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 June 7 Problem 1 For $latex m\ge 2$, there is no Dirichlet eigenfunction on $latex V_m$ with respect to $latex \lambda_m=2$. Problem 2 The Dirichlet eigenfunctions on $latex V_m$ constructed in class, with respect to $latex \lambda_m=5$, are linearly independent. If we add the three eigenfunctions chained from $latex p_1$ to $latex p_2$, from $latex p_2$ to $latex p_3$, and from $latex p_3$ to $latex p_1$, they are linearly dependent. Problem 3 Let $latex \mathcal E_m$ be a sequence of Dirichlet forms such that  $latex \displaystyle \mathcal E_m(u,v) = \sum_{j=1}^N \frac{1}{r_j}\mathcal E_{m-1}(u\circ f_j,v\circ f_j)$ and, given a function $latex u$ on $latex V_0$, $latex \min\{\mathcal E_1(v,v): v|_{V_0} = u\} = \mathcal E_0(u,u)$, then $latex \min\{\mathcal E_m(v,v): v|_{V_{m-1}} = u\} = \mathcal E_{m-1}(u,u)$ for every $latex m\ge1$ and any given function $latex u$ on $latex V_{m-1}$. Problem 4 Calculate all harmonic structures on the interval, s...

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 $latex x$.

Due May 3 Problem 1 Let $latex \gamma$ be the lower semicircle of radius $latex N$ around the origin, and $latex \xi > 0.$ Then $latex \displaystyle \int_\gamma f(z) dz = 2\pi i\text{Res}_{z=-it} f(z) = i e^{-2\pi t\xi},$ where $latex f(z)$ is the function defined by $latex f(z) = \dfrac{1}{\pi} \dfrac{z}{z^2+t^2} e^{-2\pi iz\xi}.$ Problem 2 If $latex f\in L^1(\mathbb R)$ and diferentiable at $latex x\in\mathbb R$, then the limit $latex \displaystyle \lim_{t\to 0} \int_{|y|\ge t} \frac{f(x-y)}{y} dy$ exists. ( Hint:  Use the identity, for any $latex \delta_n>0$, $latex \displaystyle \int_{t\le|y|<\delta_n} \frac{f(x-y)}{y} dy = \int_{t\le|y|<\delta_n} \frac{f(x-y) - f(x)}{y} dy + \int_{t\le|y|<\delta_n} \frac{f(x)}{y} dy,$ and take $latex \delta_n\to 0$.) Problem 3 If $latex f_n, g_n$ are sequences in $latex C_c^\infty(\mathbb R)$ that converge in $latex L^1(\mathbb R)$ to $latex f$, then $latex Hf_n$ and $latex Hg_n$ converge in measu...

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

Due April 12 Problem 1 For any $latex \xi\in\mathbb R$, $latex \displaystyle \lim_{N\to\infty} \int_{-N}^N e^{-\pi (x+i\xi)^2}dx = \lim_{N\to\infty} \int_{-N}^N e^{-\pi x^2}dx = 1.$ ( Hint:  Consider the contour integral $latex \int_\gamma e^{-\pi z^2} dz = 0$ over the rectangle $latex \gamma$ with vertices $latex N, N+i\xi, -N+i\xi$ and $latex -N$.) Problem 2 Let $latex u>0$. Then $latex \displaystyle \frac{1}{\pi} \int_{-\infty}^\infty \frac{e^{-2\pi iuv}}{1 + v^2} dv = \frac{1}{\sqrt\pi} \int_0^\infty \frac{1}{\sqrt s} e^{-s} e^{-\pi^2 u^2/s} ds$; $latex \displaystyle \frac{1}{\pi} \int_{-\infty}^\infty \frac{e^{-2\pi iuv}}{1 + v^2} dv = e^{-2\pi u}$. Problem 3 Prove the Riemann-Lebesgue Lemma : If $latex f\in L^1(\mathbb R^d)$, then $latex \hat f(\xi) \to 0$ as $latex |\xi|\to\infty$. Problem 4 If $latex \Phi(x) = e^{-\pi |x|^2}$, the collection $latex \{\Phi_t(x)\}_{t>0}$ if its dilations is a collection of better kernels. Problem 5 ...

Debido a que la hora de oficina para los estudiantes de cálculo tenía un conflicto con otra clase, las horas de oficina cambian de horario: la de Análisis Armónico será a las 4:00pm, cada miércoles.

Due April 5 Problem 1 The set $latex L^1(\mathbb R^d)$ of integrable functions is a complex vector space, and $latex f\mapsto \int f$ is a linear functional. $latex \displaystyle ||f||_1 = \int f$ is a norm on $latex L^1(\mathbb R^d)$, when $latex L^1(\mathbb R^d)$ is seen as a set of equivalence classes of $latex f\sim g$ if and only if $latex f=g$ a.e. Problem 2 Let $latex L^2(\mathbb R^d)$ be the set of measurable functions $latex f$ such that $latex \displaystyle \int |f|^2 < \infty$, seen as a set of equivalence classes of $latex f\sim g$ if and only if $latex f=g$ a.e. $latex L^2(\mathbb R^d)$ is a complex vector space. The bilinear form $latex \displaystyle \langle f, g \rangle = \int f \bar g$ is well defined on $latex L^2(\mathbb R^d)$ and is an inner product. $latex L^2(\mathbb R^d)$ is complete with the norm $latex \displaystyle ||f||_2 = \sqrt{\langle f, f \rangle}.$ ( Hint:  Proceed as in the case of $latex L^1$ seen in class.) Problem 3 If...

Due March 29 Problem 1 $latex A\subset\mathbb R^d$ is measurable if and only if, for all $latex B\subset\mathbb R^d$,  $latex |B|_* = |B\cap A|_* + |B\setminus A|_*$. Problem 2 Let $latex A\subset\mathbb R^d$. The following are equivalent. $latex A$ is measurable. $latex A = P\setminus M$, where $latex P$ is a $latex G_\delta$ set and $latex |M|=0$. $latex A = Q\cup N$, where $latex Q$ is an $latex F_\sigma$ set and $latex |N|=0$. Problem 3 Let $latex A\subset\mathbb R^d$ be a measurable set. For $latex \delta > 0$, let $latex \delta A = \{ \delta x: x\in A\}$. Then $latex \delta A$ ls measurable and $latex |\delta A| = \delta^d |A|$. For a $latex d$-tuple $latex \bar{\delta} = (\delta_1, \ldots, \delta_d)$ with each $latex \delta_j>0$, $latex j=1,\ldots,d$, define $latex \bar\delta A = \{(\delta_1 x_1, \ldots, \delta_d x_d): (x_1, \ldots, x_d)\in A\}$. Then $latex \bar\delta A$ is measurable and $latex |\bar\delta A| = \delta_1\cdots\delta_d |A|...

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