Ricardo A. Sáenz

Problem 1 Find solutions of the following initial-value problems in $latex \R^2$. $latex \partial_y u + (1+x^2)\partial_x u - u = 0; u(x,0) = \tan x$ $latex \partial_y u + \partial_x u = u^2; u(x,0) = h(x)$ Problem 2 Find a smooth function $latex a(x,y)$ in $latex \R^2$ such that the equation $latex \partial_y u + a(x,y) \partial_x u = 0$ does not have global solutions for any Cauchy data $latex \{y=0\}$. Problem 3 Let $latex \alpha\in\R$ and $latex h(x)$ a continuous function in $latex \R$, and consider que problem $latex y\partial_x u + x \partial_y u = \alpha u; \quad u(x,0) = h(x).$ Find all points where $latex \{y=0\}$ is characteristic. What is the compatibility condition on  h on those points? Find the solution of the initial-value problem away of the point in (1). What is the domain of this solution? For the cases $latex \alpha=1, h(x) = x$ and $latex \alpha=3, h(x) = x$, check whether this solution can be extended over the characteristic points. Find the chara...

Due Septembre 7 Problem 1 If $latex E, F\subset\R$ are a closed and a compact disjoint sets, then $latex \text{dist}(E,F)>0$. Problem 2 If $latex E\subset\R$ is closed, then it is measurable. Follow the next steps. Prove that it is sufficient to assume that  E is compact, and thus $latex |E|_* < \infty.$ Given $latex \e>0$, choose an open $latex U\supset E$ with $latex |U|_* < |E|_* + \e.$ Prove that we can write $latex U\setminus E = \bigcup I_j$, where the $latex I_j$ are disjoint open intervals. If $latex I,J$ are disjoint open intervals, then $latex |I\cup J|_* = |I| + |J|.$ For each  N , $latex |U|_* \ge |E|_* + \sum_{j=1}^N |I_j|$. Conclude $latex |U\setminus E|_* < \e$. Problem 3 Find a sequence of measurable sets $latex E_1 \supset E_2 \supset ...$ such that, for $latex E = \bigcap E_j$, $latex |E| \not= \lim |E_j|$. Problem 4 For $latex E\subset\R$, let $latex U_n=\{x\in\R: \text{dist}(x,E) < 1/n\}$. If  E is compact, then $latex |E| = \lim |U_n...

Problem 1 If, for $latex k=1, 2, \ldots, N$, $latex y_k(x_1, \ldots, x_{n-1},t) = u(x_1 + \ldots + x_{n-1},t)$, then $latex Y = (y_k)$ solves the system $latex \displaystyle \partial_t y_k = \frac{Mr}{r - (x_1 + \ldots + x_{n-1}) - (y_1 + \ldots + y_N)} \Big( \sum_{i=1}^{n-1}\sum_{j=1}^N \partial_{x_i}y_j + 1 \Big)$ with $latex y_k(x,0)=0$ if and only if $latex u(s,t)$ solves the equation $latex \displaystyle \partial_t u = \frac{Mr}{r - s - Nu} \big( N(n-1)\partial_su + 1\big)$ with $latex u(x,0)=0$. Problem 2 The function $latex \displaystyle u(s,t) = \frac{r - s - \sqrt{(r-s)^2 - 2MNnrt}}{Nn}$ solves the equation of the previous problem near $latex (0,0)$. Problem 3 Find a solution as a power series expansion of the initial-value problem $latex \displaystyle \begin{cases} u_{tt} - u_{xx} - u=0,\quad (x,t)\in\R\times(0,\infty)\\ u(x,0) = x,\quad \partial_t u(x,0) = -x. \end{cases}$ Identify this solution.

Due August 31 Problem 1 Let $latex f_n:[a,b]\to\R$ be Riemann-integrable and $latex f_n\rightrightarrows f$. f is Riemann-integrable on $latex [a,b]$. ( Hint:  Given $latex \e > 0$, find a partition $latex \mathscr P$ of $latex [a,b]$ such that $latex U(f,\mathscr P) - L(f,\mathscr P) < \e,$ where $latex U(f,\mathscr P), L(f,\mathscr P)$ are the upper and lower sums of  f  with respect to $latex \mathscr P$, respectively.) $latex \displaystyle \int_a^b f_n \to \int_a^b f$. Problem 2 Consider the functions $latex \displaystyle f_n(x) = \frac{x}{1 + nx^2}.$ Then $latex f_n\rightrightarrows 0$, but $latex f_n'(0)\not\to 0$. Problem 3 Let $latex f\in C^k(\mathbb S)$, a  k -continuously differentiable periodic function, with period $latex 2\pi$, and let $latex a_n$ be its  n th Fourier coefficient. There exists $latex C>0$ such that $latex |a_n| \le \dfrac{C}{|n|^k}$. The series $latex \sum a_n e^{inx}$ converges uniformly if $latex k\ge 2$. Problem 4 Let  X be a metr...

Problem 1 Let  u be a $latex C^1$ function defined on a neighborhood of the domain $latex \Omega$ such that $latex \Delta u = 0$ in $latex \Omega$. Then $latex \displaystyle \int_{\partial \Omega} \partial_\nu u \; d\sigma = 0.$ Problem 2 If the power series $latex \sum a_\alpha x^\alpha$ converges at $latex x^0$, then it converges on the cube $latex |x_j|<r$ where $latex r = \min_i |x_i^0|$. Problem 3 If  u satisfies the Cauchy problem $latex \begin{cases} \partial_t^mu = G\big(x,t,(\partial_x^\alpha\partial_t^ju)_{|\alpha|+j\le m, j<m}\big)\\ \partial_t^ju(x,0) = \phi_j(x), \quad 0\le j < m,\end{cases}$ then the functions $latex y_{\alpha j} = \partial_x^\alpha\partial_t^ju$ satisfy the system $latex \partial_t y_{\alpha j} = y_{\alpha(j+1)} \quad |\alpha|+j < m,$                              $latex \partial_t y_{\alpha j} = \partial_{x_i}y_{(\alpha-e_i)(j+1)} \quad |\alpha|+j=m, j<m,$ $latex \displaystyle \partial_t y_{0m} = \frac{\partial G}{\partial t} + \sum_{|\a...

Due August 24 Problem 1 Two metrics $latex d_1, d_2$ on a space  X are equivalent if there exist constants $latex c, C>0$ such that $latex c d_1(x,y) \le d_2(x,y) \le C d_1(x,y)$ for all $latex x,y\in X$. Suppose $latex (X,d_1), (X,d_2)$ are equivalent. They have the same convergent sequences. They have the same Cauchy sequences. $latex (X,d_1)$ is complete if and only if $latex (X,d_2)$ is complete. Problem 2 Are $latex (C([0,1]), d_u)$ and $latex (C([0,1]), d_1)$ equivalent? Problem 3 Are $latex (\R,|\cdot|)$ and $latex (\R,d_B)$ equivalent? Problem 4 Suppose $latex (X,d_1)$ and $latex (X,d_2)$ have the same convergent sequences. Are they equivalent? Is one of complete if and only if the other is complete?

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.

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Bibliografía Clase 1: métodos de sumabilidad, teorema de Fejér Elias M. Stein, Rami Shakarchi,  Fourier Analysis: An Introduction , Princeton books.google.com.mx/books?id=FAOc24bTfGkC T. W. Körner, Fourier Analysis, Cambridge books.google.com.mx/books?id=OcZ5iKsGrmoC Clase 2: transformada de Fourier, distribuciones Elias M. Stein, Guido Weiss,  Introduction to Fourier Analysis on Euclidean Spaces , Princeton books.google.com.mx/books?id=xnIwDAAAQBAJ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons books.google.com.mx/books?id=wI4fAwAAQBAJ Clase 3: multiplicador de la bola Elias M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton books.google.com.mx/books?id=ljcOSMK7t0EC Javier Enrique Sáenz Casas, La conjetura de Kakeya, Tesis e...

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Construye una función explícita $latex \eta$ como la que se necesita en la demostración del teorema de Marcinkiewicz: $latex \eta\in C^\infty(\mathbb R^d)$ tal que $latex \eta(\xi)=1$ si $latex |\xi|\le 1$ y $latex \eta(\xi)=0$ si $latex |\xi|\ge 2$. ( Sugerencia: considera la función en $latex \mathbb R$ dada por $latex \phi(t) = \begin{cases} e^{-\frac{1}{(1+t)^2}}e^{-\frac{1}{(1-t)^2}} & |t|<1\\ 0 & |t|\ge 1.) \end{cases}$ La hipótesis del teorema de Marcinkiewicz puede ser reemplazada por la condición $latex \displaystyle \sup_{R>0} R^{-d + 2|\alpha|} \int_{R\le|\xi|\le 2R} |\partial_\xi^\alpha m(\xi)|^2 d\xi \le A_\alpha$ para todo $latex 0\le|\alpha|\le l$, donde  l es el menor entero mayor que $latex d/2$. El multiplicador de Böchner-Riesz $latex m(\xi) = (1 - |\xi|^2)^\delta \chi_{\mathbb B}(\xi)$ satisface las hipótesis del teorema de Marcinkiewicz cuando $latex \delt...

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Sea $latex m\in L^\infty(\mathbb R^d)$, $latex m_R(\xi) = m(\xi/R)$ para $latex R>0$, y considera el operador de multiplicación dado por $latex \widehat{T_Rf}(\xi) = m_R(\xi)\hat f(\xi)$. Si $latex ||T_1f||_{L^p} \le M ||f||_{L^p}$, entonces $latex ||T_Rf||_{L^p} \le M||f||_{L^p}$ para todo $latex R>0$. Sean $latex T_1,T_2$ dos triángulos contiguos como en la figura, cada uno de base  b y altura  h , y sea $latex \tilde T_2$ el triángulo que resulta de trasladar a la izquierda el triángulo $latex T_2$ una distancia $latex 2(1-\alpha)b$, para $latex 1/2 < \alpha < 1$. Entonces $latex |T_1\cup\tilde T_2| = (\alpha^2 + 2(1-\alpha)^2)|T_1\cup T_2|$. $latex f(\alpha) = \alpha^2 -2 (1-\alpha)^2 < 1$ para todo $latex \alpha\in(1/2,1)$, y tiene mínimo 2/3. Para todo $latex n\in\mathbb N$, $latex A_n = \alpha^{2n} + 2(1-\alpha)^2 + 2(1-\alpha)^2\alpha^2 + \ldots + 2(1-\a...