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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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





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