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.
Category Archives: Books
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.
Martí Prats: Singular integral operators on sobolev spaces on domains and quasiconformal mappings (PhD dissertation)
In this dissertation some new results on the boundedness of Calderón-Zygmund operators on Sobolev spaces on domains in R^d. First a T(P)-theorem is obtained which is valid for W^{n,p} (U), where U is a bounded uniform domain of R^d, n is a given natural number and p>d. Essentially, the result obtained states that a convolution Calderón-Zygmund operator is bounded on this function space if and only if T(P) belongs to W^{n,p} (U) for every polynomial P of degree smaller than n restricted to the domain. For indices p less or equal than d, a sufficient condition for the boundedness in terms of Carleson measures is obtained. In the particular case of n=1 and p \leq d, this Carleson condition is shown to be necessary in fact. The case where n is not integer and 0 < n < 1 is also studied, and analogous results to the former are obtained for a larger family of function spaces, the so-called Triebel-Lizorkin spaces. The thesis contains some optimal conditions to establish when the Beurling transform of a polynomial restricted to a domain is contained in a Sobolev space W^{n,p}(U), where U is a bounded planar lipschitz domain, in terms of the Besov regularity of the boundary of U. This result, in combination with the one mentioned above, provides a condition to determine whether the Beurling transform is bounded on W^{n,p}(U) or not for p>2, which is optimal in case n=1.
Finally, an application of the aforementioned results is given for quasiconformal mappings in the complex plane. In particular, it is checked that the regularity W^{n,p}(U) of the Beltrami coefficient of a quasiconformal mapping for a bounded Lipschitz domain U with boundary parameterizations in a certain Besov space and p>2, implies that the mapping itself is in W^{n+1,p}(U).