The fall of the doubling condition in Calderón-Zygmund Theory

The most important results of standard Calderón-Zygmund Theory have recently been extended to very general non-homogeneous contexts. In this survey paper we describe these extensions and their striking applications to removability problems for bounded analytic functions. We also discuss some of the techniques that allow us to dispense with the doubling condition in dealing with singular integrals. Special attention is paid to the Cauchy Integral

Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension

Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $\mu$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $\mu$ exists has Hausdorff dimension 1. The result is extended to Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ and Riesz singular integrals of homogeneity $-\alpha$, 0 < $\alpha$ < d : the set of points where the principal value of the Riesz singular integral of $\mu$ exists has Hausdorff dimension $\alpha$. A martingale associated with the singular integral is introduced to support the proof.Comment: 14 pages, minor revision after the referee's report, to appear in Pacific J. of Mat

The regularity of the boundary of vortex patches for some non-linear transport equations

We prove the persistence of boundary smoothness of vortex patches for a non-linear transport equation in $\mathbb{R}^n$ with velocity field given by convolution of the density with an odd kernel, homogeneous of degree $-(n-1)$ and of class $C^2(\mathbb{R}^n\setminus\{0\}, \mathbb{R}^n).$ This allows the velocity field to have non-trivial divergence. The quasi-geostrophic equation in $\mathbb{R}^3$ and the Cauchy transport equation in the plane are examples.Comment: About the replacement: the main result has been significantly improved. The kernels to which the result applies are now of the more natural class conceivable. Furthermore, the proof has been adapted to the new context and an appendix has been adde

On Geometric Variational Models for Inpainting Surface Holes

Geometric approaches for filling-in surface holes are introduced and studied in this paper. The basic idea is to represent the surface of interest in implicit form, and fill-in the holes with a scalar, or systems of, geometric partial differential equations, often derived from optimization principles. These equations include a system for the joint interpolation of scalar and vector fields, a Laplacian-based minimization, a mean curvature diffusion flow, and an absolutely minimizing Lipschitz extension. The theoretical and computational framework, as well as examples with synthetic and real data, are presented in this paper

Boundary regularity of rotating vortex patches

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The rotating patch is convex if it is close enough to the bifurcation circle in the C^2 norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on H\"older spaces, certain non-convolution singular integral operators of Calder\'on-Zygmund type.Comment: Various proofs have been shortened. One added referenc

The Ellipse Law: Kirchhoff Meets Dislocations

In this paper we consider a nonlocal energy I \u3b1 whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter \u3b1 08 R. The case \u3b1 =&nbsp;0 corresponds to purely logarithmic interactions, minimised by the circle law; \u3b1 =&nbsp;1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for \u3b1 08 (0 , 1) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes 1-\u3b1 and 1+\u3b1. This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff\u2019s celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses