Ricardo A. Sáenz

Here you can find the complete notes for the course:  HarmonicAnalysis.pdf

Notes on the eigenfunctions of the Laplacian on the Sierpińsky gasket:  Chapter15.pdf

Notes on the harmonic structure and the Laplacian on the Sierpiński gasket:  Chapter14.pdf

Notes on the Hausdorff dimension and fractal sets:  Chapter13.pdf

Due May 31st Problem 1 Let $latex u(x) = e^{\omega x}$ on $latex I=[0,1]$. Then, for each $latex m\ge 1$, $latex u|_{\mathcal P_m}$ is a discrete eigenfunction of $latex \Delta_m$ with eigenvalue $latex \lambda_m = \dfrac{\omega^2}{4^m} + O(2^{-3m})$. $latex \mathcal P_m$ is the dyadic partition $latex \{0, 1/2^m, \ldots, 1\}$ of $latex [0,1]$. Problem 2 Let $latex \phi(x) = 2 - \sqrt{4-x}$, for $latex x\in[0,4]$. $latex \phi(x) = \dfrac{1}{4}x + O(x^2)$ as $latex x\to0$. The sequence defined by, for given $latex \lambda_0\in[0,2]$, $latex \lambda_m = \phi(\lambda_{m-1})$ and $latex x_m = 4^m\lambda_m$ for $latex m\ge1$ satisfies $latex x_m - x_{m-1} = O(2^{-m})$. $latex x_m$ is Cauchy and hence converges. Problem 3 Let $latex \psi(x) = \dfrac{5 - \sqrt{25- 4x}}{2}$, for $latex x\in[0,4]$. $latex \psi(x) = \dfrac{1}{5}x + O(x^2)$ as $latex x\to0$. The sequence defined by, for given $latex \lambda_0\in[0,2]$, $latex \lambda_m = \psi(\lambda_{m-1})$ and ...

Due May 17 Problem 1 Let $latex s\ge 0$ and $latex \mathcal H^s$ the Hausdorff measure with exponent $latex s$ in $latex \mathbb R^d$. If $latex A\subset B$, then $latex \mathcal H^s(A) \le \mathcal H^s(B)$. If $latex A = \bigcup_j A_j$, then $latex \displaystyle \mathcal H^s(A) \le \sum_j \mathcal H^s(A_j)$. If $latex \text{dist}(A,B)>0$, then $latex \mathcal H^s(A\cup B) = \mathcal H^s(A) + \mathcal H^s(B)$. Problem 2 If $latex A\subset\mathbb R^d$ is countable, then $latex \dim(A) = 0$. Problem 3 If $latex 0 < p < 1$, the function $latex x\mapsto x^p$ is concave: for all $latex x,y>0$ and $latex t\in[0,1]$,  $latex (tx + (1-t)y)^p \ge t x^p + (1-t) y^p$. Problem 4 Let $latex f:\mathbb R^d \to \mathbb R^d$ be a similitude with coefficient $latex \alpha >0$: for every $latex x,y\in\mathbb R^d$,  $latex |f(x) - f(y)| = \alpha |x-y|$. Let $latex g(x) = \dfrac{1}{\alpha} (f(x) - f(0))$. For all $latex x,y\in\mathbb R^d$, $latex g(x...

Notes on the Dirichlet principle:  Chapter12.pdf

Due May 13 Problem 1 Let $latex X$ be a closed subspace of the Hilbert space $latex \mathscr H$. $latex X^\perp = \{x\in\mathscr H: x\perp X \}$ is a closed subspace of $latex \mathscr H$. $latex \mathscr H \cong X\oplus X^\perp$ Problem 2 If $latex g$ is the weak derivative of $latex f\in L^2(\mathbb R^d)$ with respect to $latex x_j$, then $latex \hat g(\xi) = 2\pi i \xi_j \hat f(\xi).$ Problem 3 Let $latex \mathscr H^1(\Omega)$ be the set of equivalence classes in $latex H^1(\Omega)$ under the relation $latex f\sim g$ if and only if $latex f-g$ is a constant. $latex \mathscr H^1(\Omega)$ is a vector space. The bilinear form $latex \mathcal E$ is an inner product on $latex \mathscr H^1(\Omega)$. $latex \mathscr H^1(\Omega)$ is a Hilbert space with respect to $latex \mathcal E$. $latex H_0^1(\Omega)$ is a closed subspace of $latex \mathscr H^1(\Omega)$. Problem 4 Let $latex \Omega$ be a bounded $latex C^1$-domain in $latex \mat...

Notes on the Hilbert transform:  Chapter11.pdf Note that Chapter 9 was modified to consider averages over all balls containing a given point $latex x$, and not only those centered at $latex x$.

Notes on pointwise and uniform convergence of Fourier series of continuous functions:  Chapter05.pdf

Notes on the Hardy-Littlewood maximal functions and boundary limits of harmonic functions:  Chapter09.pdf Notes on the Fourier transform, the inversion formula and square means convergence:  Chapter10.pdf

Notes on the Poisson kernel for the upper half-space:  Chapter07.pdf Recall that you can find the bibliography in the link:  Bibliography.pdf

Notes on spherical harmonics:  Chapter06.pdf These notes don't include the bibliography, as the previous chapters had. Rather, I have prepared a link for the bibliography of the course here:  Bibliography.pdf . This link is included in the course webpage , listed as Extended bibliography .

Notes on Fourier series and square means convergence:  Chapter04.pdf

As noted in class, we gave Fejér's original proof of his theorem on the Cesàro summability of a Fourier series of a continuous function (except for the fact the he assumed the function to be even and, thus, its Fourier series contains only cosines). Fejér's theorem was published in Comptes Rendus Hebdomadaries, Seances de l’Academie de Sciences, Paris 131 (1900), and you can find the paper here:   Sur les fonctions bornées et intégrables .

Notes on the Poisson kernel and the Poisson integral:  Chapter 3 .

Notes on basic properties of harmonic functions:  Chapter02.pdf

Las notas del curso que impartí en la Escuela de análisis ya están en la página de la escuela. Pueden verlas aquí:  Notas del curso Operadores de multiplicación La idea del curso fue introducir el Análisis armónico, tomando como pretexto la teoría de operadores de multiplicación, empezando con operadores diagonalizados y su relación con el teorema espectral de álgebra lineal, además de métodos de sumabilidad de series. El objetivo final es la demostración del teorema de Marcinkiewicz de multiplicadores de Fourier, que utiliza el teorema de Calderón y Zygmund de integrales singulares. Como “paréntesis cultural”, presento un bosquejo del contraejemplo de Fefferman a la conjetura del disco, que muestra que el operador de la bola no puede ser acotado en el espacio euclideano de dimensión mayor a 1.

En clase, para demostrar el principio del buen orden, utilizamos el principio de inducción fuerte, que dice lo siguiente. Principio de inducción fuerte. Sea $latex K\subset\N$ tal que $latex 0\in K$ Si $latex 0,1,\ldots, n\in K $, entonces $latex n+1\in K $. Entonces $latex K=\N $. Para demostrar el principio de inducción fuerte, necesitaremos el siguiente lema. Lema. Para $latex n\in\N $, cualquier subconjunto no vacío de $latex \{0,1,\ldots, n\} $ tiene un mínimo. Demostración: Demostraremos este lema por inducción en $latex n $. Si $latex n=0$, el resultado es claro porque, si $latex S\subset\{0\} $ es no vacío, entonces $latex S=\{0\} $, y $latex 0$ es su mínimo. Suponemos ahora que el resultado es cierto para $latex n $, y sea $latex S\subset\{0,1,\ldots, n+1\} $ no vacío. Si $latex S\cap\{0,1,\ldots, n\} =\emptyset $, entonces $latex S=\{n+1\} $, y $latex n+1$ es el mínimo de $latex S $. Si $latex S\cap\{0,1,\ldots, n\} \not=\emptyset $, entonces es un subconjunto no vacío ...

La presentación, en PDF, de la primera clase de mi curso "Las matemáticas de los fractales", en el Taller de Ciencia para Jóvenes . En PDF:  Las matemáticas de los fractales .