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Mostrando las entradas de julio, 2018

### Problemas 2

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Si $f\in L^p(\mathbb R^d)$ y $g\in L^1(\mathbb R^d)$, entonces $f*g\in L^p(\mathbb R^d)$ y $||f*g||_{L^p} \le ||f||_{L^p} ||g||_{L^1}.$ Si $f\in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$, entonces $||\hat f||_{L^2} = ||f||_{L^2}.$ ( Sugerencia: Considera $h=f*g$, con $g(x) = \overline{f(-x)}$, y nota que $h(0) = \int \hat h$.) El operador $f\mapsto \hat f$ es sobreyectivo en $L^2(\mathbb R^d)$. ( Sugerencia: Si no, existiría $g\in L^2$ tal que $\int \hat f g = 0$ para toda $f\in L^2$. Utiliza el pasito del sombrero.) Sea $h(x) = \begin{cases} 0 & x<0\\1 & x\ge 0\end{cases}$ la función de Heaviside. Como distribuciones, $h' = \delta,$ la función delta de Dirac. Sea $\mathbf 1$ la función constante igual a 1. Como distribuciones, $\hat{\mathbf 1} = \delta$.

### Problemas 1

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Si $T:V\to V$ es simétrico y $\lambda_i\not=\lambda_j$ son eigenvalores reales distintos de $T$, con eigenvectores $u_i, u_j$ correspondientes, entonces $u_i\perp u_j$. Sea $T:\mathscr H\to\mathscr H$ un operador acotado diagonalizado con sucesión multiplicadora $\lambda_k$. T es unitario si, y solo si, $|\lambda_k|=1$ para todo k . T es una proyección ortogonal si, y solo si, todo $\lambda_k = 0\text{ o } 1$. T es un operador compacto si, y solo si, $\lambda_k \to 0$. Si $\sum a_n$ es una serie convergente, entonces es Cesàro-sumable. Si $\sum a_n$ es Cesàro-sumable, entonces es Abel-sumable.

### Problem Set 11

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Prove by induction, for the interval case, that $\min\{ \mathscr E_m(u): u|_{\{0,1\}}=v\} = \mathscr E_0(v) = (v(0)-v(1))^2,$ with the minimizer satisfying $\displaystyle u\Big(\frac{2k+1}{2^m}\Big) = \frac{1}{2}\Big(u\Big(\frac{k}{2^{m-1}}\Big) + u\Big(\frac{k+1}{2^{m-1}}\Big)\Big).$ The minimum of $f(x,y,z) = (a-x)^2 + (x-y)^2 + (y-a)^2 + (x-b)^2 + (b-z)^2 + (z-x)^2 + (y-z)^2 + (z-c)^2 + (c-y)^2$ is attained at $\displaystyle x^* = \frac{2a+2b+c}{5},\; y^* = \frac{2a+b+2c}{5},\; z^* = \frac{a+2b+2c}{5},$ with $f(x^*, y^*, z^*) = \dfrac{3}{5}\big((a-b)^2 + (b-c)^2 + (c-a)^2\big).$ Prove that one can obtain the values $u(F_2(q_1)) = x, u(F_3(q_1)) = y$ of a harmonic function in terms of the values $a,b,c$ at the points $p_2, q_1, p_3$, respectively  (as in the figure below). Use the previous problem to show that, if $u$ is a ha

### Problem Set 10

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Show that $||\cdot||_{H^1}$ is a norm, by showing it is induced by an inner product. Consider, for a connected domain $\Omega$, the energy form $\mathscr E(u,v) = \int_\Omega \nabla u\cdot \nabla v$. $\mathscr E(u,v)$ is an inner product on $H^1$ modulo constants. $\mathscr E(u,v)$ is an inner product on $H_0^1$. Show the equivalences of the Dirichlet principle.

### Problem Set 9

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Extend the result that the Hilbert transform is of weak-type $L^1$ to any operator of the form $Tf = K*f$ where $K$ satisfies: $\hat K\in L^\infty(\R)$ (say, taking the Fourier transform in the $L^2$ sense); there exists a constant $A>0$ such that $|K'(x)| \le A/x^2$ for any $x\in\R, \; x\not=0$. Show that we can replace (2) above by the condition 2'. There exists a constant $A>0$ such that $\displaystyle \int_{|x|\ge 2|t|} |K(x-t) - K(x)| dx \le A$ for any $t\in\R$. (Chebyshev Inequality) If $f\in L^p$, for some $1 < p <\infty$, then $|\{x : |f(x)|>\alpha\}| < \dfrac{1}{\alpha^p} ||f||_{L^p}^p$ Prove that we can substitute the condition $\hat K\in L^\infty$ with the boundedness of $T$ on any $L^q(\R), \; q>1$, that is, 1'. there exists a

### Problem Set 6

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Verify the integral $\displaystyle \int_{\R^d}\frac{dx}{(|x|^2+1)^\frac{d+1}{2}} = \frac{\pi^\frac{d+1}{2}}{\Gamma(\frac{d+1}{2})}.$ Use Minkowski inequality to prove that, if $K\in L^1(\R^d)$ and $f\in L^p(\R^d)$, then $||K*f||_{L^p}\le ||K||_{L^1}||f||_{L^p}.$ Prove that, if $f\in C_c(\R^d)$ and $y\in\R^d$, then $||f(\cdot - ty) - f||_{L^p} \to 0$ as $t\to 0$. Let $\Phi\in L^1(\R^d)$ with $\int \Phi = 1$, and $\Phi_t(x) = t^{-d}\Phi(x/t).$ Then $\int \Phi_t = 1$ for all $t>0.$ There exists some $M>0$ such that $\int |\Phi_t| \le M$ for all $t > 0.$ For each $\delta>0$, $\displaystyle \int_{|x|\ge\delta} |\Phi_t(x)|dx \to 0$ as $t\to 0.$ If $f\in C_c(\R^d)$, then $\Phi_t*f(x) \to f(x)$ uniformly. State conditions on $\Phi$ (as i

### Problem Set 5

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis A polynomial $p\in\mathscr P_k$ if and only if $\sum x_j \partial_j p = kp.$ $\dim \mathscr P_k = \binom{k+d-1}{k}$ Let $p\in\mathscr P_k$, and consider its orthogonal projection $|x|^2q$ onto the space $|x|^2\mathscr P_{k-2}$ with respect to the inner product seen in class. Then $p - |x|^2q$ is harmonic. ( Hint : Prove $\langle r,\Delta(p - |x|^2q\rangle = 0$ for every $r\in\mathscr P_{k-2}$.) If $p\in\mathscr H_k$, then $\dfrac{p}{|x|^{2k+d-2}}$ is harmonic on $\R^d\setminus\{0\}$. The spaces are invariant under rotations ,  i.e. if $f\in H_k$ then $f\circ R\in H_k$ for any rotation $R$.

### Problem Set 4

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis The function $x\mapsto P(x,\xi)$ is harmonic for each $\xi$, where $P(x,\xi)$ is the Poisson kernel for the ball. (Symmetry Lemma) For any $x\in\mathbb B$ and $\xi\in\mathbb S$, $\displaystyle \Big| |x|\xi - \frac{x}{|x|}\Big| = |x-\xi|$. (Harnack's inequality) If $u$ is a positive harmonic function on $\bar{\mathbb B}$, then $\displaystyle \frac{1-|x|}{(1+|x|)^{d-1}} u(0) \le u(x) \le \frac{1 + |x|}{(1-|x|)^{d-1}}u(0).$ (Hopf Lemma) If $u$ is a nonconstant harmonic function on $\bar{\mathbb B}$ and attains its maximum at $\zeta\in\mathbb S$, there exists $c>0$ such that $u(\zeta) - u(r\zeta) \ge c(1-r)$ for any $0 < r < 1.$ If $u$ is harmonic on $\bar{\mathbb B}$ and its normal derivative is 0 everywhere on $\mathbb S$, then $u$ is constant.

### Problem Set 3

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis If $u$ is harmonic in the connected domain $\Omega$ and is not constant, then $u(\Omega)$ is open in $\R$. Suppose $\Omega$ is bounded and that its boundary $\partial\Omega$ is connected. If $u$ is harmonic in $\Omega$, then $u(\Omega)\subset u(\partial\Omega)$. A radial harmonic function on $\mathbb B$ is constant. A positive harmonic function on $\R^d$ is constant.

### Problem Set 2

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Let $P:\R^d\to\R^d$ be orthogonal. $\Delta(u\circ P) = (\Delta u)\circ P$ If $u$ is harmonic, then $u\circ P$ is also harmonic. If $T$ is a translation and $u$ is harmonic, then $u\circ T$ is harmonic. Use polar coordinates to verify $\displaystyle \int_{\R^2}e^{-\pi |x|^2} dx = 1.$ Use (3.1) to prove $\displaystyle \int_{\R^d} e^{-\pi |x|^2} dx = 1$, for any $d\ge 1$. Use spherical coordinates and the previous result to prove $\omega_d = \dfrac{2\pi^{d/2}}{\Gamma(d/2)}$, where $\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t} dt$ is the Gamma function. Use the divergence theorem $\displaystyle \int_\Omega \nabla\cdot \vec F dx = \int_{\partial\Omega} \vec F \cdot \hat{\mathbf n} \; d\sigma$ to prove Green's identity. ( Hint: Consider $\vec F = u \nabla v - v\nabla u$.) The f

### Problem Set 1

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Let $R$ be a rotation in the plane. Consider the change of variables $(\xi, \eta) = R(x,y)$. Then $\displaystyle \frac{\partial^2 u}{\partial\xi^2} + \frac{\partial^2 u}{\partial\eta^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$. If $u$ is harmonic, then $u\circ R$ is also harmonic. Let $(r,\theta)$ be the polar coordinates of the plane. Then $\displaystyle \Delta u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial\theta^2}$ Let $u$ be a harmonic function on $\R^2$. Then there exists a harmonic function $v$ that is conjugate to $u$, so $f = u + iv$ is holomorphic. ( Hint: Consider a line integral of the 1-form $\displaystyle - \frac{\partial u}{\partial y} dx + \frac{\partial u}{\partial x} dy$.)