Ricardo A. Sáenz

As stated in the syllabus, 50% of the midterms grades consist of a written essay of a short research project. The projects may be worked in pairs and the essay must be turn in by June 7. Each enrolled pair of students must choose a different project. Unenrolled students sitting in the course may work on a project, and are free to choose independently of other students, but cannot work with enrolled students. The following is the list of projects to choose from. The maximal operator on $latex L\log L$ functions Karla Flores and Jaime Hernández Functions equal to their Fourier transform Bernardo Ameneyro and Gabriel Rosales The Fourier transform of radial functions Yair Castillo and Rafael Morales

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 24th Problem 1 The minimum of $latex f(x,y,z) =$ $latex (a-x)^2 + (x-y)^2 + (y-a)^2 + (x-b)^2 + (b-z)^2 + (z-x)^2 + (y-z)^2 + (z-c)^2 + (c-y)^2$ is attained at $latex \displaystyle x^* = \frac{2a+2b+c}{5},\; y^* = \frac{2a+b+2c}{5},\; z^* = \frac{a+2b+2c}{5},$ with $latex f(x^*, y^*, z^*) = \dfrac{3}{5}\big((a-b)^2 + (b-c)^2 + (c-a)^2\big).$ Problem 2 One can obtain the values $latex u(f_2(q_1)) = x, u(f_3(q_1)) = y$ of a harmonic function in terms of the values $latex a,b,c$ at the points $latex p_2, q_1, p_3$, respectively  (as in the figure below). Problem 3 If $latex u$ is a harmonic function with boundary values $latex u(p_1) = u(p_2) = 0$ and $latex u(p_3) = 1$, then its restriction to the bottom side of the Sierpinski triangle is an increasing function on $latex [0,1]$. ( Hint:  Use the previous problem.) Problem 4 If $latex u$ is harmonic on the Sierpinski gasket $latex S$, then there exists a constant $latex c>0$ such that $...

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 $latex x$.