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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\exercise}{{\bf \green EXERCISE:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Hodge theory \\[15mm] \small lecture 4: Sobolev $L^2$-spaces and Rellich lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, February 3, 2018 } \end{center} \newpage {\bf \blue Banach spaces} \definition Let $M$ be a topological space, and $\|f\|:= \sup_M |f|$ {\bf\blue the sup-norm on functions}. {\bf \blue $C^0$-topology}, or {\bf \blue uniform topology} on the space $C^0(M)$ of bounded continuous functions is topology defined by the sup-norm. \definition A {\bf \blue Banach space} is a complete normed vector space. \theorem {\bf \red A space of bounded continuous functions on $M$ with $C^0$-topology is Banach.} \proof A uniform limit of continuous functions is continuous (Weierstrass), and a limit of a Cauchy sequence of functions in $C^0(M)$ exists pointwise because $\R$ is complete. \endproof \newpage {\bf \blue Stone-Weierstrass approximation theorem} \definition Let $A\subset C^0 M$ be a subspace in the space of continuous functions. We say that $A$ {\bf\blue separates the points} of $M$ if for all distinct points $x, y\in M$, there exists $f\in A$ such that $f(x) \neq f(y)$. \theorem ({\bf \blue Stone-Weierstrass approximation theorem}) Let $M$ be a compact manifold and $A\subset C^0 M$ be a subring separating points, and $\bar A$ its closure. {\bf \red Then $\bar A= C^0M$.} \proof Handouts or the next lecture. \endproof \newpage {\bf \blue Hilbert spaces (reminder)} \definition {\bf \blue Hilbert space} is a complete, infinite-dimensional Hermitian space which is second countable (that is, has a countable dense set). \definition {\bf \blue Orthonormal basis} in a Hilbert space $H$ is a set of pairwise orthogonal vectors $\{x_\alpha\}$ which satisfy $|x_\alpha|=1$, and such that $H$ is the closure of the subspace generated by the set $\{x_\alpha\}$. \theorem {\bf \red Any Hilbert space has a basis, and all such bases are countable.} \theorem {\bf \red All Hilbert spaces are isometric}. \proof Each Hilbert space has a countable orthonormal basis. \endproof \newpage {\bf \blue Fourier series} \claim {\bf\blue ("Fourier series")} Functions $e_k(t)= e^{ 2\pi\1 k t}$, $k\in \Z$ on $S^1=\R/\Z$ {\bf \red form an orthonormal basis in the space $L^2(S^1)$} of square-integrable functions on the circle. \proof Orthogonality is clear from $\int_{S^1} e^{ 2\pi\1 k t} dt =0$ for all $k\neq 0$ (prove it). To show that the space of Fourier polynomials $\sum_{i=-n}^n a_k e_k(t)$ is dense in the space of continuous functions on circle, use the Stone-Weierstrass approximation theorem, applied to the ring $R= \langle \sin(mx), \cos (nx)\rangle$ of functions obtained from real and imaginary parts of $e^{ 2\pi\1 k t}$. \endproof \definition {\bf \blue Fourier monomials} on a torus are functions $F_{l_1, ..., l_n}:= \exp(2\pi\1\sum_{i=1}^n l_it_i)$, where $l_1, ..., l_n\in \Z$. \claim Fourier monomials {\bf \red form an orthonormal basis in the space $L^2(T^n)$} of square-integrable functions on the torus $T^n$. \proof The same. \endproof \newpage {\bf \blue $L^2$-norms on vector spaces} \theorem Let $V$ be a vector space, and $g_1$, $g_2$ two scalar products. We say that {\bf \blue $g_1$ is bounded by $g_2$} if for some $C>0$, one has $g_1 \leq C g_2$. \exercise Prove that {\bf \purple this is equivalent to the continuity of the map $(V, g_2)\arrow (V, g_1)$.} \remark Let $g_1$ be bounded by $g_2$. {\bf \purple Then the identity map extends to a continuous map on the corresponding completion spaces $L^2(V, g_2) \arrow L^2(V, g_1)$.} \remark The topology induced by $g_1$ {\bf \purple is equivalent to topology induced by $g_2$ if and only if $C^{-1} g_2 \leq g_1 \leq C g_2$.} \newpage {\bf \blue Sobolev's $L^2$-norm on $C^\infty_c(\R^n)$} \definition Denote by $C^\infty_c(\R^n)$ the space of smooth functions with compact support. For each differential monomial \[ P_\alpha = \frac {\partial ^{k_1}}{\partial x_1^{k_1}}\frac {\partial ^{k_2}}{\partial x_2^{k_2}}... \frac{\partial ^{k_n}}{\partial x_1^{k_n}} \] consider the corresponding partial derivative \[ P_\alpha(f) = \frac {\partial ^{k_1}}{\partial x_1^{k_1}} \frac{\partial ^{k_2}}{\partial x_2^{k_2}}... \frac{\partial ^{k_n}}{\partial x_1^{k_n}}f. \] Given $f\in C^\infty_c(\R^n)$, one defines {\bf \blue the $L^2_p$ Sobolev's norm $|f|_{p}$} as follows: \[ |f|_s^2= \sum_{\deg P_\alpha\leq p} \int \left|P_\alpha(f) \right |^2\Vol \] where the sum is taken over all differential monomials $P_\alpha$ of degree $\leq p$, and $\Vol= dx_1\wedge dx_2\wedge ... dx_n$ - the standard volume form. \remark Same formula defines {\bf \blue Sobolev's $L^2$-norm $L^2_p$ on the space of smooth functions on a torus $T^n$.} \newpage {\bf \blue Sobolev's $L^2$-norm on a torus} \claim The Fourier monomials $F_{l_1, ..., l_n}:= e^{2\pi\1\sum l_it_i}$ {\bf \red are eigenvectors for the differential monomials $P_\alpha = \frac {\partial ^{k_1}}{\partial x_1^{k_1}}\frac {\partial ^{k_2}}{\partial x_2^{k_2}}... \frac{\partial ^{k_n}}{\partial x_1^{k_n}}$. Moreover, $P_\alpha(F_{l_1, ..., l_n})= \prod_{i=1}^n (2\pi\1 k_i)^{l_i}$.} \endproof \corollary The Fourier monomials are orthogonal in the Sobolev's $L^2_p$-metric, and \[ |F_{l_1, ..., l_n}|^2_{2,p}= \sum_{k_1+ ...+ k_n=1}^p \prod_{i=1}^n (2\pi l_i)^{2k_i}. \] \endproof \newpage {\bf \blue Weak convergence (reminder)} \definition Let $x_i \in H$ be a sequence of points in a Hilbert space $H$. We say that $x_i$ {\bf \blue weakly converges} to $x\in H$ if for any $z\in H$ one has $\lim_i g(x_i, z) = g(x, z)$. \remark Let $y(i)= \alpha_j(i) e_j$ be a sequence of points in a a Hilbert space with orthonormal basis $e_i$. {\bf \purple Then $y(i)$ converges to $y= \sum_j \alpha_j e_j $ if and only if $\lim_i \alpha_j(i)=\alpha_i$.} \claim For any sequence $\{y(i)=\sum_j \alpha_j(i) e_j\}$ of points in a unit ball, {\bf \red there exists a subsequence $\{\tilde y(i)=\tilde \alpha_j(i) e_i\}$ weakly converging to $y\in H$.} \proof Indeed, $|\alpha_j(i)|\leq 1$, hence there exist a subsequence $\tilde y(i)= \tilde \alpha_j(i) x_j$ with $\tilde \alpha_j(i)$ converging for each $j$. The limit belongs to the unit ball because otherwise $\left|\sum_{j=1}^n \tilde \alpha_j(i) e_j\right|>1$, which is impossible. \endproof \remark Note that {\bf \red the function $x\arrow |x |$ is not continuous in weak topology.} Indeed, weak limit of $\{e_i\}$ is 0. The proof above shows that $|\cdot|$ is semicontinuous. \newpage {\bf \blue Compact operators (reminder)} \definition {\bf \blue Precompact set} is a set which has compact closure. {\bf \blue A compact operator} is an operator which maps bounded sets to precompact. \theorem Let $A:\; H \arrow H_1$ be an operator on Hilbert spaces. Then {\bf \red $A$ is compact if and only if it maps weakly convergent sequences to convergent ones.} \newpage {\bf \blue Rellich lemma for a torus} \theorem {\bf \blue (Rellich lemma for a torus)}\\ {\bf \red The identity map $L^2_p(T^n)\arrow L^2_{p-1}(T^n)$. is compact.} \pstep Consider, instead of $L^2_p$-metric, the metric $q_p$ which is orthogonal in the same basis and satisfies $|F_{l_1, ..., l_n}|_{q_p}:= 1+ (2\pi)^p\sum_{i=1}^n l_i^{p}$. Clearly, $|F_{l_1, ..., l_n}|_{q_p} \leq |F_{l_1, ..., l_n}|_{2,p}$ and $|F_{l_1, ..., l_n}|_{q_p}\geq C^{-1} |F_{l_1, ..., l_n}|_{2,p}$, where $C$ is a number of differential monomials of degree $p$. Therefore, {\bf \purple $q_p$ and $L^2_p$ induce the same topology,} and it would suffice to prove the Rellich lemma for the identity map $L^2(T^n, q_p)\arrow L^2(T^n, q_{p-1})$. {\bf \green Step 2:} Now, \[ \frac{|F_{l_1, ..., l_n}|^2_{q_p}}{|F_{l_1, ..., l_n}|^2_{q_{p-1}}}= \frac{\sum_{i=1}^n (2\pi)^p l_i^{2p}}{\sum_{i=1}^n (2\pi)^{p-1} l_i^{2p-2}} \geq \frac {n}{\max l_i^2}. \] {\bf \green Step 3:} Let $x_i\in L^2(T^n, q_p)$ be weakly converging to $x$, with $|x_i|_{q_p}< 1$. %This means that $x_i = \sum \alpha_{l_1, ..., l_n}(i) F_{l_1, ..., l_n}$ %and $\lim_i \alpha_{l_1, ..., l_n}(i) = \alpha_{l_1, ..., l_n}$. Let $x_i = y_i + z_i$, with $y_i$ being the sum of all Fourier terms with $\max |l_i| < N$, and $z_i$ the rest. Then $|z_i-z|_{q_{p-1}} < \frac {\sqrt n} N |z_i-z|_{q_{p}} < \frac {2\sqrt n} N $, and $y_i$ converges to $y$ because it is a sum of finitely many terms which all converge. {\bf \purple We obtain that $\lim_i |x_i -x|_{q_{p-1}}=0$, hence a $x_i$ (strongly) converges to $x$.} \endproof \newpage {\bf \blue Franz Rellich (1906-1955)} \centerline{\epsfig{file=Rellich.jpg,width=0.25\linewidth}} {\small After Weyl's resignation [from G\"ottingen], his former assistant, Franz Rellich, became Institute Director ... Rellich had only a low-level appointment and ... was not an established figure ... There was need for a prominent mathematical figure who was suitable politically to take over the leadership in Gottingen. Furthermore, in mid-December, Rellich was ordered to report on January 7 for ten weeks to a field-sports camp near Berlin. This was, in fact, a mistake, since Rellich, as an Austrian citizen, was not subject to such forced training regimens. When he arrived at the camp, he was not admitted on these grounds. However, on December 27, the Curator had, after some hesitation, replaced Rellich with Werner Weber as acting director of the Mathematical Institute. Rellich himself would lose his position at Gottingen six months later, on June 18. -- S. L. Segal, {\it Mathematicians under the Nazis}} \newpage {\bf \blue Rellich lemma for $C^\infty_K(\R^n)$} \corollary Let $C^\infty_K(\R^n)$ be the space of smooth functions on $\R^n$ with support in a compact set $K$. {\bf \red Then the identity map \[ L^2_p(C^\infty_K(\R^n))\arrow L^2_{p-1}(C^\infty_K(\R^n))\] is compact.} \proof We consider a quotient map $\R^n \arrow T^n$ which is bijective on $K$ for an appropriate choice of a lattice. This embeds $C^\infty_K(\R^n)$ to $C^\infty(T^n)$, and this embedding is compatible with the $L^2_p$-norms. \endproof \newpage {\bf \blue Sobolev's $L^2$-norm on a compact manifold} \definition Let $M$ be a manifold, $\{U_i\}$ a finite atlas, and $\{\psi_i\}$ the corresponding partition of unity. We will identify $U_i$ with bounded subsets in $\R^n$. Given a function $f\in C^\infty(M)$, define {\bf \blue the Sobolev $L^2_p$-metric} $|f|_{2,p}^2$ as $\sum |f\psi_i|^2_{2,p}$, where $ f\psi_i$ is considered as a function with compact support on $U_i \subset \R^n$, and $\cdot|_{2,p}$ is the Sobolev $L^2_p$-metric on $C^\infty_c(\R^n)$. \proposition The topology induced on $C^\infty(M)$ by $L^2_p$ {\bf \red is independent from the choice of $\{U_i\}$ and $\{\psi_i\}$.} \proof Let $\Psi:\; \R^n \arrow \R^n$ be a map with uniformly bounded partial derivatives up to $p$-th. From the definition of the $L^2_p$-norm and the chain rule it follows that \[ C^{-1} |f|_{2,p}^q \leq |\Psi^* f|_{2,p}^q \leq C |f|_{2,p}^q \] where the constant $C$ depends on the supremum of partial derivatives of $\Psi$. Then, for any refinement $\{V_j\}$ of $\{U_i\}$ and the corresponding partition of unity $\{\phi_j\}$, the $L^2_p$-norm of $f\psi_i$ associated with $\{V_j, \phi_j\}$ is bounded by the one associated with $\{U_i, \psi_i\}$. For the same reason the $L^2_p$-norm of $f\phi_j$ associated with $\{V_j, \phi_j\}$ is bounded by the one associated with $\{U_i, \psi_i\}$. This gives an estimate of form $C^{-1} g_2 \leq g_1 \leq C g_2$ for $L^2_p$-metrics associated with a cover and its refinement. To obtain a similar estimate for two different covers, we find a common refinement. \endproof \newpage {\bf \blue Rellich lemma for $C^\infty(M)$.} \theorem {\bf \blue (Rellich lemma)} Let $M$ be a compact manifold. {\bf \red Then the identity map $L^2_p(M)\arrow L^2_{p-1}(M)$ is compact. } \pstep Let $\{U_i\}$ be a finite atlas on $M$ and $\{\psi_i\}$ the corresponding partition of unity. We will identify $U_i$ with bounded subsets in $\R^n$. Then $|f|_{2,p}^2= \sum_i |\psi_i f|_{2,p}^2$, where the second $|\cdot|_{2,p}^2$-norm is taken on a bounded subset in $\R^n$. {\bf \green Step 2:} Let $f_j\in L^2_p(M)$ be a sequence weakly converging to $f$. Then $\psi_i f_j$ weakly converges to a function $\tilde f_i$ with support in $\Supp(\psi_i)$. Using Rellich lemma for functions on $\R^n$ with compact support, we obtain that $\psi_i f_j$ converges in $L^2_{p-1}$ to $\tilde f_i$. Then $f_j = \sum_i \psi_i f_j$ converges in $L^2_{p-1}$ to $\sum_i \tilde f_i$. \endproof \end{document}