Ricardo A. Sáenz

The final grade is distributed as follows: Final exam: 50% Final project: 50% The final exam is scheduled on June 27, 10:00 am, and will evaluate all the content of the course. The final projects extend the results from the course. Each student is assigned one project, and the projects cannot be changed. Each student must work alone, although any references may be consulted. There will be office hours for any questions related to the projects (or the material from the class) on June 14, 20 and 21, 11:00am - 12:00pm. The final projects are due June 27, 10:00 am. Yair Antonio Castillo:  Convex functions Jaime Daniel Hernández:  Sturm-Liouville operators Manuel Irán Torres:  The maximal function on L 2

Due June 8 Problem 1 Consider the Koch-type curve $latex K^l$, for $latex 1/4 < l < 1/2$, described by the diagram The function $latex t\mapsto K^l(t)$ satisfies a Hölder condition of exponent $latex \gamma = -\log l/\log 4$. $latex t\mapsto K^l(t)$ is a simple curve. $latex t\mapsto K^l(t)$ is continuous and nowhere differentiable. $latex \dim K^l = 1/\gamma$. Problem 2 If we take $latex l=1/2$ in the previous definition, we obtain a space-filling curve. Problem 3 On $latex \R^d$, define the functions, for $latex \alpha > 0$, $latex f_0(x) = \begin{cases} 1/|x|^{\alpha} & |x|<1\\0 & |x|\ge 1;\end{cases}\qquad f_\infty(x) = \begin{cases} 0 & |x|<1\\1/|x|^{\alpha} & |x|\ge 1.\end{cases}$ $latex f_0\in L^p$ if and only if $latex p\alpha < d$. $latex f_\infty \in L^p$ if and only if $latex p\alpha > d$. What happens if we replace $latex |x|^\alpha$ with $latex |x|^\alpha \big|\log 2|x|\big|$ in the previous definitions? Problem 4 Suppose ...

Due June 1 Problem 1 Let $latex f:[0,1]\to\R$ satisfy a Hölder condition of exponent $latex \gamma > 1$. Then  f is constant. Is $latex f:[0,1]\to[0,1]\times[0,1]$ is a surjective Hölder function of exponent $latex \gamma$, then $latex \gamma \le 1/2$. (Prove directly, without using Lemma 2.2 from the text.) Problem 2 Let $latex K\subset\R$ be the set $latex \displaystyle K = \Big\{ \sum_{k=1}^\infty \frac{a_k}{4^k} \in\R : a_k=0\text{ or }2\Big\}$. Then $latex \dim K = 1/2$ and $latex 0 < \mathscr H^{1/2}(K) < \infty$. Problem 3 Let $latex 2N+1$ be an odd integer and consider the "middle $latex 1/(2N+1)$th" set  K , that is, the result of the Cantor process when removing the middle interval of length $latex 1/(2N+1)$ of the previous interval. Calculate $latex \dim K$ Prove that for any $latex 0 < \alpha < 1$, there exists a totally disconnected perfect set in $latex \R$ whose dimension is larger then $latex \alpha$. Problem 4 There exists a Cantor-lik...

Due May 25 Problem 1 Suppose $latex \tau$ is measure-preserving, with $latex \mu(X) = 1$. If  E is invariant, then there exists a set  E' so that $latex E' = \tau^{-1}(E')$, and  E and  E' differ by a set of measure zero. Problem 2 Let $latex \tau$ be measure-preserving, with $latex \mu(X)=1$. Then $latex \tau$ is ergodic if and only if whenever $latex \nu$ is absolutely continuous with respect to $latex \mu$ and $latex \nu$ is invariant (that is $latex \nu(\tau^{-1}(E) = \nu(E)$ for all measurable  E ), then $latex \nu = c\mu$), then $latex \nu = c\mu$ for some constant  c . Problem 3 The Hausdorff measure $latex \mathscr H^\alpha$ is not $latex \sigma$-finite on $latex \R^d$ if $latex \alpha < d$. Problem 4 Let $latex \{E_k\}$ be a sequence of Borel sets in $latex \R^d$. If $latex \dim E_k\le\alpha$ for all  k , then $latex \displaystyle \dim \bigcup E_k \le \alpha$.

Due May 18 Problem 1 The purpose of the following exercises is to prove the following statement:  If $latex \mu$ is a translation-invariant Borel measure on $latex \R^d$ that is finite on compact sets, then $latex \mu$ is a multiple of Lebesgue measure. Let $latex Q_r$ be a translate of the cube $latex \{x\in\R^d: 0 < x_j \le r, j=1,\ldots,d\}.$ If $latex \mu(Q_1) = c$, then $latex \mu(Q_{1/n}) = c/n^d$ for each integer  n . $latex \mu$ is absolutely continuous with respect to  m , and there is a locally integrable function  f such that $latex \displaystyle \mu(E) = \int_E f dx.$ By the differentiation theorem, is follows that $latex f(x) = c$ a.e., and hence $latex \mu = cm$. Problem 2 Suppose $latex \nu, \nu_1, \nu_2$ are signed measures on $latex (X,\mathscr M)$ and $latex \mu$ a positive measure. If $latex \nu_1\perp\mu$ and $latex \nu_2\perp\mu$, then $latex \nu_1+\nu_2 \perp \mu$. If $latex \nu_1\ll \mu$ and $latex \nu_2\ll\mu$, then $latex \nu_1 + \nu_2 \ll\mu$. ...

Due May 11 Problem 1 Let $latex \rho:\R^d\to\R^d$ be a rotation. Then it induces a measure-preserving map of the sphere $latex \mathbb S^{d-1}$ with its measure $latex \sigma$. Problem 2 Use the polar coordinate formula to prove the following statements. $latex \displaystyle \int_{\R^2} e^{-\pi |x|^2} dx = 1$ $latex \displaystyle \int_{\R^d} e^{-\pi |x|^2} dx = 1$ for any  d . $latex \sigma(\mathbb S^{d-1}) = \dfrac{2\pi^{d/2}}{\Gamma(d/2)}$. $latex \displaystyle m(\mathbb B^d) = \frac{\pi^{d/2}}{\Gamma(d/2+1)}.$ Problem 3 If $latex \mu$ is a finite Borel measure on the interval $latex [a,b]$, then $latex \displaystyle f\mapsto l(f) = \int_a^b f d\mu$ is a linear functional on $latex C([a,b])$, positive in the sense that $latex l(f)\ge 0$ if $latex f\ge 0$. Conversely, if  l is a positive linear functional on $latex C([a,b])$, there exists a unique finite Borel measure $latex \mu$ on $latex [a,b]$ such that $latex l(f) = \int f d\mu$ for every $latex f\in C([a,b])$.

Due May 4 Problem 1 Let  X be a set and $latex \mathcal M$ a nonempty collection of subsets of  X  closed under complements and countable unions of disjoint sets. Then $latex \mathcal M$ is a $latex \sigma$-algebra. Problem 2 Let $latex (X,\mathcal M, \mu)$ be a measure space. Its  completion is defined as the collection $latex \overline{\mathcal M}$ of sets of the form $latex E\cup N$, where $latex E\in\mathcal M$ and $latex N\subset F$ for some $latex F\in\mathcal M$ with $latex \mu(F)=0,$ and $latex \bar\mu(E\cup N) = \mu(E).$ $latex \overline{\mathcal M}$ is the smallest $latex \sigma$-algebra containing $latex \mathcal M$ and all subsets of its elements of measure 0. The function $latex \bar\mu$ is a complete measure on $latex \overline{\mathcal M}$. Problem 3 Consider the Lebesgue exterior measure. Then a set is Caratheodory measurable if and only if is Lebesgue measurable. Problem 4 Let $latex (X,\mathcal M,\mu)$ be a measure space, $latex A,B,C$ subsets of  X such that $...

Due April 27 Problem 1 Let $latex f\in L^2(\R^d), k\in L^1(\R^d)$. $latex \displaystyle (f*k)(x) = \int_{\R^d} f(x-y)k(y) dy$ converges for a.e.  x . $latex ||f*k||_{L^2} \le ||f||_{L^2} ||k||_{L^1}$. $latex \widehat{(f*k)}(\xi) = \hat k(\xi) \hat f(\xi)$ for a.e. $latex \xi$. The operator $latex Tf = f*k$ is a Fourier multiplier operator with multiplier $latex m(\xi) = \hat k(\xi)$. Problem 2 Let $latex \Omega\subset\C$ be open, and $latex \mathscr H\subset L^2(\Omega)$ be the subspace of holomorphic functions on $latex \Omega$. $latex \mathscr H$ is a closed subspace of $latex L^2(\Omega)$. If $latex \{\phi_k\}$ is an orthonormal basis of $latex \mathscr H$, then $latex \displaystyle \sum_k |\phi_k(z)|^2 \le \frac{1}{\pi d(z,\C\setminus\Omega)}$ for $latex z\in\Omega$, where $latex d(z,\C\setminus\Omega)$ is the distance from  z to the complement of $latex \Omega$. The sum $latex B(z,w) = \sum_k \phi_k(z)\overline{\phi_k(w)}$ converges absolutely for $latex z,w\in\Omeg...

Due April 20 Problem 1 Let $latex \{\phi_k\}_{1\le k<\infty}$ be a complete orthonormal system for $latex L^2(\R^d)$. Then $latex \{\phi_{k,j}\}_{1\le k,j < \infty}$ a complete orthonormal system for $latex L^2(\R^d\times\R^d)$, where $latex \phi_{k,j}(x,y) = \phi_k(x)\phi(y)$. Problem 2 Let  P be the orthogonal projection onto a closed subspace  S of a Hilbert space. Then $latex P^2 = P\text{ and } P^*=P$. Conversely, if  P is a bounded operator such that $latex P^2 = P$ and $latex P^*=P$, then it is the orthogonal projection onto some closed subspace. Let $latex P_1\text{ and }P_2$ be the orthogonal projections onto the closed subspaces $latex S_1\text{ and }S_2$, respectively. Then $latex P_1P_2$ is an orthogonal projection if and only if they conmute, in that case, it projects onto $latex S_1\cap S_2$. Problem 3 Let $latex \{u_k\}$ be a complete orthonormal system for a Hilbert space  H , and $latex (a_k)$ a sequence of positive numbers such that $latex \sum_k a_k^2 ...

Due March 23 Problem 1 If $latex \{K_\delta\}$ is a family of better kernels, there exists a constant $latex A>0$ such that $latex \sup_{\delta>0} |f*K_\delta(x)| \le A f^*(x)$ for all $latex f\in L^1$. Problem 2 For $latex a,b>0$, let $latex \begin{cases} x^a\sin x^{-b} & 0 < x \le 1, \\ 0 & x=0.\end{cases}$ f is of bounded variation iff $latex a>b$. For each $latex 0<\alpha<1$, construct an $latex \alpha$-Hölder continuos function that is not of bounded variation. If $latex a=b=2$,  f' exists at every point but is not integrable. Problem 3 Define the  one-sided maximal function  for locally integrable functions on $latex \R$ as $latex \displaystyle f_+^*(x) = \sup_{h>0} \frac{1}{h} \int_x^{x+h} |f|.$ If $latex E_\alpha^+ = \{x\in\R: f_+^*(x)>\alpha\}$, then $latex \displaystyle m(E_\alpha^+) = \frac{1}{\alpha} \int_{E_\alpha^+} |f|.$ Problem 4 Let $latex f:\R\to\R$ be absolutely continuous. f maps sets of measure zero to sets of measu...

Due March 16 Problem 1 Consider the function on $latex \R$ given by $latex f(x) = \begin{cases}\dfrac{1}{|x|(\log |x|)^2} & |x|\le 1/2\\0 & \text{otherwise.}\end{cases}$ f is integrable. $latex f^*(x) \ge \dfrac{c}{|x|\log 1/|x|}$ for some $latex c>0$ and all $latex |x|\le 1/2$. $latex f^*$ is not locally integrable. Problem 2 Let $latex E\subset [0,1]$ be a measurable set such that there exists $latex \alpha > 0$ such that $latex m(E\cap I) \ge \alpha m(I)$ for all intervals $latex I\subset[0,1]$. Then $latex m(E)=1$. Problem 3 Let $latex F\subset\R$   be closed and $latex \delta(x)$ the distance from  x to  F . Then $latex \delta(x+y) =o(|y|)$ for almost every $latex x\in F$. Problem 4 Suppose $latex \{K_\delta\}$ is a family of kernels that satisfies $latex \displaystyle \int_\R K_\delta = 0$ for all $latex \delta>0$. For some $latex A>0$, $latex |K_\delta(x)| \le A \min\{\delta^{-d}, \delta/|x|^{d+1}\}$ for all $latex \delta > 0, x\in\R^d$. If $la...

Due March 9 Problem 1 Let  f be integrable, and for each $latex \alpha>0$ let $latex E_\alpha = \{x:|f(x)|>\alpha\}$. Then $latex \displaystyle \int |f| = \int_0^\infty m(E_\alpha)d\alpha.$ Problem 2 (Riemann-Lebesgue Lemma) For $latex f\in L^1(\R^d)$, let $latex \displaystyle \hat{f}(\xi) = \int_{\R^d} f(x) e^{-2\pi i x\cdot\xi} dx$ be its Fourier transform. Then $latex \hat f(\xi) \to 0$ as $latex |\xi|\to0$. Problem 3 Let $latex f,g\in L^1(\R^d)$. $latex (x,y) \mapsto f(x-y)g(y) \in L^1(\R^d\times\R^d)$. The convolution $latex \displaystyle f*g(x) = \int_{\R^d} f(x-y)g(y) dy$ is well defined for a.e.  x . $latex ||f*g||_{L^1} \le ||f||_{L^1} ||g||_{L^1}$. $latex \widehat{f*g}(\xi) = \hat f(\xi) \hat g(\xi)$. Problem 4 There does not exist $latex I\in L^1(\R^d)$ such that, for all $latex f\in L^1(\R^d)$, $latex f*I = f.$ Problem 5 Let $latex f_n\to f$ in measure. If $latex f_n\ge 0$, then $latex \displaystyle \int f \le \liminf \int f_n$ If there exists $latex g\in...

Due March 2 Problem 1 Let f be integrable on $latex [0,b]$ and define, on $latex [0,b]$, $latex \displaystyle g(x) = \int_x^b \frac{f(t)}{t} dt.$ Then g is integrable on $latex [a,b]$ and $latex \displaystyle \int_0^b g(x) dx = \int_0^b f(t) dt$. Problem 2 Let $latex F\subset\R$ be a closed set such that $latex m(\R\setminus F) <\infty$, and let $latex \delta(x) = d(x,F)$ be the distance from x to F . Then $latex \delta(x)$ is a Lipschitz function. Let $latex \displaystyle I(x) = \int_\R \frac{\delta(x)}{|x-y|^2} dy.$ Then $latex I(x) = \infty$ for any $latex x\in\R\setminus F$, and $latex I(x) < \infty$ for a.e $latex x\in F$. Problem 3 There exists $latex f\in L^1(\R^d)$ and a sequence $latex f_n\in L^1(\R^d)$ such that $latex f_n\to f$ in $latex L^1$, but $latex f_n(x) \not\to f(x)$ for every x . Problem 4 Consider the function defined on $latex \R$ by $latex f(x) = \begin{cases} x^{-1/2} & 0 < x < 1,\\0 & \text{otherwise.} \end{cases}$ For a fixed enumaer...

Due February 23 Probl For any two Cantor sets $latex \mathcal C_1, \mathcal C_2$, as constructed in HW2, Problem 3, there exists a continuous, bijective and increasing function $latex F:[0,1]\to[0,1]$ that maps $latex \mathcal C_1$ surjectively onto $latex \mathcal C_2$. There exists a measurable function f and a continous function $latex \Phi$ such that $latex f\circ\Phi$ is non-measurable. Problem 2 Given a collection of sets $latex E_1, E_2, \ldots, E_n$, there exists a disjoint collection $latex F_1, F_2, \ldots, F_N$, $latex N=2^n - 1$, such that $latex \bigcup E_j = \bigcup F_k$ and each $latex \displaystyle E_j = \bigcup_{F_k\subset E_j} F_k.$ Problem 3 (Tchebychev inequality) Let $latex f\ge0$ be integrable, $latex \alpha > 0$ and $latex E_\alpha = \{x:f(x)>\alpha\}$. Then $latex \displaystyle m(E_\alpha) \le \frac{1}{\alpha}\int f.$ Problem 4 Let $latex f\ge 0$ be finite almost everywhere, $latex E_k = \{ x:f(x) > 2^k\}$ and $latex F_k = \{x: 2^k < f(x) \le ...

Due February 16 Problem 1 Let $latex E\subset\R$ with $latex m_*(E)>0$. For each $latex 0 < \alpha < 1$, there exists an open interval  I such that $latex m_*(E\cap I) \ge \alpha |I|$. If E is measurable, the difference set $latex \{x\in\R: x=a-b\text{ for some }a,b\in E\}$ contains an open interval centered at the origin. Problem 2 Let  C be the Cantor set. $latex x\in C$ if and only if $latex \displaystyle x=\sum_{k=1}^\infty \frac{a_k}{3^k}$, where $latex a_k=0\text{ or } 2$. The Cantor-Lebesgue function is defined on  C by $latex \displaystyle F(x)=\sum_{k=1}^\infty \frac{a_k}{2^{k+1}}$, if $latex \displaystyle x=\sum_{k=1}^\infty \frac{a_k}{3^k}.$  F is well-defined, continuous on  C , $latex F(0)=0$ and $latex F(1)=1$, and surjective. Problem 3 Construct a compact set  D in an analogous way as the Cantor set but, at the  k th stage of the construction, we remove $latex 2^{k-1}$ central open intervals of length $latex l_k$, with $latex l_1 + 2l_2 + 4l_3 + \...

Due February 9 Problem 1 The Cantor set is totally disconnected (for any $latex x\not= y\in C$ there is $latex z\not\in C$ between  x and  y ) and perfect (compact and without isolated points). Problem 2 Let $latex E\subset \R$ and $latex O_n = \{x: d(x,E) < 1/n\}.$ If  E is compact, $latex m(E) = \lim_{n\to\infty} m(O_n)$ The previous conclusion may be false for  E  closed and unbounded, or open and bounded. Problem 3 (Borel-Cantelli lemma) Let $latex \{E_k\}$ be a sequence of measurable sets such that $latex \displaystyle \sum_{k=1}^\infty m(E_k) < \infty,$ and define $latex E = \limsup_{k\to\infty} E_k = \{x\in\R^d: x\in E_k$ for infinitely many $latex k\}$. E is measurable. $latex m(E) = 0$. Problem 4 Let, for a subset $latex E\subset\R^d$, $latex \displaystyle m_*^R(E) = \inf \sum_{j=1}^\infty |R_j|,$ where the infimum is taken over all countable covers $latex \{R_j\}$ for  E of rectangles. Then $latex m_*^R(E) = m_*(E).$ We can thus conclude that we obtain the s...