Notes (en construcció) escrites per la primera part de l’assignatura d’Anàlisi complexa i de Fourier, assignatura obligatòria de segon curs del grau en matemàtica computacional i analítica de dades de la UAB, primavera del curs 2023/2024. Els apunts es basen en el contingut d’edicions prèvies d’aquesta assignatura, així com de l’assignatura Anàlisi complexa, obligatòria de tercer curs del grau en matemàtiques de la UB, i miren de compatibilitzar els dos enfocaments, per tal que el material es pugui fer servir en aquests dos contextos, així com en l’assignatura Anàlisi complexa i de Fourier de tercer curs del grau en matemàtiques de la UAB.
All posts by mprats
Kari Astala, Martí Prats and Eero Saksman: Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale
We give sufficient conditions for quasiconformal mappings between simply connected Lipschitz domains to have Hölder, Sobolev and Triebel-Lizorkin regularity in terms of the regularity of the boundary of the domains and the regularity of the Beltrami coefficients of the mappings. The results can be understood as a counterpart for the Kellogg-Warchawski Theorem in the context of quasiconformal mappings.
Martí Prats: Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity
We study the stability of Triebel-Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel-Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain our results we provide an equivalent norm for the Triebel-Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.
El teorema de Monge
Martí Prats and Xavier Tolsa: Notes on harmonic measure
In these notes we provide a straightforward introduction to the topic harmonic measure. This is an area were many advances have been obtained in the last years and we think that this book can be useful for people interested in this topic.
In the first Chapters 2-7 we have followed classical references such as [Fol95], [Car98], [GM05], [Lan72], and [Ran95], as well as some private notes of Jonas Azzam. The content of Chapter 8 is based on Kenig’s book [Ken94], and on papers by Aikawa, Hofmann, Martell, and many others. Chapter 9 is based on a paper by Jerison and Kenig [JK82], while in some parts of Chapter 10 we follow the book of Caffarelli and Salsa [CS05] and some work by Mourgoglou and the second named author of these notes. Most of the last chapter follows [AHM+16].
We apologize in advance for possible inaccuracies or lack of citation. Anyway, we remark that this work is still under construction and we plan to add more content as well as more accurate citations in future versions of these notes.
Martí Prats: Anàlisi harmònica i teoria del senyal
Notes escrites per l’assignatura d’Anàlisi harmònica i teoria del senyal, traduint al català part del llibre de María Cristina Pereyra i Lesley A. Ward i adaptant-ne el contingut al context de l’assignatura, optativa de 4t curs del grau de matemàtiques de la Universitat de Barcelona la primavera del curs 2021/2022.
Laura Brustenga i Martí Prats: Mani’m? (Bits de matemàtiques)
Max Engelstein, Aapo Kauranen, Martí Prats, Georgios Sakellaris and Yannick Sire: Minimizers for the thin one-phase free boundary problem
We consider the “thin one-phase” free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in \mathbb{R}^{n+1}_+ plus the area of the positivity set of that function in \mathbb{R}^{n}. We establish full regularity of the free boundary for dimensions n \leq 2, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight.
While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced by Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.
Laura Brustenga i Martí Prats: Divulgació matemàtica en línia (Bits de matemàtiques)
Riemann mapping and regularity in the Sobolev and Triebel-Lizorkin scale
Presented in Spanish Network in Complex Analysis and Operator Theory, June 1st 2021, online, Spain.
Laura Brustenga i Martí Prats: LATEX per a no iniciats (Bits de matemàtiques)
Martí Prats and Xavier Tolsa: 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^+ has big pieces of uniformly rectifiable measures, 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.
Laura Brustenga i Martí Prats: Un tastet de topologia algebraica (Bits de matemàtiques)
The two-phase problem for harmonic measure in VMO via jump formulas for the Riesz transform
Presented in Harmonic Analysis Seminar, March 6th 2020, Helsinki, Finland, Seminario de analisis y aplicaciones UAM-ICMAT, March 12th 2021, Madrid, Spain, Oberseminar Discrete Harmonic and Harmonic Analysis, July 8th 2021, Würzburg, Germany and Workshop on Harmonic analysis, Singular Integrals and PDEs, February 1st 2022, Bonn, Germany.