\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 64, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/64\hfil Self-adjointness of Schr\"odinger-type operators] {Self-adjointness of Schr\"odinger-type operators with singular potentials on manifolds of bounded geometry} \author[Ognjen Milatovic\hfil EJDE--2003/64\hfilneg] {Ognjen Milatovic} \address{Ognjen Milatovic \newline 78 Apsley Street, Apt. 1\\ Hudson, MA 01749, USA } \email{omilatovic@fsc.edu} \date{} \thanks{Submitted May 13, 2003. Published June 11, 2003.} \subjclass[2000]{35P05, 58J50, 47B25, 81Q10} \keywords{Schr\"odinger operator, self-adjointness, manifold, bounded geometry, \hfill\break\indent singular potential} \begin{abstract} We consider the Schr\"odinger type differential expression $$ H_V=\nabla^*\nabla+V, $$ where $\nabla$ is a $C^{\infty}$-bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M,g)$ with metric $g$ and positive $C^{\infty}$-bounded measure $d\mu$, and $V=V_1+V_2$, where $0\leq V_1\in L_{\rm loc}^1(\operatorname{End} E)$ and $0\geq V_2\in L_{\rm loc}^1(\operatorname{End} E)$ are linear self-adjoint bundle endomorphisms. We give a sufficient condition for self-adjointness of the operator $S$ in $L^2(E)$ defined by $Su=H_Vu$ for all $u\in\operatorname{Dom}(S)=\{u\in W^{1,2}(E)\colon \int\langle V_1u,u\rangle\,d\mu<+\infty\text{ and }H_Vu\in L^2(E)\}$. The proof follows the scheme of T.~Kato, but it requires the use of more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of $u\in L^2(M)$ satisfying the equation $(\Delta_M+b)u=\nu$, where $\Delta_M$ is the scalar Laplacian on $M$, $b>0$ is a constant and $\nu\geq 0$ is a positive distribution on $M$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{rem}[theorem]{Remark} \section{Introduction and main result}\label{S:main} Let $(M,g)$ be a $C^{\infty}$ Riemannian manifold without boundary, with metric $g$, $\dim M=n$. We will assume that $M$ is connected. We will also assume that $M$ has bounded geometry. Moreover, we will assume that we are given a positive $C^{\infty}$-bounded measure $d\mu$, i.e. in any local coordinates $x^{1}, x^{2},\dots,x^{n}$ there exists a strictly positive $C^{\infty}$-bounded density $\rho(x)$ such that $d\mu=\rho(x)dx^{1}dx^{2}\dots dx^{n}$. Let $E$ be a Hermitian vector bundle over $M$. We will assume that $E$ is a bundle of bounded geometry (i.e.~it is supplied by an additional structure: trivializations of $E$ on every canonical coordinate neighborhood $U$ such that the corresponding matrix transition functions $h_{U,U'}$ on all intersections $U\cap U'$ of such neighborhoods are $C^{\infty}$-bounded, i.e.~all derivatives $\partial_{y}^{\alpha}h_{U,U'}(y)$, where $\alpha$ is a multiindex, with respect to canonical coordinates are bounded with bounds $C_{\alpha}$ which do not depend on the chosen pair $U$, $U'$). We denote by $L^2(E)$ the Hilbert space of square integrable sections of $E$ with respect to the scalar product \begin{equation}\label{E:inner} (u,v) = \int_{M}\, \langle u(x),v(x)\rangle\, d\mu(x). \end{equation} Here $\langle\cdot,\cdot\rangle$ denotes the fiberwise inner product in $E_x$. In what follows, $C^{\infty}(E)$ denotes smooth sections of $E$, and $C_{c}^{\infty}(E)$ denotes smooth compactly supported sections of $E$. Let $$\nabla\colon C^{\infty}(E)\to C^{\infty}(T^*M\otimes E)$$ be a Hermitian connection on $E$ which is $C^{\infty}$-bounded as a linear differential operator, i.e.~in any canonical coordinate system $U$ (with the chosen trivializations of $E|_{U}$ and $(T^*M\otimes E)|_{U}$), $\nabla$ is written in the form \[ \nabla=\sum_{|\alpha|\leq 1}a_{\alpha}(y)\partial_{y}^{\alpha}, \] where $\alpha$ is a multiindex, and the coefficients $a_{\alpha}(y)$ are matrix functions whose derivatives $\partial_{y}^{\beta}a_{\alpha}(y)$ for any multiindex $\beta$ are bounded by a constant $C_{\beta}$ which does not depend on the chosen canonical neighborhood. We will consider a Schr\"odinger type differential expression of the form \[ H_V=\nabla^*\nabla+V, \] where $V$ is a linear self-adjoint bundle map $V\in L_{\rm loc}^{1}(\operatorname{End} E)$. Here $$\nabla^*\colon C^{\infty}(T^*M\otimes E)\to C^{\infty}(E)$$ is a differential operator which is formally adjoint to $\nabla$ with respect to the scalar product~\eqref{E:inner}. If we take $\nabla=d$, where $d\colon C^{\infty}(M)\to \Omega^{1}(M)$ is the standard differential, then $d^*d\colon C^{\infty}(M)\to C^{\infty}(M)$ is called the scalar Laplacian and will be denoted by $\Delta_M$. We make the following assumption on $V$. \begin{itemize} \item[(A1)] $V=V_1+V_2$, where $0\leq V_1\in L_{\rm loc}^{1}(\operatorname{End} E)$ and $0\geq V_2\in L_{\rm loc}^{1}(\operatorname{End} E)$ are linear self-adjoint bundle maps (here the inequalities are understood in the sense of operators $E_x\to E_x$). \end{itemize} By $W^{1,2}(E)$ we denote the completion of the space $C^\infty_c(E)$ with respect to the norm $\|\cdot\|_{1}$ defined by the scalar product \[ (u,v)_{1} := (u,v) + (\nabla u,\nabla v) \quad u, v\in C^{\infty}_c(E). \] By $W^{-1,2}(E)$ we will denote the dual of $W^{1,2}(E)$. \section{Quadratic forms} In what follows, all quadratic forms are considered in the Hilbert space $L^2(E)$. By $h_0$ we denote the quadratic form \begin{equation}\label{E:quad-h_0} h_0(u)=\int |\nabla u|^2 \,d\mu \end{equation} with the domain $\text{D}(h_0)=W^{1,2}(E)\subset L^2(E)$. Clearly, $h_0$ is a non-negative, densely defined and closed form. By $h_1$ we denote the quadratic form \begin{equation}\label{E:quad-h_1} h_1(u)=\int\langle V_1u,u\rangle\,d\mu \end{equation} with the domain \begin{equation}\label{E:d(h_1)} \text{D}(h_1)=\big\{u\in L^2(E):\int \langle V_1u,u\rangle\,d\mu<+\infty\big\}. \end{equation} Clearly, $h_1$ is a non-negative, densely defined, and closed form. By $h_2$ we denote the quadratic form \begin{equation} \label{E:quad-h_2} h_2(u)=\int\langle V_2u,u\rangle\,d\mu \end{equation} with the domain \begin{equation}\label{E:d(h_2)} \text{D}(h_2)=\big\{u\in L^2(E):\int \left|\langle V_2u,u\rangle\right|\,d\mu<+\infty\big\}. \end{equation} Clearly, $h_2$ is a densely defined form. Moreover, $h_2$ is symmetric (but not semi-bounded below). We make the following assumption on $h_2$. \begin{itemize} \item[(A2)] Assume that $h_2$ is $h_0$-bounded with relative bound $b<1$, i.e. \begin{enumerate} \item[(i)] $\text{D}(h_2)\supset\text{D}(h_0)$ \item[(ii)] There exist constants $a\geq 0$ and $0\leq b<1$ such that \begin{equation}\label{E:h_2-h_0-domination} |h_2(u)|\leq a\|u\|^2+b|h_0(u)|,\quad\text{for all }u\in\text{D}(h_0), \end{equation} where $\|\cdot\|$ denotes the norm in $L^2(E)$. \end{enumerate} \end{itemize} \begin{rem}\label{R:Assumption-B} \rm With the above assumptions on $(M,g)$, bundle $E$ and connection $\nabla$, Assumption (A2) holds if $V_2\in L^{p}(\operatorname{End} E)$, where $p=n/2$ for $n\geq 3$, $p>1$ for $n=2$, and $p=1$ for $n=1$. The proof is given in the last section of this article. %Sec.~\ref{S:sufficient-condition}. \end{rem} %\subsection*{A realization of $H_V$ in $L^2(E)$}\label{SS:operator-S} As a realization of $H_V$ in $L^2(E)$, we define the operator $S$ in $L^2(E)$ by the formula $Su=H_Vu$ on the domain \begin{equation}\label{E:domain-S} \operatorname{Dom}(S)=\big\{u\in W^{1,2}(E): \int \langle V_1u,u\rangle\,d\mu<+\infty\text{ and }H_Vu\in L^2(E)\big\}. \end{equation} \begin{rem}\label{R:distrib-interpret} \rm For all $u\in\text{D}(h_0)=W^{1,2}(E)$ we have $\nabla^*\nabla u\in W^{-1,2}(E)$, and from Corollary~\ref{C:diagonal-2} below it follows that for all $u\in W^{1,2}(E)\bigcap\text{D}(h_1)$, we have $Vu\in L_{\rm loc}^1(E)$. Thus $H_Vu$ in~\eqref{E:domain-S} is a distributional section of $E$, and the condition $H_Vu\in L^2(E)$ makes sense. \end{rem} We now state the main result. \begin{theorem}\label{T:main} Assume that $(M,g)$ is a manifold of bounded geometry with positive $C^{\infty}$-bounded measure $d\mu$, $E$ is a Hermitian vector bundle of bounded geometry over $M$, and $\nabla$ is a $C^{\infty}$-bounded Hermitian connection on $E$. Suppose that Assumptions (A1) and (A2) hold. Then $S$ is a semi-bounded below self-adjoint operator. \end{theorem} \begin{rem} \rm Theorem~\ref{T:main} extends a result of T.~Kato, cf.~Theorem VI.4.6(a) in~\cite{Kato66} (see also remark 5(b) in~\cite{Kato78}) which was proven for the operator $-\Delta+V$, where $\Delta$ is the standard Laplacian on $\mathbb{R}^n$ with the standard metric and measure, and $V=V_1+V_2$, where $0\leq V_1 \in L_{\rm loc}^1({\mathbb{R}}^n)$ and $0\geq V_2 \in L_{\rm loc}^1({\mathbb{R}}^n)$ are as in Assumption (A1) above, and the quadratic form $h_2$ corresponding to $V_2$ is as in Assumption (A2) above. \end{rem} \section{Proof of Theorem \ref{T:main}}\label{S:proof-main} We adopt the arguments from Sec.~VI.4 in~\cite{Kato66} to our setting with the help of more general version of Kato's inequality~\eqref{E:kato}. We begin with the following variant of Kato's inequality for Bochner Laplacian (for the proof see Theorem 5.7~in~\cite{bms}). The original version of Kato's inequality was proven in Kato~\cite{Kato72}. \begin{lemma}\label{L:kato} Assume that $(M,g)$ is a Riemannian manifold. Assume that $E$ is a Hermitian vector bundle over $M$ and $\nabla$ is a Hermitian connection on $E$. Assume that $w\in L_{\rm loc}^{1}(E)$ and $\nabla^*\nabla w\in L_{\rm loc}^{1}(E)$. Then \begin{equation}\label{E:kato} \Delta_M |w| \leq \operatorname{Re}\langle \nabla^*\nabla w,\operatorname{sign} w\rangle, \end{equation} where \[ \operatorname{sign} w(x) = \begin{cases} \frac{w(x)}{|w(x)|} & \textrm{if } w(x)\neq 0 ,\\ 0 &\text{otherwise.} \end{cases} \] \end{lemma} In what follows, we will use the following Lemma whose proof is given in Appendix B of~\cite{bms}. \begin{lemma}\label{L:positivity} Assume that $(M,g)$ is a manifold of bounded geometry with a smooth positive measure $d\mu$. Assume that \[ \big(\, b+\Delta_M\, \big)\, u \ = \ \nu \ \ge \ 0, \quad u\in L^2(M), \] where $b>0$, $\Delta_M=d^*d$ is the scalar Laplacian on $M$, and the inequality $\nu\ge 0$ means that $\nu$ is a positive distribution on $M$, i.e. $(\nu,\phi)\ge 0$ for any $0\leq\phi\in C_c^\infty(M)$. Then $u\ge 0$ (almost everywhere or, equivalently, as a distribution). \end{lemma} \begin{rem} \rm It is not known whether Lemma~\ref{L:positivity} holds if $M$ is an arbitrary complete Riemannian manifold. For more details about difficulties in the case of arbitrary complete Riemannian manifolds, see Appendix B of~\cite{bms}. \end{rem} \begin{lemma}\label{L-sum-of-forms} The quadratic form $h:=(h_0+h_1)+h_2$ is densely-defined, semi-bounded below and closed. \end{lemma} \begin{proof} Since $h_0$ and $h_1$ are non-negative and closed, it follows by Theorem VI.1.31 from~\cite{Kato66} that $h_0+h_1$ is non-negative and closed. Since $h_1$ is non-negative, it follows immediately from Assumption (A2) that $h_2$ is $(h_0+h_1)$-bounded with relative bound $b<1$. Since $h_0+h_1$ is a closed, non-negative form, by Theorem VI.1.33 from~\cite{Kato66}, it follows that $h=(h_0+h_1)+h_2$ is a closed semi-bounded below form. Since $C_{c}^{\infty}(E)\subset\text{D}(h_0)\bigcap \text{D}(h_1)\subset \text{D}(h_2)$, it follows that $h$ is densely defined. \end{proof} In what follows, $h(\cdot,\cdot)$ will denote the corresponding sesquilinear form obtained from $h$ via polarization identity. \subsection*{Self-adjoint operator $H$ associated to $h$}\label{SS:operator-H} Since $h$ is densely defined, closed and semi-bounded below form in $L^2(E)$, by Theorem VI.2.1 from~\cite{Kato66} there exists a semi-bounded below self-adjoint operator $H$ in $L^2(E)$ such that \begin{enumerate} \item[(i)] $\operatorname{Dom}(H)\subset\text{D}(h)$ and \[ h(u,v)=(Hu,v)\quad\text{for all }u\in\operatorname{Dom}(H),\text{ and }v\in \text{D}(h). \] \item[(ii)] $\operatorname{Dom}(H)$ is a core of $h$. \item[(iii)] If $u\in\text{D}(h)$, $w\in L^2(E)$ and $h(u,v)=(w,v)$ holds for every $v$ belonging to a core of $h$, then $u\in\operatorname{Dom}(H)$ and $Hu=w$. The semi-bounded below self-adjoint operator $H$ is uniquely determined by the condition (i). \end{enumerate} In what follows we will use the following well-known Lemma. \begin{lemma}\label{L:diagonal} Assume that $0\leq T\in L_{\rm loc}^1(\operatorname{End} E)$ is a linear self-adjoint bundle map. Assume also that $u\in Q(T)$, where $Q(T)=\{u\in L^2(E)\colon \langle T u,u\rangle\in L^1(M)\}$. Then $Tu\in L_{\rm loc}^1(E)$. \end{lemma} \begin{proof} By adding a constant we can assume that $T\geq 1$ (in the operator sense). Assume that $u\in Q(T)$. We choose (in a measurable way) an orthogonal basis in each fiber $E_x$ and diagonalize $1\leq T(x)\in\operatorname{End}(E_x)$ to get $$ T(x)=\mathop{\rm diag}(c_1(x),c_2(x),\dots,c_m(x)), $$ where $00$, it follows that $c_j|u_j|^2\in L^1(M)$, for all $j=1,2,\dots,m$. Now, for all $x\in M$ and $j=1,2,\cdots,m$ \begin{equation}\label{E:this} 2|c_ju_j|=2|c_j||u_j|\leq |c_j|+|c_j||u_j|^2, \end{equation} The right hand side of~\eqref{E:this} is clearly in $L_{\rm loc}^1(M)$. Therefore $c_ju_j\in L_{\rm loc}^1(M)$. But $(Tu)(x)$ has components $c_j(x)u_j(x)$ ($j=1,2,\dots,m$) with respect to chosen bases of $E_x$. Therefore $Tu\in L_{\rm loc}^1(E)$, and the Lemma is proven. \end{proof} The following corollary follows immediately from Lemma~\ref{L:diagonal}. \begin{corollary}\label{C:diagonal-1} If $u\in \text{D}(h_1)$, then $V_1u\in L_{\rm loc}^1(E)$. \end{corollary} \begin{corollary}\label{C:diagonal-2} If $u\in \text{D}(h)$, then $Vu\in L_{\rm loc}^1(E)$. \end{corollary} \begin{proof} Let $u\in \text{D}(h)=\text{D}(h_0)\bigcap \text{D}(h_1)$. By Assumption (A1) we have $V=V_1+V_2$, where $0\leq V_1\in L_{\rm loc}^1(\operatorname{End} E)$ and $0\geq V_2\in L_{\rm loc}^1(\operatorname{End} E)$. By Corollary~\ref{C:diagonal-1} it follows that $V_1u\in L_{\rm loc}^1(E)$ and since $\text{D}(h)\subset \text{D}(h_2)$, by Lemma~\ref{L:diagonal} we have $-V_2u\in L_{\rm loc}^1(E)$. Thus $Vu\in L_{\rm loc}^1(E)$, and the corollary is proven. \end{proof} \begin{lemma}\label{L:H-subset-S} The following operator relation holds: $H\subset S$. \end{lemma} \begin{proof} We will show that for all $u\in\operatorname{Dom}(H)$, we have $Hu=H_Vu$. Let $u\in\operatorname{Dom}(H)$. By property (i) of operator $H$ we have $u\in\text{D}(h)$, hence by Corollary~\ref{C:diagonal-2} we get $Vu\in L_{\rm loc}^1(E)$. Then, for any $v\in C_{c}^{\infty}(E)$, we have \begin{equation}\label{E:representation} (Hu,v)=h(u,v)=(\nabla u,\nabla v)+\int\langle Vu,v \rangle\,d\mu, \end{equation} where $(\cdot,\cdot)$ denotes the $L^2$-inner product. The first equality in~\eqref{E:representation} holds by property (i) of operator $H$, and the second equality holds by definition of $h$. Hence, using integration by parts in the first term on the right hand side of the second equality in~\eqref{E:representation} (see, for example, Lemma 8.8 from~\cite{bms}), we get \begin{equation}\label{E:distr} (u,\nabla^*\nabla v)=\int\langle Hu-Vu,v\rangle\,d\mu,\quad\text{for all}\quad v\in C_{c}^{\infty}(E). \end{equation} Since $Vu\in L_{\rm loc}^1(E)$ and $Hu\in L^2(E)$, it follows that $(Hu-Vu)\in L_{\rm loc}^1(E)$, and~\eqref{E:distr} implies $\nabla^*\nabla u=Hu-Vu$ (as distributional sections of $E$). Therefore, \[ \nabla^*\nabla u+Vu=Hu, \] and this shows that $Hu=H_Vu$ for all $u\in\operatorname{Dom}(H)$. Now by definition of $S$ it follows that $\operatorname{Dom}(H)\subset\operatorname{Dom}(S)$ and $Hu=Su$ for all $u\in\operatorname{Dom}(H)$. Therefore $H\subset S$, and the Lemma is proven. \end{proof} \begin{lemma}\label{L:h0+h1-form-core} $C_{c}^{\infty}(E)$ is a core of the quadratic form $h_0+h_1$. \end{lemma} \begin{proof} We need to show that $C_{c}^{\infty}(E)$ is dense in the Hilbert space $\text{D}(h_0+h_1)=\text{D}(h_0)\bigcap\text{D}(h_1)$ with the inner product \[ (u,v)_{h_0+h_1}:=h_0(u,v)+h_1(u,v)+(u,v), \] where $(\cdot,\cdot)$ is the inner product in $L^2(E)$. Let $u\in\text{D}(h_0+h_1)$ and $(u,v)_{h_0+h_1}=0$ for all $v\in C_{c}^{\infty}(E)$. We will show that $u=0$. We have \begin{equation}\label{E:preceding-kato} 0 = h_0(u,v)+h_1(u,v)+(u,v)\\ = (u,\nabla^*\nabla v)+ \int\langle V_1u,v \rangle\,d\mu + (u,v). \end{equation} Here we used integration by parts in the first term on the right hand side of the second equality. By Corollary \ref{C:diagonal-1} it follows that $V_1u\in L_{\rm loc}^1(E)$, and from~\eqref{E:preceding-kato} we get the following equality of distributional sections of $E$: \begin{equation}\label{E:distributional-equality} \nabla^*\nabla u=(-V_1-1)u. \end{equation} From~\eqref{E:distributional-equality} we have $\nabla^*\nabla u\in L_{\rm loc}^1(E)$. By Lemma~\ref{L:kato} and by~\eqref{E:distributional-equality}, we obtain \begin{equation}\label{E:kato-inequality} \Delta_M|u| \leq \operatorname{Re}\langle\nabla^*\nabla u,\operatorname{sign} u\rangle = \langle -(V_1+1)u,\operatorname{sign} u\rangle \leq -|u|. \end{equation} The last inequality in~\eqref{E:kato-inequality} follows since $V_1\geq 0$ (as an operator $E_x\to E_x$). Therefore, \begin{equation}\label{E:pzt} (\Delta_M+1)|u|\leq 0. \end{equation} By Lemma~\ref{L:positivity}, it follows that $|u|\leq 0$. So $u=0$, and the proof is complete. \end{proof} \begin{lemma}\label{L:h-form-core} $C_{c}^{\infty}(E)$ is a core of the quadratic form $h=(h_0+h_1)+h_2$. \end{lemma} Since the quadratic form $h_2$ is $(h_0+h_1)$-bounded, the lemma follows immediately from Lemma \ref{L:h0+h1-form-core}. \begin{proof}[Proof of Theorem~\ref{T:main}] We will show that $S=H$. By Lemma~\ref{L:H-subset-S} we have $H\subset S$, so it is enough to show that $\operatorname{Dom}(S)\subset\operatorname{Dom}(H)$. Let $u\in\operatorname{Dom}(S)$. By definition of $\operatorname{Dom}(S)$, we have $u\in\text{D}(h_0)\subset \text{D}(h_2)$ and $u\in\text{D}(h_1)$. Hence $u\in\text{D}(h)$. For all $v\in C_{c}^{\infty}(E)$ we have \[ h(u,v)=h_0(u,v)+h_1(u,v)+h_2(u,v)=(u,\nabla^*\nabla v)+ \int\langle Vu,v \rangle\,d\mu=(H_Vu,v). \] The last equality holds since $H_Vu=Su\in L^2(E)$. By Lemma~\ref{L:h-form-core} it follows that $C_{c}^{\infty}(E)$ is a form core of $h$. Now from property (iii) of operator $H$ we have $u\in\operatorname{Dom}(H)$ with $Hu=H_Vu$. This concludes the proof of the Theorem. \end{proof} \begin{proof}[Proof of Remark \ref{R:Assumption-B}] %\label{S:sufficient-condition} Let $p$ be as in Remark \ref{R:Assumption-B}. We may assume that $\|V_2\|_{L^{p}(\operatorname{End} E)}$ is arbitrarily small because there exists a sequence $V_2^{(k)}\in L^{\infty}(\operatorname{End} E)\bigcap L^{p}(\operatorname{End} E)$, $k\in {\mathbb{Z}}_+$, such that \[ \|V_2^{(k)}-V_2\|_{L^{p}(\operatorname{End} E)}\to 0,\quad\text{as }k\to\infty, \] and $V_2^{(k)}$, $k\in {\mathbb{Z}}_+$, contributes to $h_2$ only a bounded form. For the rest of this article, we will assume that $\|V_2\|_{L^{p}(\operatorname{End} E)}$ is arbitrarily small. By Cauchy-Schwartz inequality and H\"older's inequality we have \begin{equation}\label{E:sobolev-domination} \left|\int\langle V_2u,u \rangle\,d\mu\right|\leq \int|\langle V_2u,u \rangle|\,d\mu \leq \int |V_2||u|^2\,d\mu\leq \|V_2\|_{L^{p}(\operatorname{End} E)}\|u\|^2_{L^t(E)}, \end{equation} where $|V_2|$ denotes the norm of the operator $V_2(x)\colon E_x\to E_x$ and \begin{equation}\label{E:sob-exp} \frac{1}{p}+\frac{2}{t}=1. \end{equation} With our assumptions on $(M,g)$, $E$ and $\nabla$, the usual Sobolev embedding theorems for $W^{1,2}({\mathbb{R}}^n)$ also hold for $W^{1,2}(E)$ (see Sec.~A1.1 in~\cite{Shubin92}). For $n\geq 3$, we know by hypothesis that $p=n/2$, so from~\eqref{E:sob-exp} we get $1/t=1/2-1/n$. By the Sobolev embedding theorem (see, for example, the first part of Theorem 2.10 in~\cite{Aubin}) we have \[ \|u\|_{L^t(E)}\leq C(\|\nabla u\|_{L^2(T^*M\otimes E) }+\|u\|_{L^2(E)}),\quad\text{for all }u\in W^{1,2}(E), \] where $C>0$ is a positive constant. For $n=2$, we know by hypothesis that $p>1$, so from~\eqref{E:sob-exp} we get $20$ is a positive constant. For $n=1$, we know by hypothesis that $p=1$, so from~\eqref{E:sob-exp} we get $t=\infty$. In this case, it is well-known (see e.g. the second part of Theorem 2.10 in~\cite{Aubin}) that \[ \|u\|_{L^{\infty}(E)}\leq C(\|\nabla u\|_{L^2(T^*M\otimes E )}+\|u\|_{L^2(E)}),\quad\text{for all }u\in W^{1,2}(E), \] where $C>0$ is a positive constant. Combining each of the last three inequalities with~\eqref{E:sobolev-domination}, we get~\eqref{E:h_2-h_0-domination} (with constant $b<1$ because $\|V_2\|_{L^{p}(\operatorname{End} E)}$ is arbitrarily small). \end{proof} \begin{thebibliography}{99} \bibitem{Aubin} T.~Aubin, \emph{Some nonlinear problems in Riemannian geometry}, Springer-Verlag, Berlin, 1998. \bibitem{bms} M.~Braverman, O.~Milatovic, M.~Shubin, \emph{Essential self-adjointness of Schr\"odinger type operators on manifolds}, Russian Math. Surveys, \textbf{57}(4) (2002), 641--692. \bibitem{Brezis-Kato79} H.~Br\'ezis, T.~Kato, \emph{Remarks on the Schr\"odinger operator with singular complex potentials}, J. Math. Pures Appl., \textbf{58}(9) (1979), 137--151. \bibitem{hsu2} H.~Hess, R.~Schrader, A.~Uhlenbrock, \emph{Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifold}, J. Differential Geom., \textbf{15} (1980), 27--37. \bibitem{Kato72} T.~Kato, \emph{Schr\"odinger operators with singular potentials,} Israel J. Math., \textbf{13} (1972), 135--148. \bibitem{Kato74} T.~Kato, \emph{A second look at the essential selfadjointness of the Schr\"odinger operators}, Physical reality and mathematical description, Reidel, Dordrecht, 1974, 193--201. \bibitem{Kato78} T.~Kato, \emph{On some Schr\"odinger operators with a singular complex potential}, Ann. Sc. Norm. Sup. Pisa, Ser. IV, Vol. 5 (1978), 105--114. \bibitem{Kato66} T.~Kato, \emph{Perturbation theory for linear operators}, Springer-Verlag, New York, 1980. \bibitem{rs} M.~Reed, B.~Simon, \emph{Methods of Modern Mathematical Physics I, II: Functional analysis. Fourier analysis, self-adjointness}, Academic Press, New York e.a., 1975. \bibitem{Shubin92} M.~A.~Shubin, \emph{Spectral theory of elliptic operators on noncompact manifolds,} Ast\'erisque No. 207 (1992), 35--108. \bibitem{tm} M.~Taylor, \emph{Partial Differential Equations II: Qualitative Studies of Linear Equations}, Springer-Verlag, New York e.a., 1996. \end{thebibliography} \end{document}