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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 8: Sobolev lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, February 17, 2018 } \end{center} \newpage {\bf \blue Fredholm operators (reminder)} \definition A continuous operator $F:\; H_1\arrow H_2$ of Hilbert spaces is called {\bf\blue Fredholm} if its image is closed and kernel and cokernel are finite-dimensional. \remark {\bf \red ``Cokernel'' of a morphism $F:\; H_1\arrow H_2$ of topological vector spaces is often defined as $\frac{H_2}{\overline {\im F}}$.} \definition An operator $F:\; H_1\arrow H_2$ {\bf \blue has finite rank} if its image has finite rank. \claim An operator $F:\; H_1\arrow H_2$ {\bf \red is Fredholm} if and only if there exists $F_1:\; H_2\arrow H_1$ such that {\bf \red the operators $\Id-FF_1$ and $\Id - F_1 F$ have finite rank.} \proof This is because $F$ defines an isomorphism $F:\; H_1/\ker F \arrow \im F$ as shown above. \endproof \newpage {\bf \blue Connections (reminder)} \definition Recall that {\bf \blue a connection} on a bundle $B$ is an operator $\nabla:\; B \arrow B \otimes \Lambda^1 M$ satisfying $\nabla(fb) = b \otimes df + f \nabla(b)$, where $f \arrow df$ is de Rham differential. When $X$ is a vector field, we denote by $\nabla_X(b)\in B$ the term $\langle \nabla(b), X\rangle$. \remark In local coordinates, connection on $B$ is a sum of differential and a form $A\in \End B \otimes \Lambda^1 M$. Therefore, $\nabla_X$ is a derivation along $X$ plus linear endomorphism. This implies that {\bf\red any first order differential operator on $B$ is expressed as a linear combination of the compositions of covariant derivatives $\nabla_X$ and linear maps.} This follows from the definition of the first order differential operator: {\bf \purple by definition, it is a linear combination of partial derivatives combined with linear maps.} \newpage {\bf \blue $L^2_p$-metrics and connections (reminder)} \definition Let $F$ be a vector bundle on a compact manifold. The {\bf \blue $L^2_p$-topology} on the space of sections of $F$ is a topology defined by the norm $|f|_p$ with $|f|^2_p=\sum_{i=0}^p \int_M |\nabla^if|^2\Vol_M$, for some connection and scalar product on $F$ and $\Lambda^1M$. \remark {\bf \red The metric $|f|^2_p$ is equivalent to the Sobolev's $L^2_p$-metric on $C^\infty(M)$.} Indeed, all partial derivatives of a function $f$ are expressed through $\nabla^if$, hence an $L^2$-bound on partial derivatives gives $L^2$-bound on $\nabla^if$, and is given by such a bound. From now on, {\bf \blue we write $(x, y)$ instead of $\int_M (x, y) \Vol_M$.} This metric is also denoted $L^2$; the space of sections of $B$ with this metric $(B, L^2)$. \definition We define the {\bf \blue Sobolev's $L^2_p$-metric on vector bundles} by $L^2_p(x, y)= \sum_{i=0}^p (\nabla^i(x), \nabla^i(y))$. \newpage {\bf \blue Properties of $L^2_p$-metric} These results were proven earlier. \claim {\bf \red The Sobolev's $L^2_p$-metric is equivalent to \\ $g(x, y):= (\nabla^p(x), \nabla^p(y)) + (x,y)$.} \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. } \theorem Let $B$ be a vector bundle, and $D:\; B \arrow B$ a differential operator which has the same symbol as $(\nabla^p)^* \nabla^p$. {\bf \red Then $D:\; (B, L^2_{2p}) \arrow (B, L^2)$ is Fredholm.} \newpage {\bf \blue Laplace operators (reminder)} \definition Let $M$ be a Riemannian manifold, and $d:\; \Lambda^*(M) \arrow \Lambda^{*+1}(M)$ de Rham differential. Then $\Delta:=dd^*+d^*d$ is called {\bf \blue the Laplacian}. \definition Let $M$ be a Riemannian manifold, and $B$ a bundle with orthogonal metric and a connection $\nabla:\; B \arrow B \otimes \Lambda^1 M$. Using the formula $\nabla(b\otimes \eta)= \nabla(b) \wedge \eta + b \otimes d\eta$, we extend $\nabla$ to an operator $\nabla:\; B \otimes \Lambda^i M \arrow B \otimes \Lambda^{i+1} M$ satisfying the Leibnitz equation. This operator is denoted $d_\nabla$ to distinguish it from the connection. {\bf \blue The Laplacian with coefficients in $B$} is $d_\nabla d^*_\nabla +d^*_\nabla d_\nabla$. \theorem {\bf \red The Laplacian has the same symbol $\sigma\in \Sym^2(TM) \otimes \End(\Lambda^*M \otimes B)$ as $\nabla^*\nabla$, and it is equal to $g^{-1} \otimes \Id_{B\otimes \Lambda^*M}$,} where $g^{-1}\in \Sym^2TM$ is the bivector which corresponds to the Riemannian metric. The following corollary is immediate. \corollary {\bf \red The Laplacian is a Fredholm map} from $(\Lambda^*(M) \otimes B, L^2_{p})$ to $(\Lambda^*(M) \otimes B, L^2_{p-2})$. \proof Indeed, {\bf \purple Laplacian is a sum of a Fredholm map $(\nabla^*)\nabla$ and a compact operator} (all lower order differential operators are compact by Rellich lemma). \endproof \newpage {\bf \blue Green operator} \claim Let $\Delta:\; (\Lambda^*(M), L^2_2) \arrow (\Lambda^*(M), L^2)$ be the Laplacian operator {\bf \red Then $\im\Delta= \ker \Delta^\bot$, taken with respect to the $L^2$-metric.} \\ \proof Since $\Delta$ is self-adjoint with respect to $L^2$-metric, for each $x, y \in (\Lambda^*(M), L^2_2)$ one has $(x, \Delta y)=0$ $\Leftrightarrow$ $(\Delta x, y)=0$. Therefore, $x\in \ker \Delta$ $\Leftrightarrow$ $x \bot \im \Delta$. \endproof \corollary The restriction $\Delta:\; (\im\Delta, L^2_2) \arrow (\im\Delta, L^2)$ {\bf \red is an isomorphism of Hilbert spaces.} \endproof \definition {\bf \blue The Green operator} $G_\Delta$ is a map $(\Lambda^*(M), L^2) \stackrel{G_\Delta}\arrow (\Lambda^*(M), L^2)$ defined as $\Delta^{-1}$ on $\im \Delta$ and as 0 on $\ker\Delta= \im \Delta^\bot$. \claim {\bf \red The Green operator $G_\Delta:\; (\Lambda^*(M), L^2) \arrow (\Lambda^*(M), L^2)$ is self-adjoint and compact} in the usual $L^2$-metric on $\Lambda^*(M)$. \proof Since $\Delta$ is self-adjoint on $\im \Delta$, the same is true for $\Delta^{-1}$. However, when $x\in \im\Delta^\bot$, one has $0 = (G_\Delta x, y)= (x, G_\Delta y)$ as shown above. Compactness follows immediately from Rellich lemma, because $G_\Delta$ is a composition of a continuous operator $(\Lambda^*(M), L^2) \stackrel{\Delta^{-1}}\arrow (\Lambda^*(M), L^2_2)$ and a compact map $(\Lambda^*(M), L^2_2) \stackrel{\Id}\arrow (\Lambda^*(M), L^2)$. \endproof \newpage {\bf \blue Green operator diagonalizes} \theorem The Green operator $G_\Delta:\; (\Lambda^*(M), L^2) \arrow (\Lambda^*(M), L^2)$ {\bf \red can be diagonalized in an orthonormal basis.} Its eigenvalues are non-negative and converge to 0, and each eigenspace is finite-dimensional. \proof Follows from the von Neumann spectral theorem. \endproof Today I will prove the following theorem. \theorem Let $\alpha \in L^2(\Lambda^*(M))$ be an eigenvector of $G_\Delta$, $G_\Delta(\alpha) = \lambda \alpha$. {\bf \red Then $\alpha$ is smooth.} \proof Notice that the identiry map $L^2_p(\Lambda^*(M))\arrow L^2_{p-i}(\Lambda^*(M))$ is continuous for all $i\geq 0$. This gives a natural chain of embeddings $L^2_p(\Lambda^*(M))\subset L^2_{p-1}(\Lambda^*(M)) \subset ... \subset L^2(\Lambda^*(M)).$ Since $\lambda^k \alpha =G_\Delta^k(\alpha)$ belongs to $L^2_{2k}(\Lambda^*(M))$, we have $\alpha \in \bigcap_p L^2_p(\Lambda^*(M))$. Then the theorem is implied by the following result of Sobolev, proven later today. \theorem {\bf \blue (Sobolev)} {\bf \red Any vector in the intersection of all $L^2_p(\Lambda^*(M))$ is represented by a smooth form: $\bigcap_p L^2_p(\Lambda^*(M)) = \Lambda^*(M)$.} \remark {\bf \purple The same arguments work for Laplacian with coefficients in a vector bundle.} \newpage {\bf \blue Sobolev lemma} \definition Let $B$ be a bundle over $M$. Recall that {\bf \blue $C^l$-topology} on the space of sections $C^l(B)$ of $B$ of class $C^l$ is defined by the norm $|b|_{C^p}= \sup_M \sum_{i=0}^l |\nabla^i b|$. \exercise Prove that {\bf \purple $C^l(B)$ is a Banach space with respect to this norm.} Sobolev's theorem $\bigcap_p L^2_p(\Lambda^*(M)) = \Lambda^*(M)$ is immediately implied by the following lemma. \theorem {\bf \blue (Sobolev lemma)}\\ Let $\{b_i\}$ be a sequence of sections of a vector bundle $B$ over a manifold $M$ with $\dim M=n$, converging to $b$ in $L^2_s$, where $s> l + \frac n 2$. {\bf \red Then it converges to a section in $C^l(B)$ in $C^l$-topology.} It is proven later today. \newpage \begin{center} \centerline{\epsfig{file=Sobolev_SL_7.jpg,width=0.50\linewidth}} \small Sergei Lvovich Sobolev \\ 6 October 1908 - 3 January 1989 \end{center} \newpage {\bf \blue Fourier series (reminder)} \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 Sobolev's $L^2$-norm on $C^\infty_c(\R^n)$ (reminder)} \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 (reminder)} \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 \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.} \newpage {\bf \blue Sobolev Lemma on a circle} \lemma Consider Fourier series on $S^1$: $f(t):= \sum_{k\in\Z} \tau_k e^{2\pi\1kt}$. Suppose that $\sum_{k\in\Z} k^{2+l} |\tau_k|^2$ converges. {\bf \red Then $\sum_{k\in\Z} \tau_k e^{2\pi\1kt}$ converges to a function of class $C^l$ in $C^l$-topology.} \pstep If $l=0$, convergence of $\sum_{k\in\Z} k^{2} |\tau_k|^2$ {\bf \purple implies that $\sum_{k\in\Z} \tau_k e^{2\pi\1kt}$ converges absolutely,} because the Cauchy-Schwarz inequality $\left(\sum a_ib_i\right)^2 \leq \sum a_i^2 \sum b_i^2$ gives after putting $a_i b_i = |\tau_i|$, $a_i = i |\tau_i|$ \[ \sum_k k^{2}|\tau_k|^2 \geq \left (\sum_k |\tau_k|\right)^2 \left(\sum_{k=0}^\infty k^{-2} \right)^{-2} \] Therefore it converges in $C^0$-topology. {\bf \green Step 2:} $\frac {d^kf} {dt^k}= \sum_{k\in\Z} k^{l}\tau_k e^{2\pi\1kt}$, and this series converges absolutely when $\sum_{k\in\Z} k^{2+l} |\tau_k|^2< \infty$ for the same reason. \endproof \corollary Let $\{f_i\}$ be a sequence of smooth functions on $S^1$ which converges in $L^2_{p+1}$. {\bf \red Then it also converges in $C^p$-topology.} \endproof \newpage {\bf \blue Sobolev Lemma on a torus} \lemma Consider Fourier series on a torus $T^n$: \[ f = \sum_{k_1, ... k_n \in \Z^n} \tau_{k_1, ... k_n} e^{2\pi\1 \sum_{i=1}^n k_i t_i} \ \ \ (*) \] Suppose that \[ \sum_{k_1, ... k_n \in \Z^n} |\tau_{k_1, ... k_n}|^2\sum_{i=1}^n k_i^{2+2n + l} \ \ \ (**) \] converges. {\bf \red Then $\sum_{k\in\Z} \tau_k e^{2\pi\1kt}$ converges to a function in $C^l$-topology.} \pstep If $l=0$, convergence of (**) implies absolute convergence of (*). Indeed, the Cauchy-Schwarz inequality $\left(\sum a_\alpha b_\alpha\right)^2 \leq \sum a_\alpha^2 \sum b_\alpha^2$ applied to $a_\alpha b_\alpha = |\tau_\alpha|$, $a_\alpha = \sum k_i^{2+2n} |\tau_\alpha|$, where $\alpha=(k_1, ..., k_n)$ is a multi-index, gives \[ \sum_{k_1, ... k_n}\sum k_i^{2+2n} |\tau_{k_1, ... k_n}|^2 \geq \left (\sum_{k_1, ... k_n} |\tau_{k_1, ... k_n}|\right)^2 \left(\sum_{k_1, ... k_n}^n k_i^{-2-2n} \right)^{-2}. \] The last sum converges, hence $\sum |\tau_{k_1, ... k_n}|$ converges in $C^0$. {\bf \green Step 2:} Same computation as above (left as an exercise). \endproof \newpage {\bf \blue Sobolev Lemma} \corollary Let $\{f_i\}$ be a sequence of smooth functions on a torus $T^n$ which converges in $L^2_{s}$, with $s> l + \frac n 2$. {\bf \red Then it also converges in $C^l$-topology.} \endproof \theorem {\bf \blue (Sobolev lemma)} Let $B$ be a bundle on a compact manifold $M$, and $\{f_i\}$ be a sequence of smooth functions which converges in $L^2_{s}$, with $s> l + \frac n 2$. {\bf \red Then it also converges in $C^l$-topology.} \pstep Let $\{U_j\}$ be a finite atlas on $M$ and $\{\psi_j\}$ the corresponding partition of unity. We will identify $U_j$ with bounded subsets in $\R^n$. Then $|f|_{p}^2= \sum_j |\psi_j f|_{p}^2$, where the second $|\cdot|_{p}^2$-norm is taken on a bounded subset in $\R^n$, considered as a subset in a torus. {\bf \green Step 2:} Let $\{f_i\}$ be a sequence of sections of a bundle $B$ converging to $f$ in $L^2_s$. Then $\{\psi_j f_i\}$ converges to $\psi_j f$ in $L^2_s$. Applying Sobolev lemma for torus, we obtain that $\{\psi_j f_i\}$ converges to $\psi_j f$ in $C^l$. \endproof \end{document}