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Joan Verdera

Publications and source records attributed to Joan Verdera.

24 records · Page 2Linked to original sources
arXiv2010

Estimates for the maximal singular integral in terms of the singular integral:the case of even kernels

The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our main result states that, for even operators, both are equivalent to a purely algebraic condition formulated in terms of the expansion of the kernel in spherical harmonics. The condition holds by higher order Riesz transforms, which then satisfy an improved version of Cotlar's inequality

math.CA↗
arXiv2010

A new characterization of Sobolev spaces on $\mathbb{R}^n$

In this paper we present a new characterization of Sobolev spaces on Euclidian spaces ($\mathbb{R}^n$). Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of $\mathbb{R}^n$ and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.

math.CA↗
arXiv2010

New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calderón-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the $L^2(\Rn)$ norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type $T^\star \circ T$ arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak $(1,1)$ estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak $L^1$ estimate for functions in $L LogL$.

math.CA↗
arXiv2008

Extra cancellation of even Calderon-Zygmund operators and quasiconformal mappings

We discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. These are of the form the characteristic function of a planar bounded domain with smooth boundary of class C 1+epsilon times a density of class Lip epsilon on the domain. The crucial fact in the argument is the special extracancellation property of even Calderon-Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far-reaching consequences. The main result may also be viewed as a Lipschitz regularity result for the Beltrami equation, and so for certain planar second order elliptic equations in divergence form.

math.CA↗
arXiv2004

The capacity associated to signed Riesz kernels, and Wolff potentials

There is a natural capacity associated to any vector valued Riesz kernel of a given homogeneity. If we are in the plane and the kernel is the Cauchy kernel, then this capacity is analytic capacity. Our main result states that if the homogeneity of the kernel is negative and larger than minus one, then the capacity is comparable to one of the well studied capacities of non-linear potential theory.

math.CA↗