalt theory [73, 74]. It formalizes visual perceptual phenomena in terms of geometric concepts, such as good continuation, orientation, or vicinity. Consequently, phenomenological models of vision have been expressed in terms of geometrical instruments and minima of calculus of variations [5, 51, 96]. On the other hand, the recent progress of medical imaging and integrative neuroscience allows to study neurological structures related to perception of space and motion. The first results that used instruments of differential geometry to model the cortex and justify macroscopical visual phenomena in terms of the microscopical behavior of the cortex were due to Hoffmann [70], Koenderink [77], and Petitot–Tondut [100, 102]. More recently, in [37] and [109], the visual cortex was modeled as a Lie group with a sub-Riemannian metric. Other models in Lie groups are proposed in [12, 22, 39, 49, 62, 63, 112, 117, 118]. We refer to these papers for a complete description of this type of problems.
Giovanna Citti, Alessandro Sarti

Chapter 2. Multilinear Calderón–Zygmund Singular Integrals

It is quite common for linear operators to depend on several functions of which only one is thought of as the main variable and the remaining ones are usually treated as parameters. Examples of such operators are ubiquitous in harmonic analysis: multiplier operators, homogeneous singular integrals associated with functions Ω on the sphere, Littlewood–Paley operators, Calderón commutators, and the Cauchy integral along Lipschitz curves. Treating the additional functions that arise in these operators as frozen parameters often provides limited results that could be thought analogous to those that one obtains by studying calculus of functions of several variables by freezing variables. In this article, we advocate a more flexible point of view in the study of linear operators, analogous to that employed in pure multivariable calculus. Unfreezing the additional functions and treating them as input variables provides a more robust approach that often yields sharper results in terms of regularity of the input functions.
Loukas Grafakos

Chapter 3. Singular Integrals and Weights

This chapter is an expanded version of the material covered in a minicourse given at the Centre de Recerca Matemàtica in Barcelona during the week May 4–8, 2009.We provide details and different proofs of known results as well as new ones. We also survey on several recent results related to the core of this course, namely weighted optimal bounds for Calderón–Zygmund operators with weights. The basic topics covered by the lectures revolved around the Rubio de Francia iteration algorithm, the extrapolation theorem with optimal bounds, the Coifman–Fefferman estimate, the Besicovitch covering lemma, and rearrangements of functions. These notes can be seen as a modern introduction to the A p theory of weights.
Carlos Pérez

Chapter 4. De Giorgi–Nash–Moser Theory

We consider the second-order, linear, elliptic equations with divergence structure
$$\mathrm{div} (\mathbb{A}(x)\nabla u(x))\;=\;\sum\limits^n_{i,j=1}\;\partial_{x_{i}}(a_{ij}(x)\partial_{x_{j}}u(x))\;=\;0.$$
Xiao Zhong
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