text:
\begin{document}
\title{Logarithmic decay for damped hypoelliptic wave and Schr\"odinger equations}
\maketitle

\begin{abstract}
We consider damped wave (resp. Schr\"odinger and plate) equations driven by a hypoelliptic ``sum of squares'' operator $\scrL$ on a compact manifold and a damping function $b(x)$. We assume the Chow-Rashevski-H\"ormander condition at rank $k$ (at most $k$ Lie brackets needed to span the tangent space) together with analyticity of $\M$ and the coefficients of $\scrL$. We prove decay of the energy at rate $\log(t)^{-\frac{1}{k}}$ (resp. $\log(t)^{-\frac{2}{k}}$ ) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Grushin-type operators.
This result follows from a perturbative argument (of independent interest) showing, in a general abstract setting, that quantitative approximate observability/controllability results for wave-type equations imply {\em a priori} decay rates for associated damped wave, Schr\"odinger and plate equations.
The adapted quantitative approximate observability/controllability theorem for hypoelliptic waves is obtained by the authors in~\cite{LL:15,LL:17Hypo}.
\end{abstract}

\begin{keywords}
Stability estimates, hypoelliptic operators, wave equation, resolvent estimates, approximate observability.

\medskip
\textbf{2010 Mathematics Subject Classification:}
35B60,

35H10,

35L05,

93B05,
93B07.
\end{keywords}

\section{Introduction and statements}
\subsection{Damped hypoelliptic evolution equations}
\label{s:intro-hypo}
We consider a smooth compact connected $d$-dimensional manifold $\M$, endowed with a smooth positive density measure $ds$. We denote by $L^2 =L^2(\M) = L^2(\M, ds)$ the space of square integrable functions with respect to this measure. Given a smooth vector field $X$, we define by $X^*$ its formal dual operator for the duality of $L^2(\M)$, that is,
 $$
 \int_\M X^*(u)(x) \overline{v(x)}ds(x) =   \int_\M u(x) \overline{X(v)(x)}ds(x), \quad \text{ for any } u, v \in C^\infty(\M) .
 $$
Given $m\in \N$ and $m$ smooth real vector fields $X_1,\cdots,X_m$, we consider the (H\"ormander's type~I) hypoelliptic operator (also called sub-Riemannian Laplacian, see e.g.~\cite[Remark~1.30]{LL:17Hypo})
\bnan
\label{def:L}
\scrL =   \sum_{i=1}^m X_i^*X_i .
\enan
Note that $\scrL$ is  formally symmetric nonnegative since $( \scrL u,v)_{L^2(\M)}=  \sum_{i=1}^m (X_i u , X_i v )_{L^2(\M)}$ for all $u,v \in C^\infty(\M)$.
Given a nonnegative (so-called damping) function $b \in L^\infty(\M ; \R_+)$, we are interested in the first place in asymptotic properties of the damped wave equation associated to $(\scrL,b)$
\begin{align}
\label{e:damped-hypo}
\begin{cases}
(\d_t^2 + \scrL  + b \d_t )u = 0, \quad \text{ on }(0, + \infty) \times \M ,\\
(u, \d_t u)|_{t=0} = (u_0 , u_1) , \quad \text{ on }\M .
\end{cases}
\end{align}
Solutions of~\eqref{e:damped-hypo} enjoy formally the following dissipation identity (obtained by taking the inner product of~\eqref{e:damped-hypo} with $\d_t u$ and integrating on $(0,T)$):
$$
E(u(T)) - E(u(0)) = -\int_0^T \int_\M b(x)|\d_t u (t,x)|^2 ds(x) \, dt ,\quad E(u) = \frac12 \left(  \sum_{i=1}^m \nor{X_i u}{L^2(\M)}^2+  \nor{\d_tu}{L^2(\M)}^2\right) .
$$
We are also interested in the damped Schr\"odinger equation associated to $(\scrL,b)$
\begin{align}
\label{e:damped-schro-hypo}
\begin{cases}
( i \d_t + \scrL  + i b )u = 0, \quad \text{ on }(0, + \infty) \times \M ,\\
u|_{t=0} = u_0 , \quad \text{ on }\M ,
\end{cases}
\end{align}
for which the $L^2$ norm is a dissipated quantity (obtained by taking imaginary part of the inner product of~\eqref{e:damped-schro-hypo} with $u$ and integrating on $(0,T)$):
$$
\frac12 \nor{u(T)}{L^2(\M)}^2 - \frac12\nor{u_0}{L^2(\M)}^2 = -\int_0^T \int_\M b(x)|u(t,x)|^2ds(x) \, dt .
$$
Another related equation with similar behavior is the damped plate equation associated to $(\scrL,b)$
\begin{align}
\label{e:damped-beam-hypo}
\begin{cases}
(\d_t^2 + \scrL^2  + b \d_t )u = 0, \quad \text{ on }(0, + \infty) \times \M ,\\
(u, \d_t u)|_{t=0} = (u_0 , u_1) , \quad \text{ on }\M .
\end{cases}
\end{align}
Solutions of~\eqref{e:damped-beam-hypo} also enjoy formally a similar dissipation identity
$$
E_P(u(T)) - E_P(u(0)) = -\int_0^T \int_\M b(x)|\d_t u (t,x)|^2 ds(x) \, dt ,\quad E_P(u) = \frac12 \left( \nor{\scrL u}{L^2(\M)}^2+  \nor{\d_tu}{L^2(\M)}^2\right) .
$$

Hence, in the three situations, ``energy'' decays, and an interesting question is to understand if it converges to zero, and if so, at which rate.

\medskip
We shall always assume throughout that the family $(X_i)$ satisfies the Chow-Rashevski-H\"ormander condition (or is ``bracket generating'').
\begin{hypo}
\label{assumLiek}
There exists $\ell \geq 1$ so that for any $x\in \M$, $\Lie^\ell(X_1,\cdots,X_m) (x)=T_x\M$. Denote then by $k \in \N^*$ the minimal $\ell$ for which this holds.
\end{hypo}
Here, $\Lie^\ell$ denotes the Lie algebra at rank $\ell$ of the vector fields. The integer $k$ is sometimes referred to as the \textit{hypoellipticity index} of $\scrL$.
Under Assumption~\ref{assumLiek}, the celebrated H\"ormander \cite{Hor67} and Rothschild-Stein \cite{RS:76} theorems (see also~\cite{BCN:82} for a simpler proof) state that $\scrL$ is
subelliptic of order $\frac{1}{k}$, that is: there is $C>0$ such that for any $u\in C^{\infty}(\M)$, we have
\bnan
\nor{u}{H^{\frac{2}{k}}(\M)}^2\leq C\nor{\scrL u}{L^2(\M)}^2+C\nor{u}{L^2(\M)}^2 . \label{estimhypo3}
\enan
As a consequence, the operator $\scrL$ is selfadjoint on $L^2(\M)$ with domain $\scrL : D(\scrL) \subset L^2(\M) \to L^2(\M)$. Since $H^2(\M) \subset D(\scrL) \subset H^{\frac{2}{k}}(\M)$, $\scrL$ has compact resolvent and thus admits a Hilbert basis of eigenfunctions $(\varphi_j)_{j \in \N}$, associated with the real eigenvalues $(\lambda_j)_{j \in \N}$, sorted increasingly, that is
\bnan
\label{e:spectral-elts}
\scrL\varphi_i=\lambda_i\varphi_i, \qquad (\varphi_i , \varphi_j)_{L^2(\M)} = \delta_{ij} , \qquad 0= \lambda_0 < \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_j \to + \infty .
\enan
This allows in particular to define adapted Sobolev spaces:
$$
 \H^s_\scrL =  \{ u \in \D'(\M) ,  \ (1+\scrL)^\frac{s}{2}   u \in L^2(\M) \} , \quad \nor{u}{\H^s_\scrL}=\nor{(1+\scrL)^\frac{s}{2} u}{L^2(\M)} , \quad  s\in \R ,
$$
where $f(\scrL) u = \sum_{j \in \N} f(\lambda_j) (u , \varphi_j)_{L^2(\M)} \varphi_j$.

In addition to Assumption~\ref{assumLiek}, we will also make the following analyticity assumption.
\begin{hypo}
\label{hypoanal}
The manifold $\M$, the density $ds$, and the vector fields $X_i$ are real-analytic.
\end{hypo}
A non-exhaustive list of classical examples of operators $\scrL$ encompassed by this frameworks is provided in~\cite[Section~1.1]{LL:17Hypo}.
Note that the damping function $b$ does not need to be analytic but only $L^\infty$; in particular our results work for $b=\mathds{1}_{\omega}$ if $\omega$  is a non-empty open subset of $\M$.

On the space $ \H^1_\scrL \times L^2$, the operator $\A =
\left(
\begin{array}{cc}
0   &  \id \\
- \scrL & - b(x)
\end{array}
\right)$ with $D(\A) = \H^2_\scrL \times \H^1_\scrL$ generates a bounded semigroup (from the Hille-Yosida theorem) and~\eqref{e:damped-hypo} admits a unique solution $u \in C^0(\R^+ ; \H^1_\scrL )\cap C^1(\R^+ ;L^2)$. Our main results for damped hypoelliptic waves are summarized in the following two theorems.
\begin{theorem}[Decay rates for damped hypoelliptic waves]
\label{e:decay}
Assume that $b\in L^\infty(\M)$ is such that $b\geq \delta>0$ a.e. on a nonempty open set, together with Assumptions~\ref{assumLiek} and~\ref{hypoanal}. Then, for all $(u_0, u_1) \in \H^1_\scrL \times L^2$, the associated solution to~\eqref{e:damped-hypo} satisfies $E(u(t)) \to 0$. Moreover, for all $j \in \N^*$, there exists $C_j>0$ such that for all $(u_0, u_1) \in D(\A^j)$, the associated solution to~\eqref{e:damped-hypo} satisfies
\begin{equation}
\label{e:log-decay}
E(u(t))^{\frac12} \leq \frac{C_j}{\log(t+2)^{j/k}} \nor{\A^j (u_0, u_1)}{\H^1_\scrL \times L^2} , \quad \text{ for all } t\geq 0 .
\end{equation}
\end{theorem}

Theorem~\ref{e:decay} is actually a consequence of the following result, together with~\cite{BD:08}.
\begin{theorem}[Spectral properties for damped hypoelliptic waves]
\label{t:spectral}
Assume that $b\geq \delta>0$ a.e. on a nonempty open set, together with Assumptions~\ref{assumLiek} and~\ref{hypoanal}. Then, the spectrum of $\A$ contains only isolated eigenvalues with finite multiplicity, and satisfies:
\begin{enumerate}
\item \label{truc} $\ovl{\Sp(\A)}=\Sp(\A)$ and $\ker(\A) = \vect\{(1,0)\}$ (where $1$ denotes the constant function),
\item \label{i:loc-spec} $
\Sp(\A) \subset \left( \big( - \frac12 \|b\|_{L^\infty(\M)} , 0 \big) + i\R\right) \cup
\left([ -  \|b\|_{L^\infty(\M)} , 0] + 0 i \right),$
\item \label{e:expo-bound} there exist $C,\kappa >0$ such that  $\nor{(is-\A)^{-1}}{\L(\H^1_\scrL \times L^2)} \leq C e^{\kappa |s|^k}$ for all $|s|\geq1$,
\item there exist $\eps ,\kappa >0$ such that $\Sp(\A) \cap \Gamma_k(\eps,\kappa ) = \{0\}$, where $\Gamma_k(\eps,\kappa ) = \{z \in \C , \Re(z) \geq - \eps e^{-\kappa |\Im(z)|^k}\}$.
\end{enumerate}
\end{theorem}
The first two points are rather standard, see~\cite{Leb:96}. Item~\ref{e:expo-bound} is the key information in the Theorem, and is a consequence of the main theorem in~\cite[Theorem~1.15]{LL:17Hypo}.
The last point of the theorem states an exponentially small spectral gap, and is a consequence of Item~\ref{e:expo-bound} and Neumann series expansion.

Combined together, Theorems~\ref{e:decay} and~\ref{t:spectral} are the counterparts to~\cite[Th\'eor\`eme~1]{Leb:96} in the case of the usual wave equation ($k=1$, in which case no analyticity is required, and boundary condition can be dealt with).

Note that the fact that $\Sp(\A)\cap i\R = \{0\}$ in Item~\ref{i:loc-spec} (which, in turn, implies that $E(u(t)) \to 0$ in Theorem~\eqref{e:decay} for all solutions to~\eqref{e:damped-hypo}) is actually a consequence of the {\em qualitative} uniqueness:
\bnan
\label{e:UCP-eig-Hypo}
\Big( \varphi \in \H_\scrL^2 , \quad z \in \C, \quad \scrL \varphi  = z \varphi \text{ on }\M , \quad \varphi = 0 \text{ on } \omega \Big) \implies \varphi \equiv 0\text{ on }\M ,
\enan
 proved by Bony~\cite{Bo:69}, as a consequence of the Holmgren-John theorem. Even this weaker property is not well understood for general hypoelliptic operators if we drop Assumption~\ref{hypoanal}, see~\cite{Ba:86}.
Here the key point is the quantification of the Holmgren-John theorem proved in~\cite{LL:15,LL:17Hypo}.

\bigskip
We present analogue results in the case of the damped hypoelliptic Schr\"odinger equation.
We set $\A_S := i \scrL - b$ with $D(\A_S)=D(\scrL)$, so that~\eqref{e:damped-schro-hypo} reformulates as $(\d_t- \A_S )u=0$. Note that $\A_S$ generates a contraction semigroup (from the Hille-Yosida theorem) and~\eqref{e:damped-schro-hypo} admits a unique solution $u \in C^0(\R^+ ; L^2(\M) )$.
Our main results for the damped hypoelliptic Schr\"odinger equation are summarized in the following two theorems.
\begin{theorem}[Decay rates for the damped hypoelliptic Schr\"odinger equation]
\label{e:decay-schro}
Assume that $b\in L^\infty(\M)$ is such that $b\geq \delta>0$ a.e. on a nonempty open set, together with Assumptions~\ref{assumLiek} and~\ref{hypoanal}. Then, for all $u_0 \in L^2(\M)$, the associated solution to~\eqref{e:damped-schro-hypo} satisfies $u(t) \to 0$ in $L^2(\M)$. Moreover, for all $j \in \N^*$, there exists $C_j>0$ such that for all $u_0\in D(\A_S^j)$, the associated solution to~\eqref{e:damped-schro-hypo} satisfies
\begin{equation}
\label{e:log-decay-schro}
\nor{u(t)}{L^2(\M)} \leq \frac{C_j}{\log(t+2)^{2j/k}} \nor{\A_S^j u}{L^2(\M)} , \quad \text{ for all } t\geq 0 .
\end{equation}
\end{theorem}
Note that when comparing~\eqref{e:log-decay-schro} to~\eqref{e:log-decay}, the decay looks better ($\log(t+2)^{-2j/k}$ instead of $\log(t+2)^{-j/k}$) but actually consumes more derivatives: for smooth $b$, $\nor{\A_S^j u}{L^2(\M)} \simeq \nor{u}{\H_\scrL^{2j}}$ whereas $\nor{\A^j U}{L^2(\M)} \simeq \nor{U_0}{\H_\scrL^{j} \times \H_\scrL^{j-1}}$. Hence both decay rates essentially coincide for data having the same regularity.
Theorem~\ref{e:decay-schro} is a consequence of the following result, together with~\cite{BD:08}.
\begin{theorem}[Spectral properties for the damped hypoelliptic Schr\"odinger equation]
\label{t:spectral-schro}
Assume that $b\geq \delta>0$ a.e. on a nonempty open set, together with Assumptions~\ref{assumLiek} and~\ref{hypoanal}. Then, the spectrum of $\A_S$ contains only isolated eigenvalues with finite multiplicity, and satisfies:
\begin{enumerate}
\item \label{i:loc-spec-schro} $
\Sp(\A_S) \subset \big[ -  \|b\|_{L^\infty(\M)} , 0 \big) + i[0,+\infty)$,
\item \label{e:expo-bound-schro} there exist $C,\kappa >0$ such that  $\nor{(is-\A_S)^{-1}}{\L(L^2)} \leq C e^{\kappa |s|^{k/2}}$ for all $s\in \R$,
\item there exist $\eps ,\kappa >0$ such that $\Sp(\A_S) \cap \Gamma_{k,S}(\eps,\kappa ) =\emptyset$, where $\Gamma_{k,S}(\eps,\kappa ) = \{z \in \C , \Re(z) \geq - \eps e^{-\kappa |\Im(z)|^{k/2}}\}$.
\end{enumerate}
\end{theorem}
Note that in the elliptic case $k=1$, the results of Theorems~\ref{e:decay-schro},~\ref{t:spectral-schro} are more or less classical, even though we did not see them written explicitely in the literature. In this situation, analyticity is not necessary and boundary value problems can be dealt with. Our abstract perturbative proof below works as well, as a consequence of~\cite{LL:15} (with Dirichlet boundary conditions). One can however start from the seminal estimates of Lebeau-Robbiano in this situation, see~\cite{LR:95,Leb:96} for Dirichlet conditions and~\cite{LR:97} for Neumann boundary conditions.

\bigskip
A similar result holds for the plate equation. The framework is quite similar to the wave equation. We will work on the space $ \H^2_\scrL \times L^2$ with the operator $\A_P =
\left(
\begin{array}{cc}
0   &  \id \\
- \scrL^2 & - b(x)
\end{array}
\right)$ with $D(\A_P) = \H^4_\scrL \times \H^2_\scrL$. It generates a bounded semigroup and~\eqref{e:damped-beam-hypo} admits a unique solution $u \in C^0(\R^+ ; \H^2_\scrL )\cap C^1(\R^+ ;L^2)$.
\begin{theorem}[Decay rates for damped hypoelliptic plates]
\label{e:decaybeam}
Assume that $b\in L^\infty(\M)$ is such that $b\geq \delta>0$ a.e. on a nonempty open set, together with Assumptions~\ref{assumLiek} and~\ref{hypoanal}. Then, for all $(u_0, u_1) \in \H^2_\scrL \times L^2$, the associated solution to~\eqref{e:damped-beam-hypo} satisfies $E_P(u(t)) \to 0$. Moreover, for all $j \in \N^*$, there exists $C_j>0$ such that for all $(u_0, u_1) \in D(\A^j)$, the associated solution to~\eqref{e:damped-beam-hypo} satisfies
\begin{equation}
\label{e:log-decaybeam}
E_P(u(t))^{\frac12} \leq \frac{C_j}{\log(t+2)^{2j/k}} \nor{\A_P^j (u_0, u_1)}{\H^2_\scrL \times L^2} , \quad \text{ for all } t\geq 0 .
\end{equation}
\end{theorem}
Similar spectral statements as Theorems~\ref{t:spectral} and~\ref{t:spectral-schro} hold for the plate equation. We leave the details to the reader. Again, by using our result~\cite{LL:15}, we could also obtain a logarithmic decay in the elliptic case $k=1$ for a compact manifold with boundary and with Dirichlet boundary conditions. We do not know if this result is new in this context. The literature is quite big, we refer to \cite{Leb:92} and \cite{K:92} for exact control results (implying exponential decay of the damped equation) and e.g. to~\cite{ADZ:14} for a spectral analysis of the decay rate.

\bigskip
Finally, we show that results of Theorems~\ref{e:decay},~\ref{t:spectral},~\ref{e:decay-schro},~\ref{t:spectral-schro} are optimal in general (in case $k>1$; this is already known in the elliptic case $k=1$, see~\cite{Leb:96,LR:97}). This is also the case for Theorem~\ref{e:decaybeam} (and the associated spectral statement); we do not state the result for the sake of brevity.
\begin{proposition}
\label{Prop:BCG-damped}
Consider the manifold with boundary $\M=[-1,1]\times(\R/\Z)$, endowed with the Lebesgue measure $dx$, and for $k \in (1, +\infty)$, define the operator $\scrL =  - \big( \partial_{x_1}^2 + x_1^{2(k-1)}\partial_{x_2}^2 \big)$, with Dirichlet conditions on $\d \M$.
Assume that $\supp(b) \cap \left\{x_1=0\right\} = \emptyset$. Then there exist  $C, \kappa>0$ and a sequence $(s_j)_{j\in \N}$ with $s_j\to + \infty$  such that
\begin{align}
\label{e:lower-resolvent-bounds}
\nor{(is_j-\A)^{-1}}{\L(\H^1_\scrL \times L^2)} \geq C e^{\kappa s_j^k} , \quad  \nor{(is_j-\A_S)^{-1}}{\L(\H^1_\scrL \times L^2)} \geq C e^{\kappa s_j^{k/2}} , \quad  \text{for all } j \in \N.
\end{align}
Moreover, if for all $(u_0, u_1) \in D(\A)$, the associated solution to~\eqref{e:damped-hypo} satisfies
\begin{equation*}
E(u(t))^{\frac12} \leq  f(t) \nor{\A(u_0, u_1)}{\H^1_\scrL \times L^2} , \quad \text{ for all }  t\geq 2 ,
\end{equation*}
then there is $C>0$ such that $f(t) \geq \frac{C}{\log(t)^{1/k}}$. Similarly, if for all $u_0 \in \H_\scrL^1$, the associated solution to~\eqref{e:damped-schro-hypo} satisfies
\begin{equation*}
\nor{u(t)}{L^2(\M)} \leq f(t)\nor{\A_S u}{L^2(\M)} , \quad \text{ for all } t\geq 2 ,

\end{equation*}
then there is $C>0$ such that $f(t) \geq \frac{C}{\log(t)^{2/k}}$.
\end{proposition}
Recall that for $k\in \N^*$, the operator $\scrL =  - \big( \partial_{x_1}^2 + x_1^{2(k-1)}\partial_{x_2}^2 \big)$ satisfies precisely Assumption~\ref{assumLiek}.
The first part is thus a consequence of \cite[Section~2.3]{BeauchardCanGugl} as reformulated in~\cite[Proposition~1.14]{LL:17Hypo}. It prove the optimality in general of Item~\ref{t:spectral} in Theorem~\ref{e:expo-bound}.
The second part is a corollary of the first together with~\cite{BD:08}, and proves optimality of~\eqref{e:log-decay}.

A reformulation of~\cite{Letrouit:20} (e.g. together with~\cite{Har:89}) in the present context states that if $\vect (X_1(x), \cdots , X_m(x)) \neq T_x\M$ for $x$ in a dense subset of $\M$, and $\M \setminus \supp(b) \neq \emptyset$, then uniform decay does not hold: there is no function $f : \R^+ \to \R^+$ with $f(t)\to 0$ such that $E(u(t)) \leq f(t)E(u(0))$. This contrasts with the Riemannian case~\cite{RT:74,BLR:92}, and gives in this context a stronger interest to the result of Theorem~\ref{e:decay} as compared to the Riemannian counterpart.

However, one may notice that logarithmic decay as in Theorem~\ref{e:decay} is not always optimal. Combining for instance~\cite[Theorem~1]{BS:19} together with~\cite[Theorem~2.3]{AL:14} implies that $\frac{C_j}{\log(t+2)^{j/k}}$ in~\eqref{e:log-decay} can be replaced by $\frac{C_j}{t^{j/2}}$ (and this is probably not optimal) in the geometric setting of Proposition~\ref{Prop:BCG-damped} if $b(x_1,x_2) = \mathds{1}_{(a,b)}(x_2)$, for any $a<b$.

Similarly, logarithmic decay in Theorem~\ref{e:decay-schro} is not always optimal. For instance~\cite[Theorem~1]{BS:19} (together with classical equivalence between observability for the conservative system and uniform stabilization for the damped system) implies that in the geometric setting of Proposition~\ref{Prop:BCG-damped} if $b(x_1,x_2) = \mathds{1}_{(a,b)}(x_2)$ for $a<b$, then uniform decay holds, that is: there are $C,\gamma>0$ such that $\nor{u(t)}{L^2} \leq C e^{-\gamma t}\nor{u_0}{L^2}$ for all solutions to~\eqref{e:damped-schro-hypo}.

\bigskip
Let us finally remark that all proofs below rely on the approximate observability/controllability of the hypoelliptic wave equation with optimal cost. The latter result is proved by the authors in~\cite{LL:17Hypo}.
It is interesting to notice that in the elliptic case ($k=1$ in the discussion above), the approximate observability/controllability of the wave equation (proved in~\cite{LL:15}) with optimal (exponential) cost allows to recover many known control results obtained with Carleman estimates.
In particular, it implies
\begin{enumerate}
\item null-controllability of the heat equation with optimal short-time behavior, as proved in~\cite{EZ:11} and~\cite[Proposition~1.7]{LL:18} (the original result is~\cite{LR:95,FI:96}),
\item approximate observability/controllability of the heat equation with optimal (exponential) cost~\cite[Chapter~4]{LL:17Hypo} (the original result is~\cite{FCZ:00}),
\item optimal logarithmic decay for the damped wave equation, see Theorem \ref{e:decay} for $k=1$ (the original result is~\cite{Leb:96,LR:97}).
\end{enumerate}
Here, we provide a proof of the last point in a general framework presented in Section~\ref{s:abstract} below, and deduce counterparts for hypoelliptic equations using~\cite{LL:17Hypo}.

\subsection{From approximate control to damped waves : abstract setting}
\label{s:abstract}

As already mentioned, we prove the results of Theorems~\ref{t:spectral} and~\ref{e:decay} in an abstract operator setting. This allows us to stress links between the cost of approximate controls and a priori decay rates for damped waves.
This follows the spirit of e.g.~\cite{Har:89,BZ:04,Phung:01,Miller:05,Miller:06b,TW:09,EZ:11,AL:14,CPSST:19},
exploring the links between different equations and their control
properties (i.e. observability, controllability,
stabilization...).
Here, we follow closely~\cite{AL:14}.

Let $H$ and $Y$ be two Hilbert spaces (resp. the state space and the observation/control space) with norms $\| \cdot \|_H$ and $\| \cdot \|_Y$, and associated inner products $( \cdot , \cdot )_H$ and $( \cdot , \cdot )_Y$.
We denote by $A : D(A)\subset H \to H$ a {\em nonnegative} selfadjoint operator with compact resolvent, and $B \in \L(Y;H)$ a control operator. We recall that $B^* \in \L(H;Y)$ is defined by $(B^* h ,y)_Y = (h, B y)_H$ for all $h \in H$ and $y \in Y$. We define $H_1 = D(A^\frac12)$, equipped with the graph norm $\nor{u}{H_1} := \|(A+\id)^\frac12 u \|_H$, and its dual $H_{-1} = (H_1)'$ (using $H$ as a pivot space) endowed with the norm $\nor{u}{H_{-1}}:=\|(A+\id)^{-\frac12} u \|_H$.

In applications to Theorems~\ref{e:decay}-\ref{t:spectral}-\ref{e:decay-schro}-\ref{t:spectral-schro}, we take $H=Y=L^2(\M)$, $A=\scrL$ and $B = B^*$ is multiplication by the function $\sqrt{b}$.

We introduce in this abstract setting the wave equation
\begin{align}
\label{eq: abstract waves}
\begin{cases}
\d_t^2 u + A u =  F  , \\
(u, \d_t u)|_{t=0} = (u_0 , u_1) ,
\end{cases}
\end{align}
the damped wave equation
\begin{align}
\label{eq: damped abstract waves}
\begin{cases}
\d_t^2 u + A u + B B^* \d_t u = 0, \\
(u, \d_t u)|_{t=0} = (u_0 , u_1) ,
\end{cases}
\end{align}
and the damped Schr\"odinger equation
\begin{align}
\label{eq: damped abstract schro}
\begin{cases}
i \d_t u + A u + i B B^* u = 0, \\
u|_{t=0} = u_0 .
\end{cases}
\end{align}

\begin{definition}
\label{d:approx-obs}
Given $T>0$ and a function $G : \R_+ \to \R_+$, we say that the wave equation~\eqref{eq: abstract waves} with $F=0$ is approximately observable from $B^*$ in time $T$ with cost $G$ if there is $\mu_0>0$ such that for all $(u_0,u_1)\in H_1 \times H$, the associated solution $u$ to~\eqref{eq: abstract waves} with $F=0$ satisfies
\bnan
\label{th-estimate-k-bis}
\nor{(u_0,u_1)}{H\times H_{-1}} \leq G(\mu) \nor{B^*u}{L^2(0,T;Y)} +\frac{1}{\mu}\nor{(u_0,u_1)}{H_1 \times H} , \quad \text{for all }\mu \geq \mu_0 .
\enan
\end{definition}
According to~\cite{Robbiano:95} or~\cite[Appendix]{LL:17approx}, this is equivalent to approximate controllability ($\eps$ close) with cost $G(1/\eps)$.
This is satisfied for the usual wave equation in a general context with $B^* = \mathds{1}_\omega$, $G(\mu) = Ce^{\kappa \mu}$, for all $T> 2\sup_{x\in \M} d_g(x,\omega)$ (where $d_g$ is the Riemannian distance), as proved in~\cite{LL:15}.
For the hypoelliptic wave equation, we proved in \cite[Theorem~1.15]{LL:17Hypo} that this is satisfied for $B^* = \mathds{1}_\omega$, $G(\mu) = Ce^{\kappa \mu^k}$, for all $T> 2\sup_{x\in \M} d_\scrL(x,\omega)$ (where $d_\scrL$ is the appropriate sub-Riemannian distance and $k$ the hypoellipticity index of $\scrL$).

\bigskip
Our main results can be divided in several steps. Firstly we have
\begin{proposition}
\label{c:res-fct-propres}
Let $G : \R_+ \to \R_+$ be such that $G(\mu) \geq \frac{c_0}{\mu}>0$ for  $\mu \geq \mu_0$. Assume that there is $T>0$ such that the wave equation~\eqref{eq: abstract waves} with $F=0$ is approximately observable from $B^*$ in time $T$ with cost $G$ in the sense of Definition~\ref{d:approx-obs}.
Then, we have
\bnan
\label{e::UCPeigenvalue}
 \big(\lambda \in \C,\  v\in D(A),\ Av = \lambda^2 v ,\ B^*v=0 \big)\implies v=0 ,
\enan
and there is $\lambda_0>0$ such that for all $\alpha >0$,
\begin{align}
\label{e:res-fct-prop}
\nor{v}{H}\leq \frac{K}{\alpha}(\lambda + \sqrt{2}+\alpha)  G(\lambda + \sqrt{2}+\alpha) \big( \nor{B^* v}{Y} + C\nor{(A - \lambda^2)v }{H}  \big) , \quad \text{ for all }v \in D(A) , \lambda \geq \lambda_0 .
\end{align}
with $K = \sqrt{T} + c_0^{-1}$ and $C>0$ a constant depending only on $B$ and $T$.
\end{proposition}
Note that in this statement, $\sqrt{2}$ can be replaced by $1$ at the cost of a slightly longer proof, and $\lambda_0$ is the $\mu_0$ given in the definition of approximate observability. In most applications we have in mind, $G(\mu) \approx e^{\kappa \mu^k}$ and the estimate is better for smaller values of $\alpha$. In a situations in which one would have $G(\mu) \approx \mu^\gamma$, then a better choice of $\alpha$ would be $\alpha \approx \lambda$, so that~\eqref{e:res-fct-prop} remains a bound of order $G(\lambda)$.

\medskip
Secondly, we assume that for some function $\mathsf{G}$ and some $\lambda_0>0$ we have
\begin{align}
\label{e:res-fct-prop-bis}
\nor{v}{H}\leq \mathsf{G}(\lambda) \big( \nor{B^* v}{Y} + \nor{(A - \lambda^2)v }{H}  \big) , \quad \text{ for all }v \in D(A) , \lambda \geq \lambda_0 .
\end{align}
This is precisely~\eqref{e:res-fct-prop} with $\mathsf{G}(\lambda) = \frac{K(1+C)}{\alpha}(\lambda + \sqrt{2}+\alpha)  G(\lambda + \sqrt{2}+\alpha)$.
From this estimate, we deduce the sought spectral properties (resolvent estimates and localization of the spectrum linked to the function $\mathsf{G}$). See Section~\ref{s:damped-schro} for the damped Schr\"odinger equation and Section~\ref{s:damped-wave} for the damped wave equation.
A direct application of Proposition~\ref{c:res-fct-propres} gives in the context of hypoelliptic operators.
\begin{corollary}
\label{corresolvL}
With the notations of Section~\ref{s:intro-hypo}, assume that $b\in L^\infty(\M)$ is such that $b\geq \delta>0$ a.e. on a nonempty open set, together with Assumptions~\ref{assumLiek} and~\ref{hypoanal}.
Then,~\eqref{e:UCP-eig-Hypo} is satisfied and there is $\nu>0$, $C>0$ and $\lambda_0>0$ such that,
\begin{align*}
\nor{v}{L^2(\M)}\leq C e^{\kappa\lambda^k}\big( \nor{b v}{L^2(\M)} + \nor{(\scrL - \lambda^2)v }{L^2(\M)}  \big) , \quad \text{ for all }v \in \H^2_{\scrL} , \lambda \geq \lambda_0 .
\end{align*}
\end{corollary}
This corollary states a stronger version of the Eigenfunction tunneling estimates of~\cite[Theorem~1.12]{LL:17Hypo} (which is the same statement for solutions to $(\scrL - \lambda^2)v=0$).
Note that the constant $\kappa$ is (essentially) the same as in the cost of approximate controls in~\cite[Theorem~1.15]{LL:17Hypo}.

\medskip
Thirdly, we deduce from the spectral properties the sought decay estimates (respectively in Sections~\ref{s:damped-schro} and~\ref{s:damped-wave} for the damped Schr\"odinger and wave equations) using the Batty-Duyckaerts theorem, which we now recall.
\begin{theorem}[Batty and Duyckaerts~\cite{BD:08}]
\label{t:batty-duyckaerts}
Let $(e^{t\B})_{t\geq 0}$ be a bounded $\Con^0$-semigroup on a Banach space $\mathcal{X}$, generated by $\B$.

Assume that $ \nor{e^{t\B}\B^{-1}}{\L(\dot{\H})} \leq f(t)$ with $f$ decreasing to $0$. Then $i \R \cap \Sp(\B) = \emptyset$ and there is $C>0$ such that
$$
 \nor{(i\lambda-\B)^{-1}}{\L(\mathcal{X})} \leq 1 + C f^{-1} \left(\frac{1}{2(|\lambda|+1)} \right).
$$

Conversely, suppose that $i \R \cap \Sp(\B) = \emptyset$ and
\begin{align}
\label{e:res-bounded-M}
 \nor{(is-\B)^{-1}}{\L(\mathcal{X})}  \leq \mathsf{M}(|s|) , \quad s \in \R  ,
\end{align}
where $\mathsf{M} : \R_+\to \R_+^*$ is a non-decreasing function on $\R_+$. Then, setting
\begin{equation}
\label{Mlog}
\mathsf{M}_{\log}(s) =  \mathsf{M}(s) \big( \log(1+ \mathsf{M}(s)) + \log(1+ s)\big),
\end{equation}
 there exists $c>0$ such that for all $j \in \N$,
$$
 \nor{e^{t\B}\B^{-j}}{\L(\mathcal{X})}   = \O \bigg(  \frac{1}{\mathsf{M}_{\log}^{-1}\left( \frac{t}{c j} \right)^j}\bigg) , \quad \text{as } t \to +\infty ,
$$
where $\mathsf{M}_{\log}^{-1} : \R^+\to \R^+$ denotes the inverse of the strictly increasing function $\mathsf{M}_{\log}$.
\end{theorem}
We refer to~\cite{Duy:15,CS:16} for alternative proofs of the result of~\cite{BD:08}.
Note that on a Hilbert space (which is the case here) $\mathsf{M}_{\log}$ in the result can be replaced by $\mathsf{M}$ if it is polynomial at infinity, according to~\cite[Theorem 2.4]{BT:10} (see also~\cite{CPSST:19} and the references therein for generalizations of~\cite{BT:10}).

\bigskip
\noindent
{\em Acknowledgements.}
The first author is partially supported by the Agence Nationale de la Recherche under grant  SRGI ANR-15-CE40-0018.
Both authors are partially supported by the Agence Nationale de la Recherche under grant ISDEEC ANR-16-CE40-0013.

\section{Proofs}
\label{s:proofs}

\subsection{From approximate observability of waves to a free resolvent estimate with an observation term: Proof of Proposition~\ref{c:res-fct-propres}}

From approximate observability, we deduce the following (seemingly more general) result, concerning equation~\eqref{eq: abstract waves} with a general right hand-side $F$.

\begin{proposition}
\label{p:extrait-hypotri}
Let $T>0$ and a function $G : \R_+ \to \R_+$. Assume that the wave equation~\eqref{eq: abstract waves} with $F=0$ is approximately observable from $B^*$ in time $T$ with cost $G$, in the sense of Definition~\ref{d:approx-obs}. Then, there are $\mu_0,C>0$ such that for all $F\in L^2(0,T; H)$ and $(u_0,u_1)\in H_1 \times H$, the associated solution $u$ to~\eqref{eq: abstract waves} satisfies
\bnan
\label{th-estimate-k-bis-rhs}
\nor{(u_0,u_1)}{H\times H_{-1}} \leq G(\mu) \big(  \nor{B^*u}{L^2(0,T;Y)}  + C \nor{F}{L^2(0,T;H)}\big) +\frac{1}{\mu}\nor{(u_0,u_1)}{H_1 \times H} , \quad \text{for all }\mu \geq \mu_0 .
\enan
\end{proposition}
Note that the constant $\mu_0$ is actually the same as in Definition~\ref{d:approx-obs} and that $C$ depends only on $T$ and $\nor{B^*}{\L(Y;H)}$.
\bnp
According to the linearity of~\eqref{eq: abstract waves}, we decompose $u$ as $u=u^0+u^F$ where $u^0$ is the solution to~\eqref{eq: abstract waves} for $F=0$ and $u^F$ is the solution to~\eqref{eq: abstract waves} with $(u_0,u_1)=(0,0)$.

First, according to the assumption, Definition~\ref{d:approx-obs} applies to the function $u^0$, so that~\eqref{th-estimate-k-bis} reads:
\bnan
\label{th-estimate-k-bis-ter}
\nor{(u_0,u_1)}{H\times H_{-1}} \leq G(\mu) \nor{B^*u^0}{L^2(0,T;Y)} +\frac{1}{\mu}\nor{(u_0,u_1)}{H_1 \times H} , \quad \text{for all }\mu \geq \mu_0 .
\enan
Second, to estimate $u^F$, we perform classical energy inequalities for~\eqref{eq: abstract waves}. We rewrite~\eqref{eq: abstract waves} as
$$
(\d_t^2 + A+\id ) u^F = u^F + F , \quad (u^F, \d_t u^F)|_{t=0} = (0,0) .
$$
Taking the inner product of this equation with $\d_tu^F$ (assuming at first that $F \in L^1_{\loc}(\R; H_1)$ and thus $u^F \in  C^0(\R;D(A))\cap C^1(\R ; H_1)\cap C^2(\R ; H)$) implies
$$
\frac12 \frac{d}{dt} \left( \nor{\d_t u^F}{H}^2  + \nor{u^F}{H_1}^2 \right) \leq \left( \nor{u^F}{H} + \nor{F}{H} \right) \nor{\d_t u^F}{H}.
$$
Writing $\tilde{E}(t) = \frac12\left( \nor{\d_t u^F}{H}^2  + \nor{u^F}{H_1}^2 \right)$, this is $\tilde{E}'(t) \leq 2\tilde{E}(t) + \nor{F}{H}^2$. The Gronwall lemma together with the vanishing initial data imply
 $$
  \sup_{t\in [0,T]} \nor{u^F(t)}{H}^2 \leq \sup_{t\in [0,T]} \tilde{E}(t)  \leq C_T \nor{F}{L^1(0,T;H)}^2 .
 $$
 As a consequence, boundedness of $B^*$ yields
 $$
 \nor{B^*u^F}{L^2(0,T;Y)} \leq \nor{B^*}{\L(Y;H)}\nor{u^F}{L^2(0,T;H)} \leq \nor{B^*}{\L(Y;H)}C_T\nor{F}{L^2(0,T; H)} .
 $$
Recalling that $u^0 = u - u^F$  and combining this estimate with~\eqref{th-estimate-k-bis-ter} yields for all $\mu \geq \mu_0$
\begin{align*}
\nor{(u_0,u_1)}{H\times H_{-1}}&  \leq G(\mu) \nor{B^*(u-u^F)}{L^2(0,T;Y)} +\frac{1}{\mu}\nor{(u_0,u_1)}{H_1 \times H} \\
&  \leq G(\mu) \left( \nor{B^*u}{L^2(0,T;Y)} + C_{B,T}\nor{F}{L^2(0,T; H)} \right) +\frac{1}{\mu}\nor{(u_0,u_1)}{H_1 \times H} ,
\end{align*}
which concludes the proof of the proposition.
\enp

From this result, we deduce a proof of Proposition~\ref{c:res-fct-propres} as a direct corollary.

\bnp[Proof of Proposition~\ref{c:res-fct-propres}]
For $v \in D(A)$ and $\lambda\in \C$, we may apply the result of Proposition~\ref{p:extrait-hypotri} to the function $u(t) = \cos(\lambda t)v$ which satisfies~\eqref{eq: abstract waves} with
$$
u_0 = v , \quad u_1 =0 , \quad F(t) = \cos(\lambda t) (-\lambda^2 + A)v .
$$
We remark that the assumption of~\eqref{e::UCPeigenvalue} implies $F=0$ and $B^*u=0$, and hence~\eqref{th-estimate-k-bis-rhs} reads $\nor{v}{H} \leq \frac{1}{\mu}\nor{v}{H_1}$ for all $\mu \geq \mu_0$. Letting $\mu$ converges to $+\infty$ yields the conclusion of~\eqref{e::UCPeigenvalue}.

\medskip
Let us now prove~\eqref{e:res-fct-prop}.  Still for $u(t) = \cos(\lambda t)v$, we have
$$
 \nor{B^*u}{L^2(0,T;Y)}^2 \leq T \nor{B^*v}{Y}^2,  \quad \nor{F}{L^2(0,T;H)}^2 \leq T \nor{(-\lambda^2 + A)v}{H}^2.
$$
Estimate~\eqref{th-estimate-k-bis-rhs} thus implies for all $\lambda \geq 0$, $\mu \geq \mu_0$
\begin{align}
\label{e:interm-interm}
\nor{v}{H}\leq G(\mu) \sqrt{T} \big( \nor{B^*v}{Y} + C\nor{(A - \lambda^2)v }{H}  \big) +\frac{1}{\mu}  \nor{v}{H_1}  .
\end{align}
We now remark that
$$
 \left( A v ,v \right)_{H} -  \lambda^2 \nor{v}{H}^2 = \left( (A - \lambda^2)v ,v \right)_{H}
 \leq \nor{(A - \lambda^2)v }{H}\nor{v}{H}
$$
Hence, we deduce
\begin{align*}
\nor{v}{H_1}^2 & =  \left( (A +1)v ,v \right)_{H}  \leq ( \lambda^2+1) \nor{v}{H}^2 +\nor{(A - \lambda^2)v }{H}\nor{v}{H} \\
& \leq  ( \lambda^2+2) \nor{v}{H}^2 +\nor{(A - \lambda^2)v }{H}^2.
\end{align*}
Plugging this into~\eqref{e:interm-interm} yields, for all $\mu \geq \mu_0$ and $\lambda \geq 0$,
\begin{align*}
\nor{v}{H}\leq G(\mu) \sqrt{T}\big(  \nor{B^*v}{Y} + C\nor{(A - \lambda^2)v }{H}  \big) +\frac{1}{\mu} \left( \nor{(A - \lambda^2)v }{H}+ (\lambda +\sqrt{2}) \nor{v}{H} \right) .
\end{align*}
We let $\alpha >0$ and choose $\mu = \mu(\lambda) = \max \{\lambda + \sqrt{2} + \alpha, \mu_0 \}$ so that to absorb the last term in the right handside, implying for all $\lambda \geq 0$,
\begin{align*}
\left(1-\frac{\lambda+\sqrt{2}}{\lambda+\sqrt{2} + \alpha}\right) \nor{v}{H}\leq G(\mu(\lambda)) \sqrt{T} \big(  \nor{B^*v}{Y} + C\nor{(A - \lambda^2)v }{H}  \big) +\frac{1}{\mu(\lambda)} \nor{(A - \lambda^2)v }{H}   .
\end{align*}
We then take $\lambda \geq \mu_0$ so that $\mu(\lambda)= \lambda + \sqrt{2} + \alpha \geq 1$. This implies $\frac{1}{\mu(\lambda)} \nor{(A - \lambda^2)v }{H}  \leq c_0^{-1}G(\mu(\lambda)) \nor{(A - \lambda^2)v }{H}$ and thus, for $\lambda \geq  \mu_0$,
\begin{align*}
\frac{\alpha}{\mu(\lambda)} \nor{v}{H}\leq G(\mu(\lambda)) \sqrt{T} \big(  \nor{B^*v}{Y} + C\nor{(A - \lambda^2)v }{H}  \big) +c_0^{-1}G(\mu(\lambda)) \nor{(A - \lambda^2)v }{H} .
\end{align*}
This concludes the proof of the proposition.
\enp

We finally give a proof of Corollary~\ref{corresolvL}.
\bnp[Proof of Corollary~\ref{corresolvL}]
By assumption, $b\geq \delta>0$ on a non empty open set $\omega$. Since $\M$ is compact, $\sup_{x\in \M} d_\scrL(x,\omega)$ is finite.
For the hypoelliptic wave equation on $H=Y=L^2(\M)$, we proved in \cite[Theorem~1.15]{LL:17Hypo} that \eqref{th-estimate-k-bis} is satisfied for $A=\scrL$, $B_\omega = B^*_{\omega} =$ multiplication by $\mathds{1}_\omega$, $G(\mu) = Ce^{\kappa \mu^k}$, for all $T> 2\sup_{x\in \M} d_\scrL(x,\omega)$ (where $d_\scrL$ is the appropriate sub-Riemannian distance and $k$ the hypoellipticity index of $\scrL$). Since $\nor{\mathds{1}_{\omega} u}{L^2(\M)}\leq \delta^{-1}\nor{b u}{L^2(\M)}$, the same inequality with different constants remains true with $B=B^*=$ multiplication by $b$. Thus, we deduce from Proposition~\ref{c:res-fct-propres} that~\eqref{e:res-fct-prop-bis} is satisfied (after having fixed $\alpha = 2-\sqrt{2}$) with $\mathsf{G} (\lambda)  =K(1+C) (\lambda +2)  G(\lambda + 2) = C(\lambda +2)  e^{\kappa (\lambda +2)^k}$.
\enp

\subsection{From free resolvent estimate with an observation term to damped resolvent estimate}
In this section, we start from an estimate for $A$ with an observation term like~\eqref{e:res-fct-prop}, and deduce associated estimates for damped operators.

Now, for later use (see Sections~\ref{s:damped-schro} and~\ref{s:damped-wave} below), we introduce the operators:
\begin{align*}
Q_\lambda & = -i(\A_S - i\lambda ) = A-\lambda  +i BB^* ,\\
P_\lambda & = P(i\lambda)= A - \lambda^2  + i \lambda BB^* ,
\end{align*}
 both with domain $D(Q_\lambda) = D(P_\lambda) = D(A)$.

\begin{proposition}
\label{p:P-lambda}
Let $G_1,G_2 \geq 0$, $\lambda >0$, and $v \in D(A)$, and assume
\begin{align}
\label{e:res-fct-prop-ter}
\nor{v}{H}\leq G_1 \nor{B^* v}{Y} + G_2 \nor{(A - \lambda^2)v }{H}  .
\end{align}
Then we have
\begin{align}
\label{e:res-fct-prop-ter-P}
\nor{v}{H} & \leq \left(  (G_1\lambda^{-\frac12} + G_2 \sqrt{2} \nor{B}{\L(Y;H)})^2 +2 \sqrt{2} G_2\right)  \nor{P_\lambda v}{H}, \\
\label{e:res-fct-prop-ter-Q}
\nor{v}{H} & \leq \left(  (G_1  + G_2 \sqrt{2} \nor{B}{\L(Y;H)})^2 +2 \sqrt{2} G_2\right)  \nor{Q_{\lambda^2} v}{H}.
\end{align}

In particular, given $\mathsf{G} : \R_+ \to \R_+$ such that $\mathsf{G}(\mu) \geq c_0>0$ on $\R_+$ and $\lambda_0\geq 1$, if~\eqref{e:res-fct-prop-bis} is satisfied, then writing $K = (1+ \sqrt{2} \nor{B}{\L(Y;H)})^2 +  2\sqrt{2} c_0^{-1}$, we have \begin{align}
\label{e:res-fct-prop-terter}
\nor{v}{H} & \leq K \mathsf{G}(|\lambda|)^2 \nor{P_\lambda v}{H} , \quad \text{ for all }v \in D(A)  ,  \lambda \in \R, |\lambda| \geq \lambda_0 , \\
\label{e:res-fct-prop-terter-Q}
\nor{v}{H} & \leq K \mathsf{G}\big(\sqrt{\lambda}\big)^2 \nor{Q_\lambda v}{H} , \quad \text{ for all }v \in D(A)  , \lambda  \geq  \lambda_0^2 .
\end{align}
\end{proposition}
Note that when passing from~\eqref{e:res-fct-prop} to~\eqref{e:res-fct-prop-terter}, we change $\mathsf{G}$ to $\mathsf{G}^2$, which is a loss in general; this is linked to the fact that the proof of Proposition~\ref{p:P-lambda} consists only in a very rough estimate, treating the damping terms $iBB^*$ and $i \lambda BB^*$ as remainders.

\bnp[Proof of Proposition~\ref{p:P-lambda}]
We only prove the result for $P_\lambda$, the analogue proof for $Q_\lambda$ is identical.

First, we remark that, under the above assumptions, we have
\begin{align}
\label{e:delta-omega}
\lambda \nor{B^*v}{Y}^2  =  \lambda \left( BB^* v , v \right)_{H}  = \Im \left(P_\lambda v , v \right)_H \leq \nor{P_\lambda v}{H}  \nor{v}{H} .
\end{align}
Second, we notice that $(A - \lambda^2)v = P_\lambda v - i\lambda BB^* v$ and thus, using~\eqref{e:delta-omega},
\begin{align*}
 \nor{(A - \lambda^2)v }{H}^2 & \leq  2\nor{P_\lambda v}{H}^2 +2 \lambda \nor{BB^* v}{H}^2 \leq  2\nor{P_\lambda v}{H}^2 +2  \nor{B}{\L(Y;H)}^2 \lambda \nor{B^*v}{Y}^2 \\
 &  \leq  2\nor{P_\lambda v}{H}^2 + 2 \nor{B}{\L(Y;H)}^2 \nor{P_\lambda v}{H}  \nor{v}{H} .
\end{align*}
Plugging the last two estimates in~\eqref{e:res-fct-prop-ter} yields
\begin{align*}
\nor{v}{H}\leq (G_1 \lambda^{-\frac12} + G_2 \sqrt{2} \nor{B}{\L(Y;H)}) \nor{P_\lambda v}{H}^{\frac12}  \nor{v}{H}^{\frac12}   +G_2 \sqrt{2} \nor{P_\lambda v}{H}  .
\end{align*}
Writing
\begin{align*}
 (G_1 \lambda^{-\frac12} + G_2\sqrt{2} \nor{B}{\L(Y;H)}) \nor{P_\lambda v}{H}^{\frac12}  \nor{v}{H}^{\frac12}
 \leq \frac12 (G_1\lambda^{-\frac12} + G_2 \sqrt{2} \nor{B}{\L(Y;H)})^2 \nor{P_\lambda v}{H} +\frac12  \nor{v}{H} ,
\end{align*}
allows to absorb the last term in the left hand-side and implies
\begin{align*}
\frac12 \nor{v}{H}\leq  \frac12 (G_1\lambda^{-\frac12} + G_2 \sqrt{2} \nor{B}{\L(Y;H)})^2 \nor{P_\lambda v}{H} +G_2 \sqrt{2}  \nor{P_\lambda v}{H}  .
\end{align*}
This concludes the proof of~\eqref{e:res-fct-prop-ter-P}, and~\eqref{e:res-fct-prop-terter} corresponds to the case $G_1=G_2 =\mathsf{G}(\lambda)$. Also, we notice that for $\lambda\in\R$, $\overline{P_{-\lambda}u}=P_{\lambda}\overline{u}$, so the statement in the case $\lambda\geq\lambda_0$ implies the case $\lambda\leq -\lambda_0$. Finally, the proof of~\eqref{e:res-fct-prop-ter-Q} is similar to that of~\eqref{e:res-fct-prop-ter-P} (beware that it should be written for $Q_{\lambda^2}$ and not $Q_{\lambda}$), and~\eqref{e:res-fct-prop-terter-Q} follows from changing $\lambda^2$ into $\lambda$.
\enp
Note that another advantage of Proposition~\ref{p:P-lambda} is that it is flexible enough to support perturbations of the operator $A$ by lower order terms. This was used in \cite{JL:20} where similar estimates were used for application to perturbed operators coming from linearization of a nonlinear equation. See also~\cite{CPSST:19,Burq:19} for recent related perturbation results.

\subsection{Damped Schr\"odinger-type equations}
\label{s:damped-schro}
There are not many references concerning the damped Schr\"odinger equation. So let us start from the beginning. We set $\A_S := i A - BB^*$ with $D(\A_S)=D(A)$, so that~\eqref{eq: damped abstract schro} reformulates as $(\d_t- \A_S )u=0$.

The compact embedding $D(A)\hookrightarrow H$ implies that $\A_S$ has a compact resolvent. First spectral properties of $\A_S$ are described in the following lemma.
\begin{lemma}
\label{l:spec-AS}
The spectrum of $\A_S$ contains only isolated eigenvalues and we have
\begin{align}
\nor{(z\id -\A_S)^{-1}}{\L(H)} & \leq \frac{1}{\Re(z)} , \quad \text{ for }\Re(z)>0 , \label{schro-res-1} \\
\nor{(z\id -\A_S)^{-1}}{\L(H)} & \leq \frac{1}{|\Im(z)|} , \quad \text{ for }\Im(z) <0 . \label{schro-res-2}
\end{align}
Moreover, assuming $(Au = zu , B^*u=0 )\implies u=0$, we have
$$
\Sp(\A_S) \subset   [ - \|B^*\|_{\L(H;Y)}^2 , 0) +  i[0,+\infty) .
$$
\end{lemma}
\bnp
The structure of the spectrum comes from the fact that $\A_S$ has a compact resolvent (since so does $A$, and $BB^*$ is bounded). Now, for a general $z \in \C$, we have
$$
\nor{(z\id -\A_S)u}{H} \nor{u}{H} \geq \Re\left((z\id -\A_S)u , u \right)_{H}  =  \Re(z)\nor{u}{H}^2 + \nor{B^*u}{H}^2  \geq \Re(z)\nor{u}{H}^2 ,
$$
which yields~\eqref{schro-res-1}. The statement~\eqref{schro-res-2} comes from
$$
\nor{(\A_S-z\id)u}{H} \nor{u}{H} \geq \Im \left((\A_S-z\id)u , u \right)_{H}  =  (A u, u)_H  -\Im(z)\nor{u}{H}^2  \geq -\Im(z)\nor{u}{H}^2 .
$$
Finally given $z \in \Sp(\A_S)$, there is $u \in D(A)\setminus\{0\}$ such that $\A_S u = zu$. Taking inner product with $u$ yields
$$
z \nor{u}{H}^2 = (\A_S u, u )_{H} = i (A u, u)_H - \nor{B^*u}{H}^2.
$$
In particular, $$
\Re(z) = -\frac{\nor{B^*u}{H}^2}{\nor{u}{H}^2} \in [ - \nor{B^*}{\L(H)}^2 , 0]  , \quad \Im(z) = \frac{(A u, u)_H}{\nor{u}{H}^2} \geq 0 .
$$
Now if $\Re(z) = 0$, this implies $B^* u=0$ and hence $ zu = \A_S u  =i A u$. The assumption then yields $u=0$, which contradicts the fact that $u$ is an eigenvector. Thus $\Sp(\A_S) \cap i\R = \emptyset$.
\enp

We then deduce straightforwardly from Lemmata~\ref{p:P-lambda} and~\ref{l:spec-AS} the following result.
\begin{theorem}
\label{c:Q-lambda-S}
Let  $\mathsf{G} : \R_+ \to \R_+$ be such that $\mathsf{G} (\mu) \geq c_0>0$ on $\R_+$, $\lambda_0\geq 1$, and assume~\eqref{e:res-fct-prop-bis}. Then there exists $K>1$ (the same as in Proposition~\ref{p:P-lambda}), such that
\begin{align*}
& \|(i\lambda \id - \A_S)^{-1}\|_{\L(\H)} \leq K\mathsf{G}\big(\sqrt{\lambda}\big)^2 ,  \quad \text{ for all } \lambda  \geq \lambda_0^2 ,  \\
& \Sp(\A_S) \cap  \Gamma_{\mathsf{G},S}  = \emptyset ,
\end{align*}
where $\Gamma_{\mathsf{G},S} = \left\{z \in \C , \Im(z)\geq \lambda_0^2 ,  \Re(z) \geq - \frac{1}{K \mathsf{G}\big(\sqrt{\Im(z)}\big)^2 } \right\}$.
Finally, assuming further~\eqref{e:res-fct-prop}, there exists another constant $\widetilde{K}\geq K$ such that
\begin{align*}
& \|(i\lambda \id - \A_S)^{-1}\|_{\L(\H)} \leq \widetilde{K}\mathsf{G}\big(\sqrt{|\lambda|}\big)^2 ,  \quad \text{ for all } \lambda  \in \R, \\
& \Sp(\A_S) \cap  \widetilde{\Gamma}_{\mathsf{G},S}  = \emptyset ,
\end{align*}
where $\widetilde{\Gamma}_{\mathsf{G},S} = \left\{z \in \C ,  \Re(z) \geq - \frac{1}{\widetilde{K} \mathsf{G}\big(\sqrt{|\Im(z)|}\big)^2 } \right\}$.
\end{theorem}
\bnp
The first point is a rewriting of~\eqref{e:res-fct-prop-terter-Q} in Lemma~\ref{p:P-lambda}.

The second point comes from the general fact that $\nor{(z\id-\A_S)^{-1}}{\L(H)}  \geq \frac{1}{\dist(z, \Sp(\A_S))}$ (following from a Neumann series expansion). Hence, we have for $\lambda \geq \lambda_0^2$,
$$
\dist(i\lambda, \Sp(\A_S)) \geq \nor{(i\lambda\id-\A_S)^{-1}}{\L(H)}^{-1} \geq  \left( K  \mathsf{G}\big(\sqrt{\lambda}\big)^2  \right)^{-1} ,
$$
which, together with the localization of the spectrum in Lemma~\ref{l:spec-AS}, proves the second point.

For the last point, Lemma~\ref{l:spec-AS} ensures that $\lambda\mapsto \|(i\lambda \id - \A_S)^{-1}\|_{\L(\H)} $ is a well defined continuous function on $\R$, which is bounded by $\frac{1}{|\lambda|}$ for $\lambda<0$. On the interval $(-\infty,\lambda_0^2]$, it is therefore bounded by a constant $C_0\leq C_0 c_0^{-2} \mathsf{G}\big(\sqrt{|\lambda|}\big)^2$. This gives the expected estimates for all $\lambda\in \R$ with another $\widetilde{K}=\max \big(K,C_0 c_0^{-2}\big)$. Again, Neumann expansion gives the localization of the spectrum.
\enp
As a consequence, we deduce the following decay.

\begin{theorem}
\label{t:decay-rates-abstractSchrod}
Let  $\mathsf{G} : \R_+ \to \R_+$ be such that $\mathsf{G} (\mu) \geq c_0>0$ on $\R_+$, $\lambda_0\geq 1$, and assume~\eqref{e:res-fct-prop-bis} and \eqref{e::UCPeigenvalue}.
Assume further that $\mathsf{G}$ is nondecreasing and set $\mathsf{M}(\lambda) =\mathsf{G}\big(\sqrt{\lambda}\big)^2 $.
Then, there exists $c>0$ such that for all $j \in \N^*$, there is $C_j>0$ such that  for all $u_0 \in D(\A_S^j)$ and associated solution $u$ of \eqref{eq: damped abstract schro},
$$
\nor{u(t)}{H} \leq  \frac{C_j}{\mathsf{M}_{\log}^{-1}\left( \frac{t}{c j} \right)^j} \nor{ \A_S^j u_0}{H},  \quad \text{ for all } t>0 ,
$$
where $\mathsf{M}_{\log}$ is defined in~\eqref{Mlog}.
\end{theorem}
Again, $\mathsf{M}_{\log}$ in the result can be replaced by $\mathsf{M}$ if it is polynomial at infinity, according to~\cite[Theorem 2.4]{BT:10}.
\bnp
This is a direct corollary of Theorem~\ref{c:Q-lambda-S} and Theorem~\ref{t:batty-duyckaerts} applied to $\B=\A_S$.
\enp

We may now conclude the proofs of Theorems~\ref{e:decay-schro} and~\ref{t:spectral-schro}.
\bnp[Proof of Theorems~\ref{e:decay-schro} and~\ref{t:spectral-schro}]

Corollary \ref{corresolvL} implies that \eqref{e:res-fct-prop-bis} is true with $\mathsf{G} (\mu) = C e^{\kappa \mu^k}$. Then, Theorem~\ref{c:Q-lambda-S} implies Theorem~\ref{t:spectral-schro}. Indeed, taking into account~\eqref{e::UCPeigenvalue}, we then obtain that the resolvent is bounded on the positive imaginary axis by a constant times
$\mathsf{M}(\lambda) =\mathsf{G}\big(\sqrt{\lambda}\big)^2 =  Ce^{2 \kappa^+ \lambda^{k/2}}$ (after having changed the constants slightly).

Finally, we obtain
$$
\mathsf{M}_{\log}(\lambda) = Ce^{2\kappa^+ \lambda^{k/2}}\left( \log \big(1 + Ce^{2\kappa^+ \lambda^{k/2}}\big) + \log (1+\lambda) \right) \leq Ce^{2\kappa^+ \lambda^{k/2}}
$$
(after having changed the constants slightly), and thus $\mathsf{M}_{\log}^{-1}(t) \geq c \log(t)^{2/k}$ for large $t$. Theorem~\ref{t:decay-rates-abstractSchrod} implies Theorem~\ref{e:decay-schro}.
\enp

\subsection{Damped wave-type equations: semigroup setting and end of the proofs}
\label{s:damped-wave}
Let us now turn this estimate into a resolvent estimate for the generator of the damped wave group, and then into a decay estimate for~\eqref{eq: damped abstract waves}.
We equip $\H=H_1\times H$ with the norm
$$
\|(u_0,u_1)\|_\H^2 = \|(A+\id)^\frac12 u_0 \|_H^2 + \| u_1 \|_H^2,
$$
and define the seminorm
$$
|(u_0,u_1)|_\H^2 = \|A^\frac12 u_0 \|_H^2 + \| u_1 \|_H^2.
$$
Of course, if $A$ is coercive on $H$, $|\cdot|_\H$ is a norm on $\H$ equivalent to $\|\cdot \|_\H$.
We define the energy of solutions of \eqref{eq: damped abstract waves} by
$$
E(u(t)) = \frac{1}{2} \big( \|A^\frac12 u\|_H^2 + \|\d_t u\|_H^2 \big)
= \frac12 |(u,\d_t u)|^2_{\H}.
$$
 The damped wave equation~\eqref{eq: damped abstract waves} can be recast on $\H$ as a first order system
\begin{equation}
\label{eq: first order eqation}
\left\{
\begin{array}{l}
\d_t U = \A U , \\
U|_{t=0} = \transp(u_0 , u_1) ,
\end{array}
\right.
\quad
U =
\left(
\begin{array}{c}
u \\
\d_t u
\end{array}
\right) , \quad
\A =
\left(
\begin{array}{cc}
0   &  \id \\
- A & - BB^*
\end{array}
\right)  , \quad
D(\A) =  D(A)\times H_1 .
\end{equation}
The compact embeddings $D(A)\hookrightarrow H_1 \hookrightarrow H$ imply that $D(\A)\hookrightarrow \H$ compactly, and that the operator $\A$ has a compact resolvent. First,  spectral properties of $\A$ are described in the following lemma borrowed from~\cite{Leb:96,AL:14}.
We define the following quadratic family of operator
\begin{align}
\label{e:def-Pz}
P(z) = A + z^2 \id + z BB^* , \quad z \in \C, \quad  D(P(z)) = D(A) .
\end{align}
\begin{lemma}[Lemma~4.2 of~\cite{AL:14}]
\label{lemma: check assumptions}
The spectrum of $\A$ contains only isolated eigenvalues and, provided \eqref{e::UCPeigenvalue} is satisfied, we have
$$
\Sp(\A) \subset \left( \big( - \frac12 \|B^*\|_{\L(H;Y)}^2 , 0 \big) + i\R\right) \cup
\left([ - \|B^*\|_{\L(H;Y)}^2 , 0] + 0 i \right)  ,
$$
with $\ker(\A) = \ker(A)\times \{0\}$.
Moreover, the operator $P(z)$ in~\eqref{e:def-Pz} is an isomorphism from $D(A)$ onto $H$ if and only if $z \notin \Sp(\A)$.

\end{lemma}
This Lemma leads us to introduce the spectral projector of $\A$ on $\ker(\A)$, given by
$$
\Pi_0 = \frac{1}{2i\pi}\int_{\gamma} (z\id - \A)^{-1} dz \in \L(\H),
$$
where $\gamma$ denotes a positively oriented circle centered on $0$ with a radius so small that $0$ is the single eigenvalue of $\A$ in the interior of $\gamma$.
We set $\dot{\H} = (\id - \Pi_0)\H$ and equip this space with the norm
$$
\|(u_0 , u_1)\|_{\dot{\H}}^2 := |(u_0 , u_1)|_{\H}^2
= \|A^{\frac12}u_0\|_{H}^2 + \|u_1 \|_{H}^2 ,
$$
and associated inner product.
This is indeed a norm on $\dot{\H}$ since $\|(u_0 , u_1)\|_{\dot{\H}} =0$ is equivalent to $(u_0 , u_1)\in \ker(A)\times \{0\} = \Pi_0\H$.
Besides, we set $\dot{\A} = \A|_{\dot{\H}}$ with domain $D(\dot{\A}) = D(\A) \cap \dot{\H}$.
Remark that $\Sp(\dot{\A}) = \Sp(\A) \setminus \{0\}$ and thus $\Sp(\dot{\A}) \cap i\R = \emptyset$.

\begin{lemma}[Lemma~4.3 of~\cite{AL:14}]
\label{lemma: semigroups}
The operator $\dot{\A}$ generates a contraction $C^0$-semigroup on~$\dot{\H}$, denoted $(e^{t\dot{\A}})_{t\geq 0}$. Moreover, the operator $\A$ generates a bounded $C^0$-semigroup on~$\H$, denoted $(e^{t\A})_{t\geq 0}$ and the unique solution to~\eqref{eq: damped abstract waves} is given by $(u, \d_t u )(t) = e^{t\A}(u_0,u_1)$. Finally,  we have
\begin{equation}
\label{eq: decomposition semigroup}
e^{t\A}= e^{t\dot{\A}} (\id -\Pi_0)  + \Pi_0  , \quad \text{for all } t\geq 0 .
\end{equation}
\end{lemma}

Once we have put the abstract damped wave equation~\eqref{eq: damped abstract waves} in the appropriate semigroup setting, it remains to:
\begin{enumerate}
\item \label{point-1} deduce from~\eqref{e:res-fct-prop}-\eqref{e:res-fct-prop-bis} a resolvent estimate for $\dot{\A}$,
\item \label{point-2} relate this resolvent estimate to a decay estimate for $e^{t\dot{\A}}$, and
\item \label{point-3} deduce the decay of the energy for~\eqref{eq: damped abstract waves}.
\end{enumerate}

\medskip
Step~\ref{point-1} is realized thanks to the following result from~\cite{AL:14}.
\begin{lemma}[Lemma~4.6 of~\cite{AL:14}]
\label{lemma: conditions stable resolvent}
There exist $C>1$ such that for $s \in \R$, $|s|\geq 1$,
\begin{align}
\label{eq: estimate A estimate dotA}
 C^{-1}\|(is\id - \dot{\A})^{-1}\|_{\L(\dot{\H})} - \frac{C}{|s|}
\leq \|(is\id - \A)^{-1}\|_{\L(\H)}
\leq  C\|(is\id - \dot{\A})^{-1}\|_{\L(\dot{\H})} + \frac{C}{|s|}  ,
\end{align}
\begin{align}
\label{eq: estimate P equiv A}
C^{-1}|s| \|P(is)^{-1}\|_{\L(H)} \leq  \|(is\id - \A)^{-1}\|_{\L(\H)}
\leq C \left( 1 + |s|\|P(is)^{-1}\|_{\L(H)} \right).
\end{align}
\end{lemma}
As a corollary of this together with Proposition~\ref{p:P-lambda}, we deduce the following result.

\begin{theorem}
\label{c:P-lambda-A}
Let  $\mathsf{G} : \R_+ \to \R_+$ be such that $\mathsf{G} (\mu) \geq c_0>0$ on $\R_+$, $\lambda_0\geq 1$, and assume~\eqref{e:res-fct-prop-bis}. Then there exists $K>1$ such that
\begin{align*}
& \|(i\lambda \id - \A)^{-1}\|_{\L(\H)} \leq K|\lambda| \mathsf{G}(|\lambda|)^2 ,  \quad \text{ for all } \lambda\in \R, |\lambda|  \geq \lambda_0 ,  \\
& \|(is\id - \dot{\A})^{-1}\|_{\L(\dot{\H})}\leq K|\lambda|  \mathsf{G}(|\lambda|)^2  ,  \quad \text{ for all } \lambda\in \R, |\lambda|  \geq \lambda_0  , \\
&\Sp(\dot{\A}) \cap \Gamma_{\mathsf{G}} =\emptyset , \qquad  \Sp(\A) \cap  \Gamma_{\mathsf{G}}  = \emptyset ,
\end{align*}
where $\Gamma_{\mathsf{G}} = \left\{z \in \C , |\Im(z)| \geq \lambda_0 ,  \Re(z) \geq - \frac{1}{K |\Im(z)|  \mathsf{G}(|\Im(z)|)^2} \right\}$.

Finally, assuming further~\eqref{e:res-fct-prop}, there exists another constant $\widetilde{K}\geq K$ such that
\begin{align*}
& \|(is\id - \dot{\A})^{-1}\|_{\L(\dot{\H})}\leq \widetilde{K}\left\langle \lambda\right\rangle  \mathsf{G}(|\lambda|)^2  ,  \quad \text{ for all } \lambda\in \R, \\
&\Sp(\dot{\A}) \cap \widetilde{\Gamma}_{\mathsf{G}} =\emptyset , \qquad  \Sp(\A) \cap  \Gamma_{\mathsf{G}}  = \{0\},
\end{align*}
where $\widetilde{\Gamma}_{\mathsf{G}} = \left\{z \in \C , \Re(z) \geq - \frac{1}{\widetilde{K}\left\langle \Im(z) \right\rangle \mathsf{G}( |\Im(z)|)^2} \right\}$.
\end{theorem}

\bnp[Proof of Theorem~\ref{c:P-lambda-A}]
The first two points are corollaries of~\eqref{e:res-fct-prop-terter} in Proposition~\ref{p:P-lambda} combined with Lemma~\ref{lemma: conditions stable resolvent}.

The last point comes from $\Sp(\dot{\A}) = \Sp(\A) \setminus \{0\}$, together with the general fact that $\nor{(z\id-\dot{\A})^{-1}}{\L(\H)}  \geq \frac{1}{\dist(z, \Sp(\dot{\A}))}$ (following from a Neumann series expansion). Hence, we have for $\lambda\in \R$, $|\lambda| \geq \lambda_0$,
$$
\dist(i\lambda, \Sp(\dot{\A})) \geq \nor{(i\lambda\id-\dot{\A})^{-1}}{\L(\H)}^{-1} \geq  \left( K |\lambda|  \mathsf{G}(|\lambda|)^2  \right)^{-1} ,
$$
which, together with the localization of the spectrum in Lemma~\ref{lemma: check assumptions}, proves the statement about the zone free of spectrum. The proof concerning the compact zone follows the same way as Theorem \ref{c:Q-lambda-S}.
\enp

\medskip
Step~\ref{point-2} is achieved as a consequence of Theorem~\ref{t:batty-duyckaerts} applied to the operator $\B=\dot{\A}$.

\medskip
Finally, Step~\ref{point-3} is a consequence of the following elementary lemma~\ref{l:AL:14-44}, linking the energy of solutions to the abstract damped wave equation~\eqref{eq: damped abstract waves} to the norm of the semigroup $\big(e^{t\dot{\A}}\big)_{t\geq0}$.
\begin{lemma}
\label{l:AL:14-44}
For all $j \in \N^*$, $U_0 \in D(\A^j)$ such that $\Pi_0 U_0 \neq U_0$, and associated solution $u$ of \eqref{eq: damped abstract waves}, we have
\begin{align*}
\frac{E(u(t))}{\frac12 | \A^j U_0 |_{\H}^2} = \frac{|e^{t\A} U_0 |_{\H}^2}{| \A^j U_0 |_{\H}^2}  =\frac{\|e^{t\dot{\A}}\dot{U}_0\|_{\dot{\H}}^2}{\|\dot{\A}^j \dot{U}_0\|_{\dot{\H}}^2} , \quad \text{where} \quad  \dot{U}_0 = (\id-\Pi_0)U_0 .
\end{align*}
In particular, setting $f_j(t) := \nor{e^{t\dot{\A}}\dot{\A}^{-j}}{\L(\dot{\H})}$ for $j \in \N^*$, we have for all $U_0 \in D(\A^j)$ and associated solution $u$ of \eqref{eq: damped abstract waves},
$$
E(u(t)) \leq \frac12 f_j(t)^2\| \A^j U_0 \|_{\H}^2 , \quad \text{ for all }t\geq0 .
$$
\end{lemma}
\bnp
This is essentially~\cite[Lemma~4.4]{AL:14}. Recalling that $\A U_0= \dot{\A} \dot{U}_0$, we have
\begin{align*}
E(u(t)) & = \frac{1}{2} \big( \|A^\frac12 u(t)\|_H^2 + \|\d_t u(t)\|_H^2 \big) =\frac12 |e^{t\A} U_0 |_{\H}^2
= \frac12 |e^{t\dot{\A}} \dot{U}_0 + \Pi_0 U_0|_{\H}^2
= \frac12 \|e^{t\dot{\A}}\dot{U}_0\|_{\dot{\H}}^2 ,  \\
\|\dot{\A}^j\dot{U}_0\|_{\dot{\H}}^2 & = | \A^j U_0 |_{\H}^2 =  \| \A^j U_0 \|_{\H}^2 ,
\end{align*}
which yields the first statement. The second one follows from the fact that $|\cdot|_{\H} \leq \nor{\cdot}{\H}$.
\enp

As a consequence, we deduce the following decay.

\begin{theorem}
\label{t:decay-rates-abstract}
Let  $\mathsf{G} : \R_+ \to \R_+$ be such that $\mathsf{G} (\mu) \geq c_0>0$ on $\R_+$, $\lambda_0\geq 1$, and assume~\eqref{e::UCPeigenvalue} and \eqref{e:res-fct-prop-bis}.
Assume further that $\mathsf{G}$ is nondecreasing and set $\mathsf{M}(\lambda) = \left\langle \lambda\right\rangle \mathsf{G}(\lambda)^2$.
Then, there exists $c>0$ such that for all $j \in \N^*$, there is $C_j>0$ such that  for all $U_0 \in D(\A^j)$ and associated solution $u$ of \eqref{eq: damped abstract waves},
$$
E(u(t))^\frac12\leq  \frac{C_j}{\mathsf{M}_{\log}^{-1}\left( \frac{t}{c j} \right)^j} \nor{ \A^j U_0}{\H},  \quad \text{ for all } t>0 ,
$$
where $\mathsf{M}_{\log}$ is defined in~\eqref{Mlog}.
\end{theorem}
Again, $\mathsf{M}_{\log}$ in the result can be replaced by $\mathsf{M}$ if it is polynomial at infinity, according to~\cite[Theorem 2.4]{BT:10}.
\bnp
This is a direct corollary of Theorem~\ref{c:P-lambda-A}, Theorem~\ref{t:batty-duyckaerts} applied to $\mathcal{X} = \dot{\H}$ and $\B=\dot{\A}$, together with Lemma~\ref{l:AL:14-44}.
\enp
We conclude this paragraph with the proofs of Theorems \ref{e:decay} and \ref{t:spectral}.
\bnp[Proof of Theorems \ref{e:decay} and \ref{t:spectral}]
Again, Corollary \ref{corresolvL} implies the unique continuation property \eqref{e:UCP-eig-Hypo} (that is \eqref{e::UCPeigenvalue} in the present context) together with~\eqref{e:res-fct-prop-bis} with $\mathsf{G} (\mu) = C e^{\kappa \mu^k}$. With this estimate at hand, Theorem~\ref{e:decay} is an application of Theorem \ref{t:decay-rates-abstract} with $\mathsf{M}(\lambda) = \left\langle \lambda\right\rangle \mathsf{G}(\lambda)^2\leq  Ce^{2 \kappa^+ \lambda^{k}}$ (after having changed the constants slightly), while Theorem \ref{t:spectral} is implied by Lemma \ref{lemma: check assumptions} and Theorem \ref{c:P-lambda-A}.
\enp

\subsection{Damped plate equation}
The plate equation actually fits into the ``wave-type'' framework. Indeed, the abstract plate equation
\begin{align}
\label{eq: damped abstract beam}
\begin{cases}
\d_t^2 u + A^2 u + B B^* \d_t u = 0, \\
(u, \d_t u)|_{t=0} = (u_0 , u_1) ,
\end{cases}
\end{align}
is actually a particular case of the abstract equation \eqref{eq: damped abstract waves} applied with the operator $A^2$ (instead of $A$) which is still nonnegative selfadjoint with compact resolvent. In this case, we define $H_2 = D(A)$, equipped with the graph norm $\nor{u}{H_2} := \|(A^2+\id)^\frac12 u \|_H$, and its dual $H_{-2} = (H_2)'$ (using $H$ as a pivot space) endowed with the norm $\nor{u}{H_{-2}}:=\|(A^2+\id)^{-\frac12} u \|_H$.

The natural space is then $\H=H_2\times H$ with the norm
$$
\|(u_0,u_1)\|_\H^2 = \|(A^2+\id)^\frac12 u_0 \|_H^2 + \| u_1 \|_H^2,
$$
and the seminorm
$$
|(u_0,u_1)|_\H^2 = \|A u_0 \|_H^2 + \| u_1 \|_H^2.
$$
The associated energy is
$$
E_P(u(t)) = \frac{1}{2} \big( \|A u\|_H^2 + \|\d_t u\|_H^2 \big)
= \frac12 |(u,\d_t u)|^2_{\H}.
$$
In order to transfer the properties of $A$ to $A^2$, we will only need the following simple lemma.
 \begin{lemma}
Assume \eqref{e:res-fct-prop-bis} is satisfied. Then, we have
\begin{align}
\label{e:res-fct-prop-beam}
\nor{v}{H}\leq \mathsf{G}(\sqrt{\lambda}) \big( \nor{B^* v}{Y} +\lambda^{-1} \nor{(A^2 - \lambda^2)v }{H}  \big) , \quad \text{ for all }v \in D(A^2) , \lambda \geq \lambda_0^2 .
\end{align}
\end{lemma}
\bnp
Since $A$ is a nonnegative operator, we have $\nor{(A + \lambda^2)w }{H}\geq  \lambda^2 \nor{w }{H}$ for all $w\in D(A)$. Applying this to $w=(A - \lambda^2)v$ gives $\nor{(A^2 - \lambda^4)v }{H}\geq \lambda^2 \nor{(A - \lambda^2)v}{H}$.
This, combined with \eqref{e:res-fct-prop-bis} implies
\begin{align}
\nor{v}{H}\leq \mathsf{G}(\lambda) \big( \nor{B^* v}{Y} + \nor{(A - \lambda^2)v }{H}  \big)\leq \mathsf{G}(\lambda) \left( \nor{B^* v}{Y} + \frac{1}{\lambda^2}\nor{(A^2 - \lambda^4)v }{H}  \right) .
\end{align}
This is the expected result up to changing $\lambda$ to $\sqrt{\lambda}$.
\enp
The previous Lemma implies that if \eqref{e:res-fct-prop-bis} is satisfied, the assumptions of Theorem \ref{t:decay-rates-abstract} are satisfied for the operator $A^2$ with $G_P(\lambda)=G(\sqrt{\lambda})$. It directly gives the following result.
\begin{theorem}
\label{t:decay-rates-abstractbeam}
Let  $\mathsf{G} : \R_+ \to \R_+$ be such that $\mathsf{G} (\mu) \geq c_0>0$ on $\R_+$, $\lambda_0\geq 1$, and assume~\eqref{e::UCPeigenvalue} and~\eqref{e:res-fct-prop-bis}.
Assume further that $\mathsf{G}$ is nondecreasing and set $\mathsf{M}(\lambda) = \lambda \mathsf{G}\big(\sqrt{\lambda}\big)^2$.
Then, there exists $c>0$ such that for all $j \in \N^*$, there is $C_j>0$ such that  for all $U_0 \in D(\A^j)$ and associated solution $u$ of~\eqref{eq: damped abstract beam},
$$
E_P(u(t))^\frac12\leq  \frac{C_j}{\mathsf{M}_{\log}^{-1}\left( \frac{t}{c j} \right)^j} \nor{ \A_P^j U_0}{\H},  \quad \text{ for all } t>0 ,
$$
where $\mathsf{M}_{\log}$ is defined in~\eqref{Mlog}.
\end{theorem}

\bnp[Proof of Theorem \ref{e:decaybeam}]
Thanks to Corollary \ref{corresolvL}, \eqref{e:res-fct-prop-bis} is true with $\mathsf{G} (\mu) = C(\mu +2)  e^{\kappa (\mu +2)^k}$. Theorem \ref{e:decaybeam} is an application of Theorem \ref{t:decay-rates-abstractbeam} with $\mathsf{M}(\lambda) = \lambda \mathsf{G}(\sqrt{\lambda})^2\leq  Ce^{2 \kappa^+ \lambda^{k/2}}$ (after having changed the constants slightly).
\enp

 \subsection{Lower bounds: proof of Proposition~\ref{Prop:BCG-damped}}

\bnp[Proof of Proposition~\ref{Prop:BCG-damped}]
According to~\cite[Proposition~1.14]{LL:17Hypo} (which relies on~\cite[Section~2.3]{BeauchardCanGugl}), since $\supp(b) \cap \left\{x_1=0\right\} = \emptyset$, there exist $C,c_0>0$ and a sequence $(\lambda_j , \varphi_j) \in \R_+ \times C^\infty(\M)$ such that
$$
\scrL\varphi_j = \lambda_j \varphi_j ,\quad  \varphi_j|_{\d\M} = 0 , \quad \|\varphi_j\|_{L^2(\M)}=1, \quad \lambda_j \to +\infty , \quad \| \varphi_j\|_{L^2(\supp(b))} \leq  Ce^{ - c_0 \lambda_j^{\frac{k}{2}}} .
$$
As a consequence, concerning the damped Schr\"odinger resolvent, we have
$$
\nor{(\A_S - i\lambda_j) \varphi_j}{L^2(\M)} = \nor{(i\scrL - b - i\lambda_j) \varphi_j}{L^2(\M)} = \nor{ b \varphi_j}{L^2(\M)}  \leq \nor{b}{L^\infty}Ce^{ - c_0 \lambda_j^{\frac{k}{2}}}  .
$$
This implies the second estimate in~\eqref{e:lower-resolvent-bounds} with $s_j = \lambda_j$.

Concerning the damped wave resolvent, recalling the definition of $P(z)$ in~\eqref{e:def-Pz}, we write
$$
\nor{P\big(i\sqrt{\lambda_j}\big)\varphi_j}{L^2} = \nor{\big(\scrL - \lambda_j + i\sqrt{\lambda_j} b\big)\varphi_j}{L^2} = \nor{\sqrt{\lambda_j} b\varphi_j}{L^2} \leq  \sqrt{\lambda_j} \nor{b}{L^\infty}Ce^{ - c_0 \lambda_j^{\frac{k}{2}}} .
$$
With $s_j = \sqrt{\lambda_j}$, this implies $\nor{P\big(is_j\big)\varphi_j}{L^2} \leq s_j Ce^{ - c_0 s_j^{k}}$, and using~\eqref{eq: estimate P equiv A} in Lemma~\ref{lemma: conditions stable resolvent} proves the first estimate in~\eqref{e:lower-resolvent-bounds}.

\medskip
The last part of the Proposition follows from~\eqref{e:lower-resolvent-bounds} together with the first implication in Theorem~\ref{t:batty-duyckaerts} (and, in case of damped waves, equivalence between the resolvents of $\A$ et $\dot{\A}$ in~\eqref{eq: estimate A estimate dotA} in Lemma~\ref{lemma: conditions stable resolvent}).
\enp

\small
\bibliographystyle{alpha}
\bibliography{bibli}

\end{document}


TITLE: Whitney embedding theorem for Hölder manifolds
QUESTION [8 upvotes]: According to a result of Whitney any $C^r$-manifold, $r\geq 1$, is $C^r$-homeomorphic to a smooth embedded submanifold of some Euclidean space; see Theorem 1 in
Hassler Whitney, "Differentiable manifolds", Annals of Mathematics  (2) 37 (1936), no. 3, pp. 645–680, JFM 62.1454.01, MR1503303, Zbl 0015.32001.
I expect that an analogous result should be true for $C^{r,\alpha}$-Hölder manifolds, $r\geq 1$, $\alpha\in[0,1]$, and I am wondering if there exists a reference for such a statement. Does perhaps anybody know such a reference?
Actually, I am not so much interested in the embedding, but in a refinement of the $C^{r,\alpha}$-Hölder structure to a smooth structure, i.e. in a $C^{r,\alpha}$-Hölder homeomorphism to a smooth manifold.
In particular, I am interested in the case in which the manifold to start with is itself a ($C^{r,\alpha}$-)embedded submanifold of some Euclidean space and in the case $\alpha=1$. Maybe it is easier to say something here?

REPLY [9 votes]: Every $C^1$ manifold admits a compatible $C^\infty$ structure. You can find a proof in Hirsch's "Differential topology". It is actually quite easy and based on a fact that smoothing a $C^1$ diffeomorphism by convolution leads to a smooth diffeomorphism (because derivatives converge uniformly) on a slightly smaller domain.
In particular $C^{r,\alpha}$ manifolds, $r\geq 1$, admit a compatible $C^\infty$-structure and hence such a manifold can be embedded as a smooth submanifold of an Euclidean space. However, the embedding will only be $C^{r,\alpha}$ smooth since the original manifold is only $C^{r,\alpha}$.


text:
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import linear_algebra.matrix.to_lin
import linear_algebra.matrix.trace
import linear_algebra.contraction
import linear_algebra.tensor_product_basis
import linear_algebra.free_module.strong_rank_condition
import linear_algebra.projection

/-!
# Trace of a linear map

This file defines the trace of a linear map.

See also `linear_algebra/matrix/trace.lean` for the trace of a matrix.

## Tags

linear_map, trace, diagonal

-/

noncomputable theory

universes u v w

namespace linear_map

open_locale big_operators
open_locale matrix
open finite_dimensional

open_locale tensor_product

section
variables (R : Type u) [comm_semiring R] {M : Type v} [add_comm_monoid M] [module R M]
variables {ι : Type w} [decidable_eq ι] [fintype ι]
variables {κ : Type*} [decidable_eq κ] [fintype κ]
variables (b : basis ι R M) (c : basis κ R M)

/-- The trace of an endomorphism given a basis. -/
def trace_aux :
  (M →ₗ[R] M) →ₗ[R] R :=
(matrix.trace_linear_map ι R R) ∘ₗ ↑(linear_map.to_matrix b b)

-- Can't be `simp` because it would cause a loop.
lemma trace_aux_def (b : basis ι R M) (f : M →ₗ[R] M) :
  trace_aux R b f = matrix.trace (linear_map.to_matrix b b f) :=
rfl

theorem trace_aux_eq : trace_aux R b = trace_aux R c :=
linear_map.ext $ λ f,
calc  matrix.trace (linear_map.to_matrix b b f)
    = matrix.trace (linear_map.to_matrix b b ((linear_map.id.comp f).comp linear_map.id)) :
  by rw [linear_map.id_comp, linear_map.comp_id]
... = matrix.trace (linear_map.to_matrix c b linear_map.id ⬝
        linear_map.to_matrix c c f ⬝
        linear_map.to_matrix b c linear_map.id) :
  by rw [linear_map.to_matrix_comp _ c, linear_map.to_matrix_comp _ c]
... = matrix.trace (linear_map.to_matrix c c f ⬝
        linear_map.to_matrix b c linear_map.id ⬝
        linear_map.to_matrix c b linear_map.id) :
  by rw [matrix.mul_assoc, matrix.trace_mul_comm]
... = matrix.trace (linear_map.to_matrix c c ((f.comp linear_map.id).comp linear_map.id)) :
  by rw [linear_map.to_matrix_comp _ b, linear_map.to_matrix_comp _ c]
... = matrix.trace (linear_map.to_matrix c c f) :
  by rw [linear_map.comp_id, linear_map.comp_id]

open_locale classical

variables (R) (M)

/-- Trace of an endomorphism independent of basis. -/
def trace : (M →ₗ[R] M) →ₗ[R] R :=
if H : ∃ (s : finset M), nonempty (basis s R M)
then trace_aux R H.some_spec.some
else 0

variables (R) {M}

/-- Auxiliary lemma for `trace_eq_matrix_trace`. -/
theorem trace_eq_matrix_trace_of_finset {s : finset M} (b : basis s R M)
  (f : M →ₗ[R] M) :
  trace R M f = matrix.trace (linear_map.to_matrix b b f) :=
have ∃ (s : finset M), nonempty (basis s R M),
from ⟨s, ⟨b⟩⟩,
by { rw [trace, dif_pos this, ← trace_aux_def], congr' 1, apply trace_aux_eq }

theorem trace_eq_matrix_trace (f : M →ₗ[R] M) :
  trace R M f = matrix.trace (linear_map.to_matrix b b f) :=
by rw [trace_eq_matrix_trace_of_finset R b.reindex_finset_range,
    ← trace_aux_def, ← trace_aux_def, trace_aux_eq R b]

theorem trace_mul_comm (f g : M →ₗ[R] M) :
  trace R M (f * g) = trace R M (g * f) :=
if H : ∃ (s : finset M), nonempty (basis s R M) then let ⟨s, ⟨b⟩⟩ := H in
by { simp_rw [trace_eq_matrix_trace R b, linear_map.to_matrix_mul], apply matrix.trace_mul_comm }
else by rw [trace, dif_neg H, linear_map.zero_apply, linear_map.zero_apply]

/-- The trace of an endomorphism is invariant under conjugation -/
@[simp]
theorem trace_conj (g : M →ₗ[R] M) (f : (M →ₗ[R] M)ˣ) :
  trace R M (↑f * g * ↑f⁻¹) = trace R M g :=
by { rw trace_mul_comm, simp }

end

section

variables {R : Type*} [comm_ring R] {M : Type*} [add_comm_group M] [module R M]
variables (N : Type*) [add_comm_group N] [module R N]
variables {ι : Type*} [fintype ι]

/-- The trace of a linear map correspond to the contraction pairing under the isomorphism
 `End(M) ≃ M* ⊗ M`-/
lemma trace_eq_contract_of_basis (b : basis ι R M) :
  (linear_map.trace R M) ∘ₗ (dual_tensor_hom R M M) = contract_left R M :=
begin
  classical,
  apply basis.ext (basis.tensor_product (basis.dual_basis b) b),
  rintros ⟨i, j⟩,
  simp only [function.comp_app, basis.tensor_product_apply, basis.coe_dual_basis, coe_comp],
  rw [trace_eq_matrix_trace R b, to_matrix_dual_tensor_hom],
  by_cases hij : i = j,
  { rw [hij], simp },
  rw matrix.std_basis_matrix.trace_zero j i (1:R) hij,
  simp [finsupp.single_eq_pi_single, hij],
end

/-- The trace of a linear map correspond to the contraction pairing under the isomorphism
 `End(M) ≃ M* ⊗ M`-/
lemma trace_eq_contract_of_basis' [decidable_eq ι] (b : basis ι R M) :
  (linear_map.trace R M) =
  (contract_left R M) ∘ₗ (dual_tensor_hom_equiv_of_basis b).symm.to_linear_map :=
by simp [linear_equiv.eq_comp_to_linear_map_symm, trace_eq_contract_of_basis b]

variables (R M N)
variables [module.free R M] [module.finite R M] [module.free R N] [module.finite R N] [nontrivial R]

/-- When `M` is finite free, the trace of a linear map correspond to the contraction pairing under
the isomorphism `End(M) ≃ M* ⊗ M`-/
@[simp] theorem trace_eq_contract :
  (linear_map.trace R M) ∘ₗ (dual_tensor_hom R M M) = contract_left R M :=
trace_eq_contract_of_basis (module.free.choose_basis R M)

@[simp] theorem trace_eq_contract_apply (x : module.dual R M ⊗[R] M) :
  (linear_map.trace R M) ((dual_tensor_hom R M M) x) = contract_left R M x :=
by rw [←comp_apply, trace_eq_contract]

open_locale classical

/-- When `M` is finite free, the trace of a linear map correspond to the contraction pairing under
the isomorphism `End(M) ≃ M* ⊗ M`-/
theorem trace_eq_contract' :
  (linear_map.trace R M) =
  (contract_left R M) ∘ₗ (dual_tensor_hom_equiv R M M).symm.to_linear_map :=
trace_eq_contract_of_basis' (module.free.choose_basis R M)

/-- The trace of the identity endomorphism is the dimension of the free module -/
@[simp] theorem trace_one : trace R M 1 = (finrank R M : R) :=
begin
  have b := module.free.choose_basis R M,
  rw [trace_eq_matrix_trace R b, to_matrix_one, module.free.finrank_eq_card_choose_basis_index],
  simp,
end

/-- The trace of the identity endomorphism is the dimension of the free module -/
@[simp] theorem trace_id : trace R M id = (finrank R M : R) :=
by rw [←one_eq_id, trace_one]

@[simp] theorem trace_transpose : trace R (module.dual R M) ∘ₗ module.dual.transpose = trace R M :=
begin
  let e := dual_tensor_hom_equiv R M M,
  have h : function.surjective e.to_linear_map := e.surjective,
  refine (cancel_right h).1 _,
  ext f m, simp [e],
end

theorem trace_prod_map :
  trace R (M × N) ∘ₗ prod_map_linear R M N M N R =
  (coprod id id : R × R →ₗ[R] R) ∘ₗ prod_map (trace R M) (trace R N) :=
begin
  let e := ((dual_tensor_hom_equiv R M M).prod (dual_tensor_hom_equiv R N N)),
  have h : function.surjective e.to_linear_map := e.surjective,
  refine (cancel_right h).1 _,
  ext,
  { simp only [dual_tensor_hom_equiv, tensor_product.algebra_tensor_module.curry_apply,
  to_fun_eq_coe, tensor_product.curry_apply, coe_restrict_scalars_eq_coe, coe_comp,
  linear_equiv.coe_to_linear_map, coe_inl, function.comp_app, linear_equiv.prod_apply,
  dual_tensor_hom_equiv_of_basis_apply, map_zero, prod_map_apply, coprod_apply, id_coe, id.def,
  add_zero, prod_map_linear_apply, dual_tensor_hom_prod_map_zero, trace_eq_contract_apply,
  contract_left_apply, fst_apply] },
  { simp only [dual_tensor_hom_equiv, tensor_product.algebra_tensor_module.curry_apply,
  to_fun_eq_coe, tensor_product.curry_apply, coe_restrict_scalars_eq_coe, coe_comp,
  linear_equiv.coe_to_linear_map, coe_inr, function.comp_app, linear_equiv.prod_apply,
  dual_tensor_hom_equiv_of_basis_apply, map_zero, prod_map_apply, coprod_apply, id_coe, id.def,
  zero_add, prod_map_linear_apply, zero_prod_map_dual_tensor_hom, trace_eq_contract_apply,
  contract_left_apply, snd_apply], },
end

variables {R M N}

theorem trace_prod_map' (f : M →ₗ[R] M) (g : N →ₗ[R] N) :
  trace R (M × N) (prod_map f g) = trace R M f + trace R N g :=
begin
  have h := ext_iff.1 (trace_prod_map R M N) (f, g),
  simp only [coe_comp, function.comp_app, prod_map_apply, coprod_apply, id_coe, id.def,
  prod_map_linear_apply] at h, exact h,
end

variables (R M N)

open tensor_product function

theorem trace_tensor_product :
  compr₂ (map_bilinear R M N M N) (trace R (M ⊗ N)) =
  compl₁₂ (lsmul R R : R →ₗ[R] R →ₗ[R] R) (trace R M) (trace R N) :=
begin
  apply (compl₁₂_inj
    (show surjective (dual_tensor_hom R M M), from (dual_tensor_hom_equiv R M M).surjective)
    (show surjective (dual_tensor_hom R N N), from (dual_tensor_hom_equiv R N N).surjective)).1,
  ext f m g n,
  simp only [algebra_tensor_module.curry_apply, to_fun_eq_coe, tensor_product.curry_apply,
  coe_restrict_scalars_eq_coe, compl₁₂_apply, compr₂_apply, map_bilinear_apply,
  trace_eq_contract_apply, contract_left_apply, lsmul_apply, algebra.id.smul_eq_mul,
  map_dual_tensor_hom, dual_distrib_apply],
end

theorem trace_comp_comm :
  compr₂ (llcomp R M N M) (trace R M) = compr₂ (llcomp R N M N).flip (trace R N) :=
begin
  apply (compl₁₂_inj
    (show surjective (dual_tensor_hom R N M), from (dual_tensor_hom_equiv R N M).surjective)
    (show surjective (dual_tensor_hom R M N), from (dual_tensor_hom_equiv R M N).surjective)).1,
  ext g m f n,
  simp only [tensor_product.algebra_tensor_module.curry_apply, to_fun_eq_coe,
      linear_equiv.coe_to_linear_map, tensor_product.curry_apply, coe_restrict_scalars_eq_coe,
      compl₁₂_apply, compr₂_apply, flip_apply, llcomp_apply', comp_dual_tensor_hom, map_smul,
      trace_eq_contract_apply, contract_left_apply, smul_eq_mul, mul_comm],
end

variables {R M N}

@[simp]
theorem trace_transpose' (f : M →ₗ[R] M) : trace R _ (module.dual.transpose f) = trace R M f :=
by { rw [←comp_apply, trace_transpose] }

theorem trace_tensor_product' (f : M →ₗ[R] M) (g : N →ₗ[R] N) :
  trace R (M ⊗ N) (map f g) = trace R M f * trace R N g :=
begin
  have h := ext_iff.1 (ext_iff.1 (trace_tensor_product R M N) f) g,
  simp only [compr₂_apply, map_bilinear_apply, compl₁₂_apply, lsmul_apply,
    algebra.id.smul_eq_mul] at h, exact h,
end

theorem trace_comp_comm' (f : M →ₗ[R] N) (g : N →ₗ[R] M) :
  trace R M (g ∘ₗ f) = trace R N (f ∘ₗ g) :=
begin
  have h := ext_iff.1 (ext_iff.1 (trace_comp_comm R M N) g) f,
  simp only [llcomp_apply', compr₂_apply, flip_apply] at h,
  exact h,
end

@[simp] theorem trace_conj' (f : M →ₗ[R] M) (e : M ≃ₗ[R] N) : trace R N (e.conj f) = trace R M f :=
by rw [e.conj_apply, trace_comp_comm', ←comp_assoc, linear_equiv.comp_coe,
  linear_equiv.self_trans_symm, linear_equiv.refl_to_linear_map, id_comp]

theorem is_proj.trace {p : submodule R M} {f : M →ₗ[R] M} (h : is_proj p f)
  [module.free R p] [module.finite R p] [module.free R f.ker] [module.finite R f.ker] :
  trace R M f = (finrank R p : R) :=
by rw [h.eq_conj_prod_map, trace_conj', trace_prod_map', trace_id, map_zero, add_zero]

end

end linear_map

subset_name: curated


TITLE: Bakry-Emery Laplacian and Hodge Decomposition
QUESTION [3 upvotes]: I have a question about the Hodge Decomposition theorem. Let $(X,\omega)$ be a compact Fano Kaeler manifold, we know the Hodge theorem works very well with respect to the $\bar\partial-$Laplacian $\Box_{\bar\partial}=\bar\partial\bar\partial^*+\bar\partial^*\bar\partial$. However, on Fano manifold, we have $Ric-\omega=\sqrt{-1}\partial\bar\partial f$ for some smooth function. we can use this $f$ to modify the $L^2-$ metric on the space of $(p,q)-$form, say $<\varphi, \psi>=\int (\varphi,\psi)e^{f}$, correspondingly, we can define $\bar\partial^*_f$ and Laplacian
$\Box_f=\bar\partial\bar\partial^*_f+\bar\partial^*_f\bar\partial$. So My question is if the Hodge decomposition with respect to $\Box_f$ still works? Thank you in advance.

REPLY [4 votes]: I don't know much about the complex geometry behind your question, but the Hodge decomposition theorem works in more general contexts such as the one you have described. Let me describe this generalization.
Let $M$ be a smooth manifold and $E_0, \dots, E_m$ be smooth vector bundles on $M$. The sequence of differential operators
$$0 \longrightarrow \Gamma(E_0) \xrightarrow{~D_0~} \Gamma(E_1) \xrightarrow{~D_1~} \cdots \xrightarrow{~D_{m-1}~} \Gamma(E_m) \longrightarrow 0$$
is called an elliptic complex if

The sequence is a cochain complex, i.e. $D_i \circ D_{i-1} = 0$ for $1 \leq i \leq m$.
The associated symbol complex
$$0 \longrightarrow (E_0)_x \xrightarrow{~\sigma_{D_0}(x, \xi)~} (E_1)_x \xrightarrow{~\sigma_{D_1}(x,\xi)~} \cdots \xrightarrow{~\sigma_{D_{m-1}}(x,\xi)~} (E_m)_x \longrightarrow 0$$
is exact for all $x \in M$, $\xi \in T_x^\ast M \setminus \{0\}$.

Now let each $E_i$ be equipped with an inner product and let
$$D_i^\ast: \Gamma(E_{i+1}) \longrightarrow \Gamma(E_i)$$
be the formal adjoint of $D_i$ with respect to this inner product, for $1 \leq i \leq m$. Define also
$$\Delta_i = D_{i-1} \circ D_{i-1}^\ast + D_i^\ast \circ D_i.$$
Then the following generalization of the Hodge theorem holds.

Theorem. Let $M$ be a compact, oriented manifold and suppose $(E_\ast, D_\ast)$ is an elliptic complex over $M$. Then the following hold.

There is an orthogonal direct sum decomposition
$$\Gamma(E_i) = \ker(\Delta_i) \oplus \mathrm{Im}(D_{i-1}) \oplus \mathrm{Im}(D_i^\ast).$$
$H^i(E_\ast, D_\ast) \cong \ker(\Delta_i)$.
$H^i(E_\ast, D_\ast)$ is finite-dimensional for all $i$.

In your case, you are still considering the usual Dolbeault complex
$$0 \to \Omega^{p,0}(X) \xrightarrow{~\bar{\partial}~} \Omega^{p,1}(X) \xrightarrow{~\bar{\partial~}} \cdots,$$
which is elliptic, and instead considering a different inner product on your bundles. The above generalized Hodge theorem tells you that you still get the expected Hodge decomposition
$$\Omega^{p,q}(X) = \ker(\square_f) \oplus \mathrm{Im}(\bar{\partial}) \oplus \mathrm{Im}(\bar{\partial}_f^\ast).$$
For the full details of the generalized Hodge theorem, you can see Chapter IV of Wells's Differential Analysis on Complex Manifolds.


TITLE: Minimum $a+b$ such that $ab=M$
QUESTION [2 upvotes]: This question is related to the fleablood's answer to another question.
Let $ab=M$ where $0<a,b,M\in\mathbb N$. He wanted to show that the sum $a+b$ is the smallest when $a,b$ are closest together. But he defined $a=\sqrt{M}r$, $b=\sqrt{M}/r$ with $r=1+e$ (where I suppose that $0<e\in\mathbb R$).
My question. Is it always possible to define the positive integers in $ab=M$ as $a=\sqrt{M}r$ and $b=\sqrt{M}/r$ with $r=1+e$? How would you show it?
Thanks in advance.

REPLY [2 votes]: Wow.  I wrote that a long time ago!
What I see now that I didn't make clear was that when I introduced the variables $a,b$ the where not supposed to be integers.  They were just supposed to be so that $ab = M > 0;a>0;b>0$.
What I wanted to prove was the concept that $a+b$ will be least when $a$ and $b$ are close together and largest when $a$ and $b$ are far apart.  (Indeed the very least $a$ and $b$ can be is $2\sqrt{M}$ when $a = b = \sqrt M$.)
(The idea was that if $a+b$ is always smallest when $a$ and $b$ are closest together then to find the integers with the least sum would be a matter of finding the pair of complementary factors that are closest together.  However to prove my concept, restricting to integers would not be relevant.)
However in order to prove my concept the variables so that $a\cdot b = M$ were not the best ones to prove $a+b$ is least when $a$ and $b$ are close.  A better choice of variables would be an $r> 0$ where $(\sqrt M\cdot r)(\frac {\sqrt M}r) = M$ and we would need to prove that $(M\cdot r)+(\frac{\sqrt M}r)$ is least when $r$ and $\frac 1r$ are each closest to $1$.
So...
Now it should be clear that for any $M > 0$ we can find $a,b$ so that $ab =M$ (just let $a> 0$ be anything and let $b =\frac Mb$).  And it should be clear we can always find an $r$ so that $(\sqrt M r)(\frac {\sqrt M}r)=M$ (just let $r$ be anything positive).  And it should be clear that for any pair of $a,b$ (integers or not) we can convert to $r$ by letting $r = \frac a{\sqrt M}$. And for any $r> 0$ we con convert to $a,b$ by letting $a =\sqrt M\cdot r$ and $b=\frac {\sqrt M}r$.
However for any $r$ we have absolutely no reason to believe those  $a $ and $b$ will be integers.
So the answer to your question is: No.  $a$ and $b$ will only be integers for some values of $r$.  If $M$ is an integer there will always be some $r$ where they are both integers, but for any $r$ in general, the resulting $a,b$ will not be integers.
....
When I finished proving that the closer $r$ is to $1$ then the smaller $(\sqrt Mr) + (\frac {\sqrt M}r)$, I completed the argument by noting to find the integers $a,b$ so that $ab=M$ and $a+b$ is the least possible value, we need to find the integer factor pairs of $M$ that are closest together.


TITLE: Integral $\int_0^\infty \log \frac{1+x^3}{x^3} \frac{x \,dx}{1+x^3}=\frac{\pi}{\sqrt 3}\log 3-\frac{\pi^2}{9}$
QUESTION [11 upvotes]: I am trying to prove this interesting integral
$$
I:=\int_0^\infty \log \frac{1+x^3}{x^3} \frac{x \,dx}{1+x^3}=\frac{\pi}{\sqrt 3}\log 3-\frac{\pi^2}{9}.
$$
I tried using $y=1+x^3$ but that didn't help.
  We can possibly try
$$
I=\int_0^\infty \frac{\log(1+x^3) x}{1+x^3} \,dx-\int_0^\infty \frac{\log(x^3) x}{1+x^3}\,dx.
$$ These integrals would be much easier had the bounds been from $0 $\ to $\infty$, however they are not.  Perhaps partial integration will work but I didn't find the way if we try
$$
dv=\frac{x}{1+x^3}, \quad u= \log(1+x^3)
$$
but I ran into a divergent integral. Thanks how can we prove I?

REPLY [1 votes]: Letting $x\mapsto \frac{1}{x}$ simplify the integral as
$$
I=\int_0^{\infty} \ln \left(\frac{1+x^3}{x^3}\right) \frac{x}{1+x^3} d x =\int_0^{\infty} \frac{\ln \left(x^3+1\right)}{x^3+1} d x
$$
Next consider another integral
$$
I(a)=\int_0^{\infty}\left(x^3+1\right)^a d x
$$
and transform $I(a)$, by putting $y=\frac{1}{x^3+1}$, into a beta function
$$
\begin{aligned}
I(a) &=\frac{1}{3} \int_0^1 y^{-a-\frac{4}{3}}(1-y)^{-\frac{2}{3}} d y \\
&=\frac{1}{3} B\left(-a-\frac{1}{3}, \frac{1}{3}\right)
\end{aligned}
$$
Differentiating $I(a)$ w.r.t. $a$ yields
$$
I^{\prime}(a)=\frac{1}{3} B\left(-a-\frac{1}{3}, \frac{1}{3}\right)\left(\psi\left(-a-\frac{1}{3}\right)-\psi(-a)\right)
$$
Then putting $a=1$ gives our integral$$
\begin{aligned}
I&=I^{\prime}(-1) \\&=\frac{1}{3} B\left(\frac{2}{3}, \frac{1}{3}\right)\left(\psi\left(\frac{2}{3}\right)-\psi(1)\right) \\
&=\frac{1}{3} \cdot \frac{2 \pi}{\sqrt{3}}\left(\gamma-\frac{\pi}{2 \sqrt{3}}+\frac{3 \ln 3}{2}-\gamma\right) \\
&=\frac{\pi}{\sqrt{3}} \ln 3-\frac{\pi^2}{9}
\end{aligned}
$$


text:
\begin{document}
\baselineskip5.6mm

\title{Lipschitz retraction of finite subsets of Hilbert spaces}

\author{Leonid V. Kovalev}
\address{215 Carnegie, Mathematics Department, Syracuse University, Syracuse, NY 13244-1150}
\email{lvkovale@syr.edu}
\thanks{Supported by the National Science Foundation grant DMS-1362453.}

\subjclass[2010]{Primary 54E40; Secondary 54B20, 54C15, 54C25}

\begin{abstract}
Finite subset spaces of a metric space \(X\) form a nested sequence under natural isometric embeddings \(X=X(1)\subset X(2)\subset\dots\).
We prove that this sequence admits Lipschitz retractions \(X(n)\to X(n-1)\) when \(X\) is a Hilbert space.
\end{abstract}

\maketitle

\section{Introduction}

Let $X$ be a metric space. For $n\ge 1$, $X(n)$ denotes the set of all nonempty subsets of $X$ with cardinality at most $n$. Equipped with the Hausdorff metric, $X(n)$ becomes a metric space which is sometimes called
a symmetric product or symmetric power of $X$. Following Tuffley~\cite{Tu1} we use the descriptive term
\emph{finite subset space} for $X(n)$. This space is related to, but different from $X^n/S_n$, the space
of unordered $n$-tuples of points in $X$.

One feature that distinguishes $X(n)$ from the Cartesian power $X^n$ and the quotient $X^n/S_n$ is the existence of canonical isometric embeddings $X(n)\subset X(n+1)$. The embeddings $X^n\subset X^{n+1}$ are not canonical: they depend on the choice of a base point in $X$. Furthermore, the geometry of embedding
$X(n)\subset X(n+1)$ is far richer than the geometry of $X^n\subset X^{n+1}$. For example~\cite{Mo}, $S^1(1)\subset S^1(3)$ is a trefoil knot embedded in $S^3$ which bounds a M\"obuis band, namely $S^1(2)$.
This example shows that in general the canonical embeddings $\iota\colon X(n)\to X(n+1)$ do not split:
there need not be a continuous map $r\colon X(n+1)\to X(n)$ such that $r\circ \iota=\mathrm{id}$.

On the other hand, there is a Lipschitz retraction of $\mathbb R(n+1)$ onto $\mathbb R(n)$ for every $n\ge 1$.
This observation, made in~\cite{Ko}, was used to show the bi-Lipschitz embeddability of $\mathbb R(n)$ into a Euclidean space. Our main result shows that such a Lipschitz retraction exists for all Hilbert spaces, either finite-dimensional or infinite-dimensional. In this context, there is no loss of generality in assuming the vector spaces are real.

\begin{theorem}\label{thm1}
Let $\HH$ be a Hilbert space.  Then for every $n\ge 2$ there exists a Lipschitz retraction $r_n\colon \HH(n )\to \HH(n-1)$.
\end{theorem}

By Remark 4.5 in~\cite{Ko}, combining the case $X=\mathbb R^d$ of Theorem~\ref{thm1} with the results of~\cite{Ko} yields the following corollary.

\begin{corollary}\label{embedcor} For $d,n\ge 1$, the space $\mathbb R^d(n)$ is an absolute  Lipschitz retract.
\end{corollary}

Since the existence of Lipschitz retractions $r\colon X(n)\to X(n-1)$  is a bi-Lipschitz invariant of $X$, Theorem~\ref{thm1} applies also to Banach spaces that are isomorphic to a Hilbert space. However, it remains unclear whether such retractions exist for general Banach spaces, beyond the trivial case $X(2)\to X(1)$ given by the midpoint map $\{a,b\}\mapsto \{(a+b)/2\}$.

Of particular interest here is the case $X=\ell^\infty$, because $\ell^\infty$ is an absolute $1$-Lipschitz retract, i.e., admits a $1$-Lipschitz retraction from any larger metric space containing it. Indeed, it remains unknown whether the property of being an absolute Lipschitz retract is inherited by finite subset spaces in general. See~\cite{BU} for the topological version of this problem, and ~\cite{Go}, \cite{AIMPL} for the Lipschitz version.

Another setting to which Theorem~\ref{thm1} could be conceivably extended is CAT(0) metric spaces. The existence of $1$-Lipschitz retraction $X(2)\to X(1)$ for such spaces is a well-known consequence of the convexity of the metric in CAT(0) spaces (e.g.,~\cite{BH}): the map sending each pair of points to the midpoint of the geodesic connecting them provides such a retraction.

\section{Proof of Theorem~\ref{thm1}}

Let $\HH^n$ be the Cartesian power of $\HH$, equipped with the metric
\[
d((x_1,\dots,x_n),(y_1,\dots,y_n))=\left(\sum_{k=1}^n \|x_k-y_k\|^2 \right)^{1/2}
\]
The product $\HH^n$  is also a Hilbert space.
Define a function $\Phi \colon \HH^n\to \mathbb R$ by
\begin{equation*}
\Phi(x_1,\dots,x_n)= \sum_{1\le i<j\le n} \|x_i-x_j\|
\end{equation*}
It is easy to see that $\Phi$ is a convex function on $\HH^n$.
Let $D=\{x\in \HH^n \colon x_i=x_j \text{ for some } i\ne j\}$. The function $\Phi$ is Fr\'echet differentiable at every point of $\HH^n\setminus D$, with the derivative
\begin{equation}\label{e2grad}
\nabla \Phi(x) =\left( \sum_{j\ne i}\frac{x_i-x_j}{\|x_j-x_i\|} \right)_{i=1}^n
\end{equation}
By~\eqref{e2grad}, $\Phi$ satisfies the upper gradient bound
\begin{equation}\label{uppergrad}
\|\nabla \Phi\| \le (n-1)\sqrt{n}, \quad x\in \HH^n\setminus D
\end{equation}

Given a set in $\HH(n)\setminus \HH(n-1)$, enumerate its elements as $\{x_1,\dots,x_n\}$ (in arbitrary
order), thus associating to it a point  $x\in \HH^n\setminus D$. Since $x$ uniquely identifies the set $\{x_i\}$, we sometimes write $x$ instead of $\{x_i\}$ to simplify notation.

Consider the ODE system
\be{ode}
\frac{du_i}{dt} = \sum_{j\ne i}\frac{u_j-u_i}{\|u_j-u_i\|},\quad i=1,\dots,n
\ee
with the initial conditions $u_i(0)=x_i$. In view of~\eqref{e2grad}, the system~\eqref{ode} can be seen as the gradient flow of the function $\Phi$. Note that the right hand side of~\eqref{ode} belongs to the finite-dimensional subspace spanned by $x_1,\dots,x_n$. Hence, the solution remains in this subspace as long as it exists. By the   Picard existence and uniqueness theorem, there is a unique solution  until $u$ reaches the set $D$.

Let $[0,T(x))$ be the maximal interval of existence of solution of~\eqref{ode}.  Denote $\delta(x) = \min_{i<j}\|x_i-x_j\|$. Since
\be{lower}
\left\|\frac{du_i}{dt}\right\|  \le n-1 \quad \text{for all }i
\ee
it follows that
\[
T(x)\ge \frac{\delta(x)}{2(n-1)}
\]

The following inequality provides an estimate for $T(x)$ in the reverse direction; it turns out that $T(x)$ is comparable to $\delta(x)$.

\be{quick}
T(x) \le \frac{\delta(x)}{2}
\ee

\begin{proof}[Proof of~\eqref{quick}]
A map $F\colon\HH\to\HH$ is called monotone if
\[\langle F(a)-F(b),a-b\rangle \ge 0 \quad \text{ for all } a,b\in\R^d\]
It is a well-known fact~\cite{Ro}*{\S 24} that the gradient of any convex function is monotone. In particular, $F(x)=x/\|x\|$ is a monotone map, being the gradient of convex function $x\mapsto \|x\|$.

Renumbering the points $x_i$, we may assume $\|x_1-x_2\|=\delta(x)$. Consider the function $\varphi(t)=\|u_1(t)-u_2(t)\|$, $0<t<t_c$. Differentiation yields
\[\varphi'(t)=\|u_1-u_2\|^{-1}\left \langle \frac{du_1}{dt}-\frac{du_2}{dt} , u_1-u_2\right \rangle\]
The inner product on the right consists of the term
\[
\langle F(u_2-u_1)-F(u_1-u_2), u_1-u_2\rangle = -2\|u_1-u_2\|
\]
and the sum over $j=3,\dots,n$ of
\[\begin{split} &\langle F(u_j-u_1)-F(u_j-u_2), u_1-u_2\rangle \\
&= - \langle F(u_j-u_1)-F(u_j-u_2), (u_j-u_1)-(u_j-u_2)\rangle \\ & \le 0.
\end{split} \]
Thus, $\varphi'(t) \le -2$ for $0<t<T(x)$, and since $\varphi(t)\ge 0$ by definition, it follows that $T(x)\le \varphi(0)/2 = \delta(x)/2$.
\end{proof}

We are now ready to define the retraction $r\colon X(n)\to X(n-1)$. On the subset $X(n-1)\subset X(n)$ it is the identity map. For a set $x=\{x_1,\dots,x_n\}\in X(n)\setminus X(n-1)$
let $r(\{x_i\})=\{u_i(T(x))\}$. This is well-defined because a different enumeration of the elements
$\{x_1,\dots,x_n\}$ would only result in a different enumeration of the elements $\{u_i(T(x))\}$.

It remains to prove that $r$ is a Lipschitz
retraction of $\HH(n)$ onto $\HH(n-1)$ in the Hausdorff metric $d_H$. Specifically,
\be{Lipgoal}
d_H(r(x),r(y))\le \max(n^{3/2},2n-1) \,d_H(x,y)
\ee
for all $x,y\in\HH(n)$.

\begin{proof}[Proof of ~\eqref{Lipgoal}]
Let $(u_i)$ and $(v_i)$ be the solutions of~\eqref{ode}  with initial data $(x_i)$ and $(y_i)$, respectively.

Combining ~\eqref{quick} and~\eqref{lower}, we obtain that
\be{case1}
d_H(r(x),x) \le \frac{n-1}{2} \delta(x)
\ee
and similarly for $y$.

\begin{lemma}\label{stable}
 $\sum_{i=1}^n \|u_i(t)-v_i(t)\|^2$ is a  nonincreasing function of $t$ for
 $0<t< \min(T(x),T(y))$.
\end{lemma}

\begin{proof} The point $(u_1(t),\dots,u_n(t))\in \HH^n $ evolves under the gradient flow of the convex function $\Phi(u_1,\dots,u_n) = \sum_{i<j}\|u_i-u_j\|$. Since the gradient of a convex function is monotone, we have
\[\left\langle \frac{du}{dt}-\frac{dv}{dt}, u-v \right \rangle\le 0\]
The left hand side is $1/2$ of the derivative of $\|u(t)-v(t)\|^2$ with respect to $t$, which proves the claim.
 \end{proof}

As a consequence of Lemma~\ref{stable},
\be{stableH}
d_H(\{u_i\},\{v_i\}) \le \sqrt{n} \max_{i}\|x_i-y_i\|
\ee
for all $0<t<\min(T(x),T(y))$. Let $\rho=d_H(x,y)$.

\textbf{Case 1}: $\delta(x)+\delta(y)\le 4\,\rho$. Then from~\eqref{case1}
we get
\[
d_H(r(x),r(y)) \le \rho+ d_H(r(x),x)+d_H(r(y),y)  \le \rho + 2(n-1)\rho
\]
which implies~\eqref{Lipgoal}.

\textbf{Case 2}: $\delta(x)+\delta(y)>  4\rho$. We may assume
$\delta(x)> 2\rho$. Since the function $\delta$ is $2$-Lipschitz in the Hausdorff metric, it follows that $\delta(y)>0$.

The geometric meaning of $\delta(x)> 2\rho$ is that the points $x_i$ are separated by more than $2\rho$, yet each of them is within $\rho$ of some point $y_j$. Therefore, we can enumerate the points $x_i$ and $y_i$ in such a way that
 \be{close}
 \|x_i-y_i\|\le \rho \quad\text{ for } \ i=1,\dots,n
 \ee
From now on we use only~\eqref{close}, in which the roles of $x$ and $y$  can be
interchanged. Thus, we may assume that $T(x)\le T(y)$.

By definition, $r(x) = \{u_i(T(x))\}$. Let $z =  \{v_i(T(x))\}$. By~\eqref{stableH} we have
\be{goal1}
d_H(r(x),z)\le \sqrt{n}\,\rho
\ee
Since $\delta$ is $2$-Lipschitz and $\delta(r(x))=0$, it follows that
\[\delta(z)\le 2d_H(r(x),z)\le 2\sqrt{n}\,\rho\]
The estimate~\eqref{case1} yields
\be{goal2}
d_H(r(z),z)\le (n-1)\sqrt{n}\,\rho
\ee
Now~\eqref{Lipgoal} follows from~\eqref{goal1} and~\eqref{goal2}:
\[
d_H(r(x),r(y))\le d_H(r(x),z)+d_H(r(z),z)  \le n^{3/2}\rho
\qedhere \]
\end{proof}

\section{An example and open questions}

Since the midpoint map $\HH(2)\to\HH(1)$ is Lipschitz with constant $1$, it is natural to ask whether a $1$-Lipschitz retraction of $\HH(n)$ onto $\HH(n-1)$ exists  for $n\ge 3$. The following example, given by the referee of an earlier version of this paper, shows that the answer is negative already for  $n=3$.

\begin{example} There is no $1$-Lipschitz retraction from $\R^2(3)$ onto $\R^2(2)$.
\end{example}

\begin{proof}
Let $A=\{(0,0),(1,0),(1/2,\sqrt{3}/2)\}$ be the set of vertices of an equilateral triangle of sidelength $1$ in the plane $\R^2$. Also let $B = \{(-1,0),(0,0)\}$ and $C=\{(1,0),(2,0)\}$; these sets lie on the line extending the base of the triangle. Then
$
d_H(A,B)=d_H(A,C)=1
$
and $d_H(B,C)=2$. If there was a $1$-Lipschitz retraction of $\R^2(3)$ onto $\R^2(2)$, the image of $A$ would be some set $E\in \R^2(2)$ such that
$d_H(E,B)\le 1$ and $d_H(E,C)\le 1$. The only such set is $\{(0,0),(1,0)\}$, formed by the vertices of the base of the triangle $A$. However, the above argument   also  applies to two other sides of $A$, which yields  a contradiction.
\end{proof}

\begin{question}\label{q1}
Do there exist   retractions $\HH(n)\to  \HH(n-1)$ with the Lipschitz constants bounded independently of $n$?
\end{question}

In conclusion we state  the questions mentioned in the introduction.

\begin{question} \label{q3}
If $X$ is a CAT(0) metric space, do there exist Lipschitz retractions $X(n)\to  X(n-1)$ for every $n\ge 2$?
\end{question}

\begin{question} \label{q2}
If $X$ is a Banach space, do there exist Lipschitz retractions $X(n)\to  X(n-1)$ for every $n\ge 2$?
\end{question}

Although the linear span of every $n$-subset of a Banach space $X$ can be given an equivalent inner product metric (thus allowing for a Lipschitz retraction within this subspace), the retraction depends on the choice of renorming. Thus, it seems that Theorem~\ref{thm1} cannot be used to answer Question~\ref{q2}.

\end{document}

config: arxiv


TITLE: Combinatorics-multinomial theorem
QUESTION [2 upvotes]: Prove that:
$$\sum_{n_1+n_2+n_3 = n} \binom{n}{n_1,n_2,n_3} \cdot (-1)^{n_2} = 1$$
The sum is over all positive integer solutions of $n_1 + n_2 + n_3 = n$.
Use the multinomial theorem:
$$(x_1 + x_2 + x_3)^n = \sum_{n_1 + n_2 + n_3 = n} \binom{n}{n_1,n_2,n_3} x_1^{n_1} x_2^{n_2} x_3^{n_3}$$
Let $x_2 = z$, $x_1 = x_3 = 0$ and plug into the above:
$$(0 + z + 0)^n = \sum_{n_1 + n_2 + n_3 = n} \binom{n}{n_1,n_2,n_3} 0^{n_1} z^{n_2} 0^{n_3} = \sum_{n_1 + n_2 + n_3 = n} \binom{n}{n_1,n_2,n_3} z^{n_2}$$
Let $z = -1$ and plug in:
$$\sum_{n_1 + n_2 + n_3 = n} \binom{n}{n_1,n_2,n_3} (-1)^{n_2}$$
But can't $n_2$ be both even and odd, so it could be either $1$ or $-1$?

REPLY [4 votes]: HINT: From the multinomial theorem, we have
$$(x_1 + x_2 + x_3)^n = \sum_{n_1 + n_2 + n_3 = n} \binom{n}{n_1,n_2,n_3} x_1^{n_1} x_2^{n_2} x_3^{n_3}$$
Now try plugging in $x_1 = x_3 = 1$ and $x_2 = -1$...


TITLE: Fitting an absolute-value formula to a collection of points in the plane
QUESTION [0 upvotes]: I need an equation to satisfy the following data points and I can't come up with anything. I believe it needs an absolute value as that changes the slope of the line.
Best yet: $y=2x-|x-4|$
Points are: $(1,1)$; $(2,3)$; $(3,5)$; $(4,7)$; $(5,8)$; $(6,10)$; $(7,11)$; $(8,12)$; $(9,13)$; $(10,14)$; $(11,15)$
Thanks in advance

REPLY [1 votes]: It is not clear if the goal is to find a function $y(x)$ which fits EXACTLY, or approximately.
If exactly is required :
$$\begin{cases}
y=2x-1 &\text{ if } x\leq 4 \\
y=8 &\text{ if } x=5 \\
y=x+4 &\text{ if } x\geq 6
\end{cases} \quad\to\quad y=8+(2x-9)H(4.5-x)+(x-4)H(x-4.5)$$
$H$ is the Heaviside step function.
If the fourth point is $(5,9)$ instead of $(5,8)$ a simpler form is : $$y=\frac{1}{2}(3+3x-|x-5|)$$.
If an approximate fit is wanted :
A usual regression calculus is sufficient, for example with the function $y=ax^2+by+c$. This is a linear regression for $a, b, c $ which leads to :
$$a\simeq -0.0711$$
$$b\simeq 2.226$$
$$c\simeq -1.085$$
The graph of the function $y=ax^2+by+c$ is shown :

This was computed with the fourth point $(5,8)$.