Content-Type: multipart/mixed; boundary="-------------0403261108439" This is a multi-part message in MIME format. ---------------0403261108439 Content-Type: text/plain; name="04-93.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-93.keywords" Rellich inequality, Aharonov-Bohm magnetic field ---------------0403261108439 Content-Type: application/x-tex; name="Rellichinequality.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Rellichinequality.tex" \documentclass[reqno,a4paper,12pt]{amsart} \usepackage{amssymb} %\usepackage{a4wide} \numberwithin{equation}{section} \date{\today} \def\tr{\mathop{\mathrm{tr}}\nolimits} \def\supp{\mathop{\mathrm{supp}}\nolimits} % Tr"ager \newcommand{\lsim}{{\underset{\sim}{<}}} \newcommand{\gsim}{{\underset{\sim}{>}}} \newcommand{\q}{\qquad} \newcommand{\e}{\mathbf{e}} \newcommand\x{\mathbf{x}} \newcommand{\er}{\mathbf{e}_r} \newcommand{\etheta}{\mathbf{e}_\theta} \newcommand{\tpsi}{{\tilde{\Psi}}} \newcommand{\rz}{\mathbb{R}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\bbS}{\mathbb{S}} \newcommand{\gd}{\mathfrak{d}} \newcommand{\gq}{\mathfrak{q}} \newcommand{\gp}{\mathfrak{p}} \newcommand{\Con}{C_0^\infty} \newcommand{\Cons}{C_{\mathcal {S},0}^\infty} \newcommand{\grad}{\nabla} \newcommand{\cs}{\mathcal {S}} \newcommand{\cg}{\mathcal {G}} \newcommand{\sa}{{{\pmb{a}}}} \newcommand{\ab}{{{\mathbf{A}}}} \newcommand{\AAA}{{\text{\aa}}} \newcommand{\AAAA}{{\text{\bf{\aa}}}} \newcommand{\B}{{\mathbf{B}}} \newcommand{\al}{{\pmb{\alpha}}} \newcommand{\dv}{{\mathbb{D}_{V}}} \newcommand{\pw}{{\mathbb{P}_{W}}} \newcommand{\dvo}{{\mathbb{D}_{V_0}}} \newcommand{\rvo}{{\mathbb{R}_{V_0}}} \newcommand{\tva}{{T(V,\pmb{a})}} \newcommand{\bsig}{{\pmb{\sigma}}} \newcommand{\Da}{{\mathcal D_\sa}} \newcommand{\K}{{\mathfrak{K}}} \newcommand{\sao}{{\beta^0_{\mathbf{a}}}} \newcommand{\sac}{{\beta^C_{\mathbf{a}}}} \def\dbar{{\mathchar'26\mkern-12mud}} \newtheorem{definition}{Definition} \newtheorem{Lemma}{Lemma} \newtheorem{Theorem}{Theorem} \newtheorem{Corollary}{Corollary} \newtheorem{Proposition}{Proposition} \newtheorem{Remark}{Remark} \begin{document} \title[The Rellich inequality]{On the Rellich inequality with magnetic potentials} \author[W.~D.~Evans]{W.~D. Evans} \address{School of Mathematics\\ Cardiff University\\ 23 Senghennydd Road\\ Cardiff CF2 4YH\\ UK} \email{EvansWD@cardiff.ac.uk} \author[R.~T.~Lewis]{R.~T. Lewis} \address{Department of Mathematics\\ University of Alabama at Birmingham\\ Birmingham, AL 35294-1170\\ USA} \email{lewis@math.uab.edu} \thanks{The first author gratefully acknowledges the hospitality and support of the Mathematics Department at UAB where much of this work was done.} \maketitle \section{Introduction} Rellich's inequality is that \begin{equation}\label{1.1} \frac{n^2(n-4)^2}{16} \int_{ \mathbb{R}^n} \frac{|f(\x)|^2}{|\x|^4} d\x \le \int_{ \mathbb{R}^n} |\Delta f(\x)|^2 d\x \end{equation} for all $f\in C_0^{\infty}(\mathbb{R}^n \setminus \{0\})$ and $n\neq 2$, where the constant $n^2(n-4)^2/16 $ is sharp. It first appeared in print in \cite{RL1} but it had been given in 1953 in lectures at New York University which were published posthumously in \cite{RL2}. In \cite{DB} an extension of Rellich's inequality to higher-order derivatives was derived through the application of a technique introduced in \cite{LE}. This technique, and Bennett's development, were the basis of the method used by Davies and Hinz in \cite{DH} in their comprehensive study of Rellich-type inequalities and their higher-order counterparts in $L^p(\Omega)$, for any $p\in [1,\infty)$ and $\Omega$ a complete Riemannian manifold with smooth boundary. The papers \cite{DB} and \cite{DH} were influenced by the work of Schmincke in \cite{Sch}. In particular, Davies and Hinz obtain sharp constants $C$ for inequalities of the form \begin{equation}\label{1.2} \int_{ \mathbb{R}^n} \frac{|f(\x)|^p}{|\x|^{\beta}} d\x \le C \int_{ \mathbb{R}^n} \frac{|\Delta^m f(\x)|^p}{|\x|^{\alpha}} d\x \end{equation} for suitable values of $\alpha, \beta, m, n, p$ and $f\in C_0^{\infty}( \mathbb{R}^n\setminus \{0\})$. In Rellich's original inequality there are two cases of special interest, and these have their counterparts in the work of Davies and Hinz. Firstly, when $n=2$ the inequality still holds but only for functions $f\in \Con(\R^2\setminus \{0\})$ which also satisfy \[ \int_0^{2\pi}f(r,\theta)\cos \theta d\theta = \int_0^{2\pi}f(r,\theta)\sin \theta d\theta=0. \] Secondly, when $n=4$ there is no non-trivial inequality. Our primary concern will be with these special cases, but our general results have wider implications. Our approach has been influenced by the Laptev-Weidl inequality in \cite{LW} in which, {\em{inter alia}}, the Hardy-type inequality \begin{equation}\label{1.3} \int_{ \mathbb{R}^2} \frac{|f(\x)|^2}{|\x|^2}d\x \le C \int_{ \mathbb{R}^2} |\nabla_{\ab}f(\x)|^2 d\x \end{equation} is established for $f\in C_0^{\infty}( \mathbb{R}^2\setminus \{0\})$, where $\nabla_{\ab}$ is the {\em{magnetic gradient}} \[ \nabla_{\ab} = \nabla - i \ab, \] $\ab$ is of the form (in polar co-ordinates) \begin{equation}\label{1.4} \ab(r,\theta) = \frac{\Psi(\theta)}{r} (-\sin \theta, \cos \theta),\ \ \ \Psi \in L^{\infty}(0,2\pi) \end{equation} and the magnetic flux $\tpsi := \frac{1}{2\pi} \int_0^{2\pi} \Psi(\theta) d\theta \notin \mathbb{Z}$. The magnetic field $\rm{curl} \ab = 0$ in $ \mathbb{R}^2\setminus \{0\}$ and is of Aharonov-Bohm type. The optimal constant in (1.3) is $ C= \rm{dist} (\tpsi, \mathbb{Z})^{-2}$. Recall that when there is no magnetic potential $\ab$ (or, equivalently, the magnetic flux $\tpsi \in \mathbb{Z}$ by gauge invariance) there is no valid Hardy inequality. A method to establish an $L^p$-version of (1.3) is developed in \cite{BE} and this has also influenced the present work. The $L^2$-version of our result is of the form \begin{equation}\label{1.5} C \int_{ \mathbb{R}^n}\frac{|f(\x)|^2}{|\x|^{\alpha +4}}d\x \le \int_{ \mathbb{R}^n}\frac{|\Delta_{\ab}f(\x)|^2}{|\x|^{\alpha }}d\x , \end{equation} where $\Delta_{\ab} = \nabla_{\ab}\cdot\nabla_{\ab}$ and $\ab$ is a vector field in Poincar\'{e} (or transverse) gauge (i.e. $\ab(\x) \cdot \x =0)$ which is an exact 1-form in the simply-connected domain $\mathbb{R}^n \setminus \mathcal{L}_n$, where in appropriate polar co-ordinates $\x=(r,\theta_1, \cdots,\theta_{n-1})$, $ \mathcal{L}_n$ is the co-ordinate axis \[ \mathcal{L}_n = \{\x: r \sin \theta_1 \sin \theta_2 \cdots \sin \theta_{n-2} =0 \}. \] As in the Laptev-Weidl inequality, our constant depends on the distance of the magnetic flux $ \tpsi$ to the set of integers $ \mathbb{Z}$. The $L^p$-version of our result takes a different form which emphasises the dependence of the inequalities on the behaviour of the functions $f$ in a transverse direction. For instance, when $n=2$, we prove that \begin{equation}\label{1.6} \|f/|\cdot|^2\|_{L^p( \mathbb{R}^2)} \le B_p \|\frac{1}{r^2} \left(i\frac{\partial}{\partial \theta} + \Psi(\theta)\right) f \|_{L^p( \mathbb{R}^2)}, \end{equation} where $ B_p \le C_p/d^2 (\cos (\pi d) \big)^{1/p'}, d:= \rm{dist}(\tpsi, \mathbb{Z}) \neq 0,1/2$ and there are similar inequalities for $n\ge 3$. Such an inequality clearly does not hold when $\Psi = 0$ since radial functions are excluded. When $p=2$ there is a constant $C$ depending on $d$ such that \begin{equation}\label{1.7} \|\frac{1}{r^2} \left(i\frac{\partial}{\partial \theta} + \Psi(\theta)\right) f \|_{L^2( \mathbb{R}^2)} \le C \| \Delta^2_{\ab} f \|_{L^2( \mathbb{R}^2)}. \end{equation} We also investigate an analogue of (1.7) for $n=3.$ \section{The main result in $L^2(\R^n)$.} Throughout this paper, polar coordinates in $\R^n$ will be denoted by $(r,\omega)$ with $r:=|\x|$, $\omega = \x/|\x|$ for $\x\in\R^n$. The unit ball in $\R^n$ is denoted by $\bbS^{n-1}$ and $\R_+:=(0,\infty)$. We shall also denote the $L^2( \mathbb{R}^n)$ norm by $\|\cdot\|$. \begin{Theorem}\label{Thm2.1} Let $\Lambda_{\omega}$ be a non-negative self-adjoint operator in \newline$L^2(\bbS^{n-1};d\omega)$ whose spectrum is discrete and consists of eigenvalues $\lambda_m, m \in \mathcal{I}$, where $\mathcal{I}$ is a countable index set. Let \begin{equation}\label{Eq2.1} L_r:= -\frac{\partial^2}{\partial r^2}-\frac{n-1}{r}\frac{\partial}{\partial r} \end{equation}\ and define the operator $D:= L_r +\frac{1}{r^2}\Lambda_{\omega}$ on the domain \begin{align}\label{2.2} \mathcal{D}_0 := \{ f: &f \in C_0^{\infty}(\mathbb{R}^n\setminus \{0\}), f(r,\cdot) \in \mathcal{D}(\Lambda_{\omega}) \nonumber \\ & \text{for}\ 00$. Then, there exists a positive constant $C(n)$ such that in the notation of Theorem~\ref{Thm2.1} \begin{equation}\label{Eq2.16} \||\cdot|^{-2}\Lambda_{\omega} f\|\le C(n)\|D f\|. \end{equation} \end{Corollary} \begin{proof} From (2.13) and (2.15) with $\alpha =0$, we have \[ \int_{\R^n}|L_rf|^2d\x \ge \big[\frac{n(n-4)}{4}\big]^2\int_{\R^n}\frac{|f(\x)|^2}{|\x|^4}d\x \] and \begin{align*} 2\Re{e}&\big[\int_{\R^n} L_rf\overline{\Lambda_{\omega} f}\frac{d\x}{|\x|^{2}}] \ge \frac12 n(n-4)\int_{\R^n}\big(|\x|^{-2}\Lambda_{\omega}f\big)\big(|\x|^{-2} \overline{f}\big)d\x \\ & \ge -\frac14 n[n-4]_-\{\epsilon\||\cdot|^{-2}\Lambda_{\omega} f\|^2 +\frac{1}{\epsilon}\||\cdot|^{-2}f\|^2\}, \end{align*} where $ \varepsilon >0$ is arbitrary and $[a]_-:= \max \{0,-a\}$. Hence, from (\ref{Eq2.11}) with $\alpha =0$, \begin{align*} \|Df\|^2 \ge & \frac14 [\frac{n(n-4)}{4}]^2\||\cdot|^{-2}f\|^2 \\ &-\frac14 n[n-4]_-\{\epsilon\||\cdot|^{-2}\Lambda_{\omega} f\|^2 +\frac{1}{\epsilon}\||\cdot|^{-2}f\|^2\} +\||\cdot|^{-2}\Lambda_{\omega} f\|^2\\ =& \{\frac14[\frac{n(n-4)}{4}]^2 -\frac{1}{4\epsilon}n[n-4]_-\}\||\cdot|^{-2}f\|^2\\ &+\{1-\frac{\epsilon}{4}n[n-4]_-\}\||\cdot|^{-2}\Lambda_{\omega} f\|^2. \end{align*} When $n\ge 4$, this gives $$\begin{array}{rl} \|Df\|^2 \ge & \frac14[\frac{n(n-4)}{4}]^2\||\cdot|^{-2}f\|^2 +\||\cdot|^{-2}\Lambda_{\omega} f\|^2 \end{array} $$ whence (2.11) with $C(n)= 1$. For $n\le 3$, we set $\epsilon =\frac{4}{n[n-4]_-}\delta$, $\delta\in (0,1)$, and use Theorem~\ref{Thm2.1} to get $$\begin{array}{rl} (1-\delta)\||\cdot|^{-2}\Lambda_{\omega} f\|^2\le& \|Df\|^2 +[\frac{n(n-4)}{4}]^2(\frac{1}{\delta}-\frac14)\||\cdot|^{-2}f\|^2\\ \le& \{1+[\frac{n(n-4)}{4}]^2C(n,0)^{-1}(\frac{1}{\delta}-\frac14)\} \|Df\|^2 \end{array} $$ which again yields (2.17). \end{proof} \begin{Remark}\label{Remark2.3} There is a valid inequality (2.3) (i.e. $C(n,\alpha) >0 $) for all $n> \alpha+4$. \end{Remark} \begin{Remark}\label{Remark2.4} It follows from the proof of Theorem~\ref{Thm2.1} that for $f\in \mathcal{D}_0$, \[ Df(r,\omega) = \sum_{m\in \mathcal{I}} \big(L_r+r^{-2}\lambda_m\big) F_m(r) u_m(\omega) \] in the $ L^2( \mathbb{S}^{n-1})$ sense and $F_m \in C_0^{\infty}( \mathbb{R}_+)$. Hence, on identifying $L^2( \mathbb{R}^n)$ with $ L^2( \mathbb{R}_+; r^{n-1}dr) \bigotimes L^2( \mathbb{S}^{n-1};d\omega)$, we have that \[ \mathcal{D}_0 = \bigoplus_{m\in \mathcal{I}} \big\{C_0^{\infty}( \mathbb{R}_+)\bigotimes\{u_m\}\big\} \] and \[ D = \bigoplus_{m\in \mathcal{I}}\big \{ \big(L_r+\lambda_m r^{-2} \big) \bigotimes I_m, \] where $\bigoplus$ denotes orthogonal sum and $I_m$ is the identity on $\{u_m\}$. The map $ J: F \mapsto r^{(n-1)/2} F :L^2( \mathbb{R}_+; r^{n-1}dr)\rightarrow L^2( \mathbb{R}_+)$is an isometry and if $D_m := L_r+\lambda_m r^{-2}$ on $C_0^{\infty}( \mathbb{R}_+)$, then $D_m = J^{-1}T_mJ$, where $T_m$ also has domain $C_0^{\infty}( \mathbb{R}_+)$ and \[ T_m g = -\frac{d^2g}{dr^2} + r^{-2}\big[\lambda_m+\frac{(n-1)(n-3)}{4}\big]g. \] By Hardy's inequality, for $g\in C_0^{\infty}( \mathbb{R}_+)$, \[ -\int_0^{\infty} \overline{g}g'' dr = \int_0^{\infty} |g'|^2 dr \ge \frac{1}{4}\int_0^{\infty} r^{-2}|g|^2 dr \] and so \[ \int_0^{\infty} \overline{g}T_m g dr \ge \big[ \lambda_m + \big(\frac{n-2}{2}\big)^2 \big]\frac{1}{4}\int_0^{\infty} r^{-2}|g|^2 dr. \] Thus $T_m$ and $D_m$ are non-negative. \end{Remark} \section{Applications of Theorem~\ref{Thm2.1}} We shall apply Theorem~\ref{Thm2.1} to the operator $D=-\Delta_\ab$, the magnetic Laplacian associated with a magnetic potential $\ab$ of Aharonov-Bohm type when $n=2$ and which has analogous characteristics for other values of $n$. Although the analysis continues to be effective for unrestricted values of $n$, we shall concentrate on the cases $n=2,4,$ which are the singular cases when $D=-\Delta$ and (2.3) is the standard Rellich inequality. To handle the case $n=4$, we shall also need to discuss the case $n=3$. \subsection{The case $n=2$} A magnetic field $B = \rm{curl}\ab $ of Aharonov-Bohm type vanishes in $ \mathbb{R}^2\setminus \{0\}$ and in the simply connected domain $ \mathbb{R}^2 \setminus [0,\infty) $ the 1-form associated with $\ab$ is exact. Hence there exists a potential $P$ such that $\ab=\nabla P$. If we assume, as we can without loss of generality, that $\ab\cdot\x =0$ (Poincar\'{e} gauge) (see \cite{Th}, Section 8.4.2), then in polar coordinates $\mathbf{x}=r(\cos\theta,\sin\theta)$, $$ \ab(r,\theta):=\frac{1}{r}\Psi(\theta)\etheta,\ \ \ \Psi \in L^\infty(\bbS^1),\q \Psi(0)=\Psi(2\pi), $$ where $\etheta:=(-\sin\theta,\cos\theta)$. We have, with $\er:=(\cos\theta,\sin\theta)$, $$ \grad = \er \frac{\partial}{\partial_r} +\etheta \frac{1}{r}\frac{\partial}{\partial\theta} $$ and define \begin{equation}\label{Eq3.1} \grad_\ab := \grad -i\ab= \er \frac{\partial}{\partial r} +\etheta\frac{1}{r}\left(\frac{\partial}{\partial \theta} - i\Psi(\theta)\right) \end{equation} and \begin{equation}\label{Eq3.2} -\Delta_\ab = -\frac{\partial^2}{\partial r^2} -\frac{1}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\left(i\frac{\partial}{\partial\theta}+\Psi(\theta)\right)^2. \end{equation} Thus, in the notation of Theorem~\ref{Thm2.1}, $\Lambda_{\omega} = \Lambda_{\theta}$ is the non-negative self-adjoint operator in $L^2(0,2\pi)$ defined by $\Lambda_{\theta}=K_{\theta}^2$, where \begin{equation}\label{3.3} K_{\theta} u(\theta) = i\frac{du}{d\theta} + \Psi(\theta)u(\theta) \end{equation} with domain \[ \big \{u: u\in C^1[0,2\pi], u(0)=u(2\pi) \big \}. \] It is readily seen that $K_{\theta}$ has eigenvalues $m+ \widetilde{\Psi}, m\in \mathbb{Z}$, where $ \widetilde{\Psi}= (1/2\pi)\int_0^{2\pi} \Psi(\theta)d \theta$ is the magnetic flux, and the corresponding normalised eigenfunctions are \begin{equation}\label{3.4} u_m(\theta)=\frac{1}{\sqrt{2\pi}}\exp [-i(\theta[m+\tpsi]-\int_0^\theta\Psi(\eta)d\eta)] \end{equation} which constitute an orthonormal basis of $L^2( \mathbb{S}^1)$. For $m\in \mathbb{Z}, U: f\mapsto e^{-im\theta}f$ is unitary on $L^2( \mathbb{R}^2)$ and \[ U^{-1} \nabla_{\ab}U = \nabla_{ \widetilde{\ab}}, \] where $\widetilde{\ab}= \frac{(\Psi+m)}{r}\etheta.$ Since $\rm{curl} \ab = \rm{curl} \widetilde{\ab},$ the operators $\nabla_{\ab}, \nabla_{ \widetilde{\ab}}$, and also $ \Delta_{\ab}, \Delta_{\widetilde{\ab}}$ are gauge equivalent. Furthermore the associated magnetic fluxes differ by $m.$ Therefore we may assume that $ \tpsi \in [0,1).$ Since $ \mathcal{D}_0 = C_0^{\infty}( \mathbb{R}^2 \setminus \{0\})$ we have from Theorem~\ref{Thm2.1} \begin{Corollary}\label{Cor3.1} For all $f\in C_0^{\infty}( \mathbb{R}^2 \setminus \{0\})$ and $\alpha \ge -1$, \begin{equation}\label{3.5} \int_{ \mathbb{R}^2} |\Delta_{\ab}f(x)|^2\frac{dx}{|x|^\alpha} \ge C(2,\alpha)\int_{\R^2}|f(x)|^2\frac{dx}{|x|^{\alpha +4}}, \end{equation} where \begin{equation}\label{Eq3.6} C(2,\alpha) = \inf_{m\in\mathbb{Z}}\{(m+\tpsi)^2-(\frac{\alpha +2}{2})^2\}^2. \end{equation} \bigskip If $\tpsi \notin \mathbb{Z}$ ($\tpsi \in[0,1)$ without loss of generality), we have \begin{equation}\label{Eq3.7}\begin{array}{rl} C(2,0)=&\min\{(\tpsi^2-1)^2,\tpsi^2(\tpsi-2)^2\}\\ =& \left\{\begin{array}{lcr} (\tpsi^2-1)^2 &\text{if}&\tpsi\in[\frac12,1),\\ \tpsi^2(\tpsi-2)^2&\text{if}&\tpsi\in [0,\frac12).\end{array}\right. \end{array} \end{equation} \end{Corollary} \begin{Remark}\label{Remark 3.2} If $\tpsi\in\mathbb{Z}$, then $C(2,0)=0$. However, if $F_1=F_{-1} =0$ (see (2.5)), the infimum in (3.6) is over $m\in \mathbb{Z}\setminus \{-1,1\}$ and this gives $C(2,0)=1.$ Hence, Rellich's result in ~(\cite{RL2}, p.100) for $n=2$ is recovered. \end{Remark} \subsection{The case $n=3$} In spherical polar coordinates, we define the orthonormal vectors $$\begin{array}{rl} \mathbf{e}_0:=&\frac{\mathbf{x}}{|\mathbf{x}|}=(\cos\theta_1,\sin\theta_1 \cos\theta_2,\sin\theta_1\sin\theta_2),\\ \mathbf{e}_1:=& (-\sin\theta_1,\cos\theta_1\cos\theta_2,\cos\theta_1\sin\theta_2),\\ \mathbf{e}_2:=& (0,-\sin\theta_2,\cos\theta_2), \end{array} $$ where $r=|\mathbf{x}|\in(0,\infty)$, $\theta_1\in (0,\pi)$, and $\theta_2\in (0,2\pi)$. We now take $$ \ab:=\frac{1}{r\sin\theta_1}\Psi(\theta_2)\mathbf{e}_2,\q \Psi\in L^\infty(0,2\pi),\q \Psi(0)=\Psi(2\pi), $$ on $\R^3\setminus \mathcal{L}_3$, where $ \mathcal{L}_3=\{(r,\theta_1,\theta_2):r\sin\theta_1=0\}$. It is in Poincar\'{e} gauge and $\rm{curl} \ab =0$ in $\R^3\setminus \mathcal{L}_3$. Then \begin{equation}\label{Eq3.8} \grad_\ab=\grad - i\ab=\e_0\frac{\partial}{\partial r} +\e_1\frac{1}{r}\frac{\partial}{\partial \theta_1} +\e_2 \frac{1}{r\sin\theta_1}[\frac{\partial}{\partial\theta_2}-i\Psi(\theta_2)] \end{equation} and \begin{equation}\label{Eq3.9} -\Delta_\ab = -\frac{\partial^2}{\partial r^2}-\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\Lambda(\theta_1,\theta_2) \end{equation} where \begin{equation}\label{Eq3.10} \Lambda(\theta_1,\theta_2)=-\frac{\partial^2}{\partial \theta_1^2} -\cot\theta_1\frac{\partial}{\partial\theta_1} +\frac{1}{\sin^2\theta_1}K_{\theta_2}^2 \end{equation} and \begin{equation}\label{Eq3.11} K_{\theta_2}=i\frac{\partial}{\partial \theta_2}+\Psi(\theta_2). \end{equation} As above for (3.3), the self-adjoint operator $K_{\theta_2}$ in $L^2(\bbS^1)$ has eigenvalues $k+\tpsi$, $k\in\mathbb{Z}$, with $\tpsi:=\frac{1}{2\pi}\int_0^{2\pi}\Psi(\theta_2)d\theta_2$, and eigenvectors $u_k(\theta_2)$ which form an orthonormal basis of $L^2(\bbS^1)$. Identifying $L^2( \mathbb{S}^2)$ with \newline $ \bigoplus_{k\in \mathbb{Z}}\big( L^2(0,\pi; \sin\theta_1 d\theta_1\big) \bigotimes \{u_k\}\big)$ we shall take the operator $\Lambda_{\omega}$ of Theorem~\ref{Thm2.1} to be \begin{equation}\label{3.12} \Lambda_{\omega} = \bigoplus_{k\in \mathbb{Z}}\left(\Lambda_k(\theta_1) \bigotimes I_k \right ), \end{equation} where $I_k$ denotes the identity on $\{u_k\}$ and $\Lambda_k(\theta_1)$ is a self-adjoint realisation of the operator $ \Lambda^0_k(\theta_1)$ defined on $C_0^{\infty}(0,\pi)$ by \begin{equation}\label{3.13} \Lambda^0_k(\theta_1)u = \big(-\frac{d^2}{d\theta_1^2}- \cot \theta_1 \frac{d}{d\theta_1}+ \frac{(k+ \tpsi)^2}{\sin^2\theta_1}\big)u. \end{equation} Before we are able to apply Theorem~\ref{Thm2.1} we must first make a suitable choice of the operators $\Lambda_k(\theta_1)$ for all $k\in \mathbb{Z}$ and determine their eigenvalues. The information required is contained in the next two lemmas. \begin{Lemma}\label{Lemma 3.3} For $\mu\in[0,\infty)$, the associated Legendre equation \begin{equation}\label{3.14} \frac{d^2u}{d\theta ^2}+ \cot \theta \frac{du}{d\theta}+ \left( \lambda -\frac{\mu^2}{\sin^2\theta}\right)u =0 ,\ \ \ \lambda \in \mathbb{C}, \end{equation} is in the limit-circle case at $0$ and $\pi$ if $\mu \in [0,1)$ and in the limit-point case at $0$ and $\pi$ otherwise. Let \begin{equation}\label{3.14B} \mathcal{D}_{\mu}:=\big\{u: u, \sin \theta \frac{du}{d\theta} \in AC_{loc}(0,\pi), u, L_{\mu}u \in L^2(0,\pi; \sin \theta d\theta) \big \}, \end{equation} where \[ L_{\mu} := -\frac{d^2}{d\theta^2}- \cot \theta \frac{d}{d\theta}+ \frac{\mu^2}{\sin^2\theta} \] and denote the restriction of $L_{\mu}$ to $C_0^{\infty}(0,\pi)$ by $ \Lambda_{\mu}^0$. Then $ \Lambda_{\mu}^0$ is non-negative. It is essentially-self-adjoint if and only if $\mu \in [1,\infty)$ and for $\mu\in [0,1)$ its Friedrichs extension $\Lambda_{\mu}$ is the realisation of $L_{\mu}$ on the following domains: \begin{itemize} \item if $\mu \in(0,1)$, \begin{equation}\label{3.15} \mathcal{D}(\Lambda_{\mu})= \big\{ u: u\in \mathcal{D}_{\mu}, \sin^{\mu}\theta\ u(\theta) \rightarrow 0 \ \textrm{as}\ \theta \rightarrow0,\pi\big \}; \end{equation} \item if $\mu = 0$, \begin{equation}\label{3.16} \mathcal{D}(\Lambda_{\mu})= \big\{ u: u\in \mathcal{D}_{\mu}, u(\theta)/ \ln (\cot \theta) \rightarrow 0 \ \ \textrm{as}\ \theta \rightarrow0,\pi\big \}; \end{equation} \item if $\mu \in [1,\infty), \Lambda_{\mu}$ is the closure of $\Lambda_{\mu}^0$ and \begin{equation}\label{3.17} \mathcal{D}(\Lambda_{\mu})= \mathcal{D}_{\mu}. \end{equation} \end{itemize} For $\mu \in [0,1), \Lambda_{\mu} \ge \mu(\mu+1)$. \end{Lemma} \begin{proof} On substituting $x=\cos \theta$, (3.14) becomes \begin{equation}\label{3.18} \tau_{\mu}u:=-\frac{d}{dx}\big\{(1-x^2)\frac{du}{dx}\big\} + \frac{\mu^2}{1-x^2}u = \lambda u, \ \ x\in (-1,1). \end{equation} Denote the restriction of $\tau_{\mu}$ to $C_0^{\infty}(-1,1)$ by $T_{\mu}^0$. Clearly $T_{\mu}^0 \ge 0$. Define the functions \begin{eqnarray}\label{3.19} a(x) &=& (1-x^2)^{\mu/2}, \nonumber \\ b(x) &=& a(x)h(x),\ \ \ h(x)=|\int_0^x(1-t^2)^{-1-\mu}dt|. \end{eqnarray} Note that \begin{equation}\label{3.20} h(x)= \left\{ \begin{array}{ll} =& \frac{1}{2}\left|\ln \left(\frac{1+x}{1-x} \right)\right| \ \ \textrm{for} \ \ \mu =0, \\ \approx & (1-x^2)^{-\mu} \ \ \ \textrm{for}\ \ \mu >0, \end{array} \right. \end{equation} where $f \approx g$ means that $f(x)/g(x)$ is bounded above and below by positive constants. We also have \begin{align*} a(x)/b(x) \rightarrow 0 & \ \textrm{as}\ x \rightarrow \pm 1;\\ \int_0^1 \frac{dt}{(1-t^2)a^2(t)}&= \int_{-1}^0 \frac{dt}{(1-t^2)a^2(t)}= \infty; \\ \int_{\delta}^1 \frac{dt}{(1-t^2)b^2(t)}&<\infty,\ \ |\int_{-1}^{-\delta} \frac{dt}{(1-t^2)b^2(t)}|< \infty, \end{align*} for any $\delta >0$. Furthermore \begin{eqnarray} \tau_{\mu} a &=& \mu(\mu+1) a,\nonumber \\ \tau_{\mu} b &=& \mu(\mu+1) b. \end{eqnarray} Hence $a,b$ are respectively {\em{principal}} and {\em{non-principal}} solutions of $\tau_{\mu}u=\mu(\mu+1) u$. For $\mu \in [0,1), a,b \in L^2(-1,1)$ and so (3.19) is in the limit-circle case at $-1$ and $+1$. If $\mu \ge 1, b$ is neither in $L^2(-1,0)$ nor $L^2(0,1)$ and hence (3.19) is limit-point at both the end points $\pm 1$. Thus $T_{\mu}^0$ is essentially-self-adjoint for $\mu \ge 1$. To characterise the Friedrichs extension $T_{\mu}$ of $T_{\mu}^0$ when $\mu \in [0,1)$ we apply Rosenberger's Theorem~3 in \cite{R} to get \begin{equation}\label{3.22} \mathcal{D}(T_{\mu}) = \big\{ u: u\in \mathcal{D_{\mu}}, \lim_{x\rightarrow-1} u(x)/b(x)= \lim_{x\rightarrow 1} u(x)/b(x) =0 \big \}, \end{equation} where \[ \mathcal{D_{\mu}}= \big \{u: u, (1-x^2)u' \in AC_{loc}(-1,1), u,\tau_{\mu}u \in L^2(-1,1)\big \}. \] For $\mu \ge 1 $ we have seen that $T_{\mu}^0$ is essentially-self-adjoint and so its unique self-adjoint extension has the domain of its adjoint, namely $ \mathcal{D}_{\mu}$. To prove that $T_{\mu} \ge \mu(\mu+1)$ when $\mu \in [0,1)$, we appeal to results of Kalf in \cite{K}. In Remark 3 he gives the following characterisation of $T_{\mu}$: \begin{align}\label{3.23} \mathcal{D}(T_{\mu}) = \big\{u: & u\in \mathcal{D}_{\mu}, \int^x (1-t^2)b^2(t) \big|\big(\frac{u}{b}\big)'\big|^2dt < \infty \ \ \textrm{at}\ \ \pm 1 \nonumber \\ &\textrm{and}\ \ \lim_{x\rightarrow \pm 1} |u(x)/b(x)| = 0 \big\}. \end{align} On using Jacobi's factorisation identity (5) in \cite{K}, it follows from (3.22) (cf \cite{K} (6)) that for $u\in \mathcal{D}_\mu$ and $-10. $$ \end{Corollary} When $\tpsi=0, C(4,0)=0$ and the inequality is trivial. If $F_1=F_2 =0$ (see (2.5)), then the infimum is attained for $m=\pm2$ giving $C(4,0)=9.$ \section{Rellich-type inequalities in $L^p(\R^n)$, $1z. \end{array}\right. \end{array} \end{equation} Hence, for all $u\in W_\delta^p$ and $x\in[0,2\pi]$ \begin{equation}\label{Eq4.8} |u(x)|\le \|H(x,\cdot)\|_{L^{p'}(0,2\pi)}\| u''\|_{L^p(0,2\pi)}, \end{equation} when $10.$ The inequality (4.17) follows on choosing $ \varepsilon $ sufficiently small. We also have from (4.27) \begin{equation} \|\Lambda(\theta_1,\theta_2)f\|^2 \ge \sum_{m\in \mathbb{Z}}K_m \int _{-1}^1 \left |\frac{F_m}{(1-x^2)}\right|^2 dx, \end{equation} where \[ K_m = \left( c^{-4} +\lambda_m^2 - 2( 2-c^{-2})\lambda_m \right). \] A calculation of the constant $c$ gives \[ c= 1.19967864 \] to 8 decimal places. On recalling that $\lambda_m =(m+\tpsi)^2$, it follows that the $K_m$ are positive except for the following exceptions: \begin{itemize} \item if $0\le \tpsi < 0.447566, K_1, K_{-1} <0;$ \item if $0.447566 < \tpsi < 0.552434, K_{-2}, K_{-1}, K_0, K_{1} <0;$ \item if $0.552434 < \tpsi <1, K_0, K_{-2} <0.$ \end{itemize} This completes the proof. \end{proof} \bibliographystyle{amsalpha} %\bibliographystyle{plain} \begin{thebibliography}{22} \bibitem{BA}{H. Bateman. {\it Higher Transcendental Functions}, McGraw-Hill, New York, Toronto, London, 1953.} \bibitem{DB}{D.M.~Bennett. An extension of Rellich's inequality. {\it Proc. Amer. Math. 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