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\lhead{\tiny Hogde theory, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{7}{Hodge theory 7: Sobolev $L^2_p$ norms}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\scriptsize
  {\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra stuff to work.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sobolev $L^2_p$-spaces}
\label{_L^2_s-on-torus_Subsection_}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf Hilbert basis} in a Hilbert space $H$ is a set of
linearly independent, orthogonal vectors generating
a space $H_0$ which is dense in $H$.
\ed

\exercise
Consider the space of complex-valued functons
on an $n$-dimensional compact torus
$T^n = \R^n /\Z^n$  with the metric given
by $|f|^2=\int_{T^n} |f|^2\Vol$, where $\Vol$
is the standard volume form on a torus, induced 
from $\R^n$. Denote by $L^2(T^n)$ the Hilbert space
obtained as a completion of $C^\infty(T^n)$ 
with this metric. Prove that the following
functions constitute an orthonormal basis in 
$L^2(T^n)$
\begin{equation} \label{_Fourier_mono_def_Equation_}
\exp\left({2\pi\1 \sum_{i=1}^n k_i t_i}\right)
\end{equation}
where $k_1, ..., k_n$ run through the set $\Z^n$ of
all integer $n$-tuples.
\ez

\definition
The functions \eqref{_Fourier_mono_def_Equation_}
are called {\bf the Fourier monomials} on a torus.
\ed

\definition
Let $f:\; \R^n \arrow \R$ be a smooth function with compact support.
For any 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.
$
{\bf The $L^2_p$ Sobolev norm} $|f|_p$ is 
defined as follows:
\[
|f|_p^2= \sum_{\deg P_\alpha\leq p} 
 \int \left|P_\alpha(f) \right |^2\Vol
\]
where the sum runs through all differential monomials of degree $\leq p$,
and $\Vol$ is the standard volume form. This is a positive definite
quadratic form on the space $C^\infty_c(\R^n)$ of functions with 
compact support, and its square root gives the norm. The Sobolev
$L^2_p$ norm on sections of trivial bundle is defined the same way.
Also, the same formula can be used to define the $L^2_p$-norm
on functions on a torus.
\ed

\exercise
Consider the Fourier series for the function $f$:
\[
f = \sum_{k_1, ... k_n \in \Z^n} \tau_{k_1, ... k_n} 
e^{2\pi\1 \sum_{i=1}^n k_i t_i}
\]
\enum
\ite
Prove that the $L^2_p$-norm can be written as
\[
|f|_s^2 = \sum_{k_1, ... k_n \in \Z^n} 
  \bigg(|\tau_{k_1, ... k_n}|^2 \sum_{i=1}^n\Psi(k_1, ..., k_n)\bigg),
\]
where $\Psi(k_1, ..., k_n)=\sum_\alpha (2\pi)^{2d} |P_\alpha(k_1, ..., k_n)|^2$,
where $P_\alpha$ runs through all monomials of degree $d\leq p$.
\ite[!] Prove that $L^2_p$-norm is equivalent
to the norm 
\[
|f|_{s,\bullet}^2 = \sum_{k_1, ... k_n \in \Z^n} 
  \bigg(|\tau_{k_1, ... k_n}|^2 \sum_{i=1}^n 1+ k_i^{2p} \bigg)
\]
\ee
\ez

\exercise[!]\label{_Rellich_torus_Exercise_} 
(Rellich lemma) \\
Prove that the
identity map $L^2_s(\R^n)\arrow L^2_{s-1}(\R^n)$ 
is compact on the Hilbert space generated by
functions with support in an open ball of radius $R$,
for any  given $R>0$.
\ez

\hint Use the previous exercise. \eh

\exercise[*]
Is the map $L^2_s(\R^n)\arrow L^2_{s-1}(\R^n)$ 
compact on the Hilbert space generated by all functions with compact support?
\ez

\exercise
Consider the Fourier series of one variable
\begin{equation}\label{_Fourier_1_dim_Equation_}
\sum_{k\in\Z} \tau_k e^{2\pi\1kt}
\end{equation}
Supppose that $\sum_{k\in\Z} k^{2+2l} |\tau_k|^2$ converges.
Prove that \eqref{_Fourier_1_dim_Equation_}
converges to a function of class $C^l$.
\ez

\exercise
Prove an inequality
\[
\bigg(\sum_{k_1, ... k_n \in \Z^n\backslash 0} 
\frac{|\gamma_{k_1, ... k_n}|}{\sum \left|k_i^{n  + 1}\right|}\bigg)^2
\leq 
\bigg(\sum_{k_1, ... k_n \in \Z^n}|\gamma_{k_1,
... k_n}|^2\bigg)
\bigg(\sum_{k_1, ... k_n \in \Z^n}\frac 1 {\sum |k_i|^{n+2}})\bigg)
\]
\ez

\hint Use the Cauchy-Schwarz inequality.
\eh

\exercise
Consider the Fourier series
\begin{equation}\label{_Fourier_n_dim_Equation_}
\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}.
\end{equation}
\enum
\ite
Suppose that the series
\[ 
   \sum_{k_1, ... k_n \in \Z^n} \bigg(|\tau_{k_1, ... k_n}|^2
   \sum_{i=1^n} k_i^{n  + 1}\bigg)
\]
converges. Prove that \eqref{_Fourier_n_dim_Equation_}
converges to a continuous function.
\ite[!] 
Suppose that the series \[
  \sum_{k_1, ... k_n \in \Z^n} 
   |\tau_{k_1, ... k_n}|^2\sum_{i=1}^n |k_i|^{n + 2
   + 2l} 
\]
converges. Prove that  \eqref{_Fourier_n_dim_Equation_}
converges to a function from $C^l(T^n)$ ($l$ times differentiable).
\ee
\ez

\hint Use the Cauchy-Schwarz inequality.
\eh

\exercise[!] \label{_Sobolev_torus_Exercise_}
(Sobolev's lemma) \\
Let $\{f_i\}$ be a sequence of smooth functions with support in a ball
$B\subset \R^n$ converging in the Sobolev $L^2_s$-norm, and
$s> l+\frac n 2$. Prove that $\{f_i\}$ converges
in $L^2_s(\R^n)$-topology to a function from $C^l(\R^n)$.
\ez

\hint Use the previous exercise.
\eh

\exercise[*]
Is this statement true for any $\{f_i\}$ in $C^\infty_c(\R^n)$,
without assuming that $f_i$ are supported in a ball $B$?
\ez

\exercise[!]
Let $\sum\psi_i =1$ be a partition of unity on $\R^n$,
with $\Supp(\psi_i)\subset U_i$, where $U_i$ is a bounded set.
Let $C^\infty_c(\R^n)$ denote the space of functions with compact
support, and $C^\infty_{U_i}(\R^n)$ the space of functions with support in $U_i$.
Consider the map $C^\infty_c(\R^n)\arrow \bigoplus_i C^\infty_{U_i}(\R^n)$
mapping $f$ to $\bigoplus_i \psi_i f$. Prove that (for an appropriate
partition of unity) it can be
extended to the $L^2_p$-completions of the relevant spaces
$\Psi:\; L^2_p(\R^n)\arrow \bigoplus_i L^2_p(\R^n)_{U_i}$,
where $L^2_p(\R^n)_{U_i}$ denotes the completion of $C^\infty_{U_i}(\R^n)$.
Prove that $\Psi$ has closed image in $\bigoplus_i L^2_p(\R^n)_{U_i}$.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sobolev $L^2_p$-norms on a manifold}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $g$ be a Riemannian metric on a manifold $M$,
and $F$ a vector bundle on $M$ with connection and a
Euclidean or Hermitian metric. Define {\bf the Sobolev $L^2_s$-norm
associated with the metric and the connection} by the formula
\begin{equation}\label{_L^2_s_via_connect_Equation_}
|f|^2_s= \sum_{i=0}^s \int |\nabla^i f|^2\Vol
\end{equation}
where $\nabla^i: F \arrow F\otimes 
   \underbrace{\Lambda^1M\otimes ...}_\text{$i$ times}$ is the
$i$-th power of connection defined in Handout 6, and $|\cdot|^2$
the standard Euclidean (or Hermitian) metric. 
\ed

\exercise
Suppose that $M$ is a torus, $B$ trivial, the metrics
on $B$ and $TM$ are standard, and $\nabla$, $\nabla_0$ 
the standard connections written in stanard coordinates as
$\nabla\left(\sum f_i \xi_i\right) = \sum df_i \xi_i$.
Prove that in this case the $L^2_s$-norm defined 
in \eqref{_L^2_s_via_connect_Equation_}
is equivalent to the one defined in Subsection \ref{_L^2_s-on-torus_Subsection_}.
\ez

\exercise[!]
Let $\nu$ and $\nu'$ be Sobolev $L^2_s$-norms
on $B$ associated with some connections and metrics
(generally speaking, different). Prove that the
corresponding topologies are equivalent. Assume that the
base manifold is compact.
\ez

\hint 
Express arbitrary differential operator through iterated
connections like in Handout 6.
\eh


\exercise[!]
Let $\sum\psi_i =1$ be a partition of unity on a compact manifold $M$,
with $\Supp(\psi_i)\subset U_i$, where $U_i$ is a bounded set.
Let  $C^\infty_{U_i}(M)$ the space of functions with support in $U_i$.
Consider the map $C^\infty(M)\arrow \bigoplus_i C^\infty_{U_i}(M)$
mapping $f$ to $\bigoplus_i \psi_i f$. Prove that it can be
extended to the $L^2_p$-completions of the relevant spaces,
$\Psi:\; L^2_p(M)\arrow \bigoplus_i L^2_p(M)_{U_i}$,
where $L^2_p(M)_{U_i}$ denotes the completion of $C^\infty_{U_i}(M)$.
Prove that $\Psi$ has closed image in $\bigoplus_i L^2_p(M)_{U_i}$.
\ez

\exercise[!] (Rellich lemma)\\
Let $M$ be a compact manifold, $F$ a vector bundle on $M$, and
$L^2_s(F)\arrow L^2_{s-i}(F)$ the identity map. Prove that
this is a compact operator for all $i>0$.
\ez

\hint Use the Rellich lemma for torus (Exercise \ref{_Rellich_torus_Exercise_})
and a partition of unity.
\eh

\exercise[*]
Is this statement true for 
the $L^2_s$-space obtained as a completion
of sections with compact support on a non-compact manifold?
\ez

\exercise[!]
Let $\{f_i\}$ be a sequence of smooth sections of a bundle $B$
on a compact manifold $M$ converging in the Sobolev $L^2_s$-norm, and
$s> l+\frac n 2$. Prove that $\{f_i\}$ converges
in $L^2_s$-topology to a section which is $l$ times differentiable.
\ez

\hint Use the Sobolev lemma for a torus (Exercise \ref{_Sobolev_torus_Exercise_})
and a partition of unity.
\eh 

\exercise[*]
Prove the Sobolev lemma for the space of functions with 
compact support on a non-compact manifold.
\ez

\exercise[!]
Using the injective maps $L^2_s(B) \arrow L^2_{s-i}(B)$, we can consider
all these spaces as subspaces in $L^2(B)$. 
Prove that $\bigcap_s L^2_s(B)$ is the space of smooth sections of $B$.
\ez

\exercise
Let $D_1$, $D_2:\; F \arrow F$ be differential operators
of order $i, j$ on a vector bundle $F$ on a compact manifold.
Prove that the commutator  $[D_1, D_2]:\; L^2_s(F) \arrow L^2_{s-i-j}(F)$
is compact, or find a counterexample.
\ez


\end{document}