A T(P) theorem for Sobolev spaces on domains

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given $0<s\leq1$, $12$ and a Lipschitz domain $\Omega\subset \mathbb{C}$, the Beurling transform $Bf=- {\rm p.v.}\frac1{\pi z^2}*f$ is bounded in the Sobolev space $W^{s,p}(\Omega)$ if and only if $B\chi_\Omega\in W^{s,p}(\Omega)$. In this paper we obtain a generalized version of the former result valid for any $s\in \mathbb{N}$ and for a larger family of Calder\'on-Zygmund operators in any ambient space $\mathbb{R}^d$ as long as $p>d$. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for $p\leq d$. In the particular case $s=1$, this condition is in fact necessary, which yields a complete characterization.Comment: 35 pages, 6 figure

Exploration through drawings in the conceptual stages of product design

This paper argues that sequences of exploratory drawings - constructed by designer's movements and decisions - trace systematic and logical paths from ideas to designs. This argument has three parts. First, sequences of exploratory sketches produced by product designers, against the same task specification, are analyzed in terms of the cognitive categories of reinterpretation, emergence and abstraction. Second, a computational model is outlined for the process of exploration through drawing and third the model is applied to elucidate the logic in the sequences of exploratory sketches examined earlier

The two-phase problem for harmonic measure in VMO

Let $\Omega^+\subset\mathbb R^{n+1}$ be an NTA domain and let $\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}$ be an NTA domain as well. Denote by $\omega^+$ and $\omega^-$ their respective harmonic measures. Assume that $\Omega^+$ is a $\delta$-Reifenberg flat domain for some $\delta>0$ small enough. In this paper we show that $\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)$ if and only if $\Omega^+$ is vanishing Reifenberg flat, $\Omega^+$ and $\Omega^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $\Omega^+$ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.Comment: Minor correction

Harish-Chandra integrals as nilpotent integrals

Recently the correlation functions of the so-called Itzykson-Zuber/Harish-Chandra integrals were computed (by one of the authors and collaborators) for all classical groups using an integration formula that relates integrals over compact groups with respect to the Haar measure and Gaussian integrals over a maximal nilpotent Lie subalgebra of their complexification. Since the integration formula a posteriori had the same form for the classical series, a conjecture was formulated that such a formula should hold for arbitrary semisimple Lie groups. We prove this conjecture using an abstract Lie-theoretic approach.Comment: 10 page

Sobolev regularity of the Beurling transform on planar domains

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z)$. It is shown that if the outward unit normal vector $N$ of the boundary of the domain is in the trace space of $W^{n,p}(\Omega)$ (i.e., the Besov space $B^{n-1/p}_{p,p}(\partial\Omega)$) then $\mathcal{B} \chi_\Omega \in W^{n,p}(\Omega)$. Moreover, when $p>2$ the boundedness of the Beurling transform on $W^{n,p}(\Omega)$ follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.Comment: 33 pages, 8 figures. arXiv admin note: text overlap with arXiv:1507.0433

Assessment of the feasible CTA windows for efficient spacing with energy-neutral CDO

Continuous descent operations (CDO) with con- trolled times of arrival (CTA) at one or several metering fixes could enable environmentally friendly procedures at the same time that terminal airspace capacity is not compromised. This paper focuses on CTA updates once the descent has been already initiated, assessing the feasible CTA window (and associated fuel consumption) of CDO requiring neither thrust nor speed-brake usage along the whole descent (i.e. energy modulation through elevator control is used to achieve different times of arrival at the metering fixes). A multiphase optimal control problem is formulated and solved by means of numerical methods. The minimum and maximum times of arrival at the initial approach fix (IAF) and final approach point (FAP) of an hypothetical scenario are computed for an Airbus A320 descent and starting from a wide range of initial conditions. Results show CTA windows up to 4 minutes at the IAF and 70 seconds at the FAP. It has been also found that the feasible CTA window is affected by many factors, such as a previous CTA or the position of the top of descent. Moreover, minimum fuel trajectories almost correspond to those trajectories that minimise the time of arrival at the metering fix for the given initial conditionPeer ReviewedPostprint (published version

Beltrami equations in the plane and Sobolev regularity

New results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation $\bar{\partial} f = \mu \partial f + \nu \overline{\partial f}$ for discontinuous Beltrami coefficients $\mu$ and $\nu$ are obtained, using Kato-Ponce commutators, obtaining that $\overline \partial f$ belongs to a Sobolev space with the same smoothness as the coefficients but some loss in the integrability parameter. A conjecture on the cases where the limitations of the method do not work is raised.Comment: 13 pages, 12 figure