Laplace Beltrami operator in the Baran metric and pluripotential equilibrium measure: the ball, the simplex and the sphere

TL;DR

This paper explores the relationship between the Laplace Beltrami operator, orthogonal polynomials, and pluripotential equilibrium measures on specific geometric domains, revealing new connections in pluripotential and differential geometry.

Contribution

It demonstrates that eigenfunctions of the Laplace Beltrami operator on the ball, simplex, and sphere are orthogonal polynomials with respect to the pluripotential equilibrium measure, linking geometry and polynomial theory.

Findings

Eigenfunctions are orthogonal polynomials w.r.t. equilibrium measure

The Laplace Beltrami operator is linked to orthogonal polynomials in specific domains

Conjecture on broader applicability of these relationships

Abstract

The Baran metric $\delta_E$ is a Finsler metric on the interior of $E\subset \R^n$ arising from Pluripotential Theory. We consider the few instances, namely $E$ being the ball, the simplex, or the sphere, where $\delta_E$ is known to be Riemaniann and we prove that the eigenfunctions of the associated Laplace Beltrami operator (with no boundary conditions) are the orthogonal polynomials with respect to the pluripotential equilibrium measure $\mu_E$ of $E.$ We conjecture that this may hold in a wider generality. The considered differential operators have been already introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work instead we highlight the relationships between orthogonal polynomials with respect to $\mu_E$ and the Riemaniann structure naturally arising from Pluripotential Theory

Equations

$$\{\begin{array}{ll}\mathcal{S}[\phi ](x):=(1-x^2){\phi }^{\prime \prime }(x)-x{\phi }^{\prime }(x)=-\lambda \phi (x),& x\in ]-1,1[\\ {\phi }^{\prime }(x_0)=0,& x_0\in \{-1,1\}\end{array}.$$

$$\frac{1}{\frac{1}{\sqrt{1-x^2}}}\frac{d}{dx}\left(\frac{1}{\sqrt{1-x^2}}(1-x^2){\phi }^{\prime }(x)\right)=-n^2\phi (x),x\in ]-1,1[.$$

$${\Delta }_{LB}f=\frac{1}{\sqrt{\det g}}\sum _{i=1}^n{\partial }_{x_i}\left(\sqrt{\det g}\sum _j{g}^{i,j}{\partial }_{x_j}f\right),$$

$${\mathcal{C}}_b^2(B^n):=\left\{u\in {\mathcal{C}}^2(B^n):\underset{|\alpha |\le 2}{\max }\underset{x\in B^n}{\sup }|{\partial }^{\alpha }u(x)|<\infty \right\},$$

$$G_{B^{n}}(x):=\left[\begin{array}[]{cccccc}1+\frac{x_{1}^{2}}{1-\sum_{i=1}^{n}x_{i}^{2}}&\frac{x_{1}x_{2}}{1-\sum_{i=1}^{n}x_{i}^{2}}&\frac{x_{1}x_{3}}{1-\sum_{i=1}^{n}x_{i}^{2}}&\dots&\dots&\frac{x_{1}x_{n}}{1-\sum_{i=1}^{n}x_{i}^{2}}\\ \frac{x_{1}x_{2}}{1-\sum_{i=1}^{n}x_{i}^{2}}&1+\frac{x_{2}^{2}}{1-\sum_{i=1}^{n}x_{i}^{2}}&\frac{x_{2}x_{3}}{1-\sum_{i=1}^{n}x_{i}^{2}}&\dots&\dots&\frac{x_{2}x_{n}}{1-\sum_{i=1}^{n}x_{i}^{2}}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{x_{n}x_{1}}{1-\sum_{i=1}^{n}x_{i}^{2}}&\frac{x_{n}x_{2}}{1-\sum_{i=1}^{n}x_{i}^{2}}&\dots&\dots&\dots&1+\frac{x_{n}^{2}}{1-\sum_{i=1}^{n}x_{i}^{2}}\end{array}\right].$$

$$\sigma (\Delta )=\{{\lambda }_s:=s(s+n-1):s\in \mathbb{N}\}$$

$${\mu }_{B^n}:=\frac{1}{\sqrt{1-|x{|}^2}}{\chi }_{B^n}{\mathrm{Vol}}_{{\mathbb{R}}^n}={\mathrm{Vol}}_{g_{B^n}}.$$

$$\mathcal{D}(\Delta ):=\left\{u\in L^2(B^n,g_{B^n}):\sum _{s=0}^{\infty }{\lambda }_s^2\sum _{|\alpha |=s}|{\hat{u}}_{\alpha }{|}^2<\infty \right\}\subset H^1(B^n,g_{B^n})$$

$$\mathcal{D}({\Delta }^{1/2}):=\left\{u\in L^2(B^n,g_{B^n}):\sum _{s=0}^{\infty }{\lambda }_s\sum _{|\alpha |=s}|{\hat{u}}_{\alpha }{|}^2<\infty \right\}=H^1(B^n,g_{B^n})$$

$${\mathcal{C}}_b^2(S^n):=\left\{u\in {\mathcal{C}}^2(S^n):\underset{|\alpha |\le 2}{\max }\underset{x\in S^n}{\sup }|{\partial }^{\alpha }u(x)|<\infty \right\},$$

$$G_{S^{n}}(x):=\left[\begin{array}[]{ccccc}x_{1}^{-1}&0&\dots&\dots&0\\ 0&x_{2}^{-1}&0&\dots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\dots&\dots&0&x_{n}^{-1}\end{array}\right]+\frac{1}{1-\sum_{i=1}^{n}x_{i}}\left[\begin{array}[]{ccccc}1&1&\dots&\dots&1\\ 1&1&\dots&\dots&1\\ 1&1&\dots&\dots&1\\ 1&1&\dots&\dots&1\\ 1&1&\dots&\dots&1\end{array}\right].$$

$$\sigma (\Delta )=\{{\lambda }_s:=s(s+\frac{n-1}{2}:s\in \mathbb{N}\}$$

$${\mu }_{S^n}:=\frac{1}{\sqrt{(1-{\sum }_{i=1}^n{x}_i){\prod }_{i=1}^n{x}_i}}{\chi }_{S^n}(x){\mathrm{Vol}}_{{\mathbb{R}}^n}={\mathrm{Vol}}_{g_{S^n}}.$$

$$\mathcal{D}(\Delta ):=\left\{u\in L^2(S^n,g_{S^n}):\sum _{s=0}^{\infty }{\lambda }_s^2\sum _{|\alpha |=s}|{\hat{u}}_{\alpha }{|}^2<\infty \right\}\subset H^1(S^n,g_{S^n})$$

$$\mathcal{D}({\Delta }^{1/2}):=\left\{u\in L^2(S^n,g_{S^n}):\sum _{s=0}^{\infty }{\lambda }_s\sum _{|\alpha |=s}|{\hat{u}}_{\alpha }{|}^2<\infty \right\}=H^1(S^n,g_{S^n})$$

$$g(v,w):=F^{\ast }{g}_{{\mathbb{H}}_{+}^n}(v,w)=g_{{\mathbb{H}}_{+}^n}(dFv,dFw),\forall v,w\in T{B}^n.$$

$$\mathrm{Ric}=kg.$$

$${\mathrm{Ric}}_{i,j}:=\sum _{l=1}^n({\partial }_l{\Gamma }_{j,i}^l-{\partial }_j{\Gamma }_{l,i}^l)+\sum _{l,k=1}^n({\Gamma }_{l,k}^l{\Gamma }_{j,i}^k-{\Gamma }_{j,k}^l{\Gamma }_{l,i}^k)$$

$$\partial :=\sum _{j=1}^n\frac{\partial }{\partial z_j}\cdot d{z}_j,\bar{\partial }:=\sum _{j=1}^n\frac{\partial }{\partial {\bar{z}}_j}\cdot d{\bar{z}}_j.$$

$$(\mathrm{d}{\mathrm{d}}^{\mathrm{c}}u{)}^n:=\mathrm{d}{\mathrm{d}}^{\mathrm{c}}u\wedge \mathrm{d}{\mathrm{d}}^{\mathrm{c}}u\wedge \cdots \wedge \mathrm{d}{\mathrm{d}}^{\mathrm{c}}u=c_n\det (\mathrm{d}{\mathrm{d}}^{\mathrm{c}}u)d{\mathrm{Vol}}_{{\mathbb{C}}^n}.$$

$$\begin{array}{rl}& V_E(\zeta ):=\sup \{u(\zeta ),u\in \mathcal{L}({\mathbb{C}}^n),u{|}_E\le 0\},\\ & V_E^{\ast }(z):=\underset{\zeta \to z}{\mathrm{limsup}}{V}_E(\zeta ).\end{array}$$

$$V_E(\zeta )=\sup \left\{\frac{1}{\mathrm{deg}p}{\log }^{+}|p(\zeta )|,p\in \mathcal{P}({\mathbb{C}}^n),\Vert p{\Vert }_E\le 1\right\}.$$

$${\mu }_E:=(\mathrm{d}{\mathrm{d}}^{\mathrm{c}}{V}_E^{\ast }{)}^n,$$

$${\delta }_E(x,v):=\underset{t\to 0^{+}}{\mathrm{limsup}}\frac{V_E^{\ast }(x+itv)}{t}$$

$$d_E(x,y):=\inf \left\{{\int }_0^1{\delta }_E(\gamma (s),{\gamma }^{\prime }(s))ds,\gamma \in Lip([0,1],E),\gamma (0)=x,\gamma (1)=y\right\},$$

$${\delta }_E(x,v)=\sqrt{v^t{G}_E(x)v}$$

$$u^T{G}_E(x)v=\frac{{\delta }_E^2(x,u+v)-{\delta }_E^2(x,u-v)}{4}.$$

$$\begin{array}{rl}& \frac{\left|\frac{d}{dt}p(\gamma ((t))\right|}{\sqrt{1-p^2(\gamma (t))}}\le \mathrm{deg}p{\delta }_E(\gamma (t),{\gamma }^{\prime }(t)),\\ & \forall t\in [0,1],\gamma \in {\mathcal{C}}^1([0,1],E),p\in {\mathcal{P}}^k({\mathbb{C}}^n),\Vert p{\Vert }_E\le 1,\end{array}$$

$$\nabla :$$

$$(X,V)$$

$$\nabla u=\sum _j{g}^{ij}\frac{\partial u}{\partial x_j}d{x}_j.$$

Full text

The considered differential operators have been already introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work instead we highlight the relationships between orthogonal polynomials with respect to $\mu_{E}$ and the Riemaniann structure naturally arising from Pluripotential Theory.

1. 1 Introduction

2. 1.1 Potential Theory and polynomials

3. 1.2 Two motivational examples

4. 1.2.1 Chebyshev polynomials

5. 1.2.2 Spherical harmonics

6. 1.3 Our results and conjecture

7. 2 Preliminaries and tools

8. 2.1 The Pluripotential Theory framework

9. 2.2 Differential operators and Sobolev spaces on a Riemaniann manifold

10. 2.2.1 Differential operators

11. 2.2.2 Sobolev Spaces

12. 2.3 Unbounded linear operators on Hilbert spaces, some tools.

13. 3 Proofs

14. 3.1 Orthogonal polynomials in $L^{2}_{\mu_{B^{n}}}$

15. 3.2 Proof of Theorem 1

16. 3.3 Orthogonal polynomials in $L^{2}_{\mu_{S^{n}}}$

17. 3.4 Proof of Theorem 2

18. 3.5 Proof of Proposition 2.1

19. A Pluripotential Theory on the complexified sphere and spherical harmonics

1. Introduction

1.1. Potential Theory and polynomials

The study of Approximation Theory in the complex plane and on the real line (by polynomials and rational functions) is deeply related to the Logarithmic Potential Theory (i.e., the study of subharmonic functions and Laplace operator); this is a classical and well established topic whose study goes back to Fekete, Leja, Szegö, Walsh and many others.

These relations between Logarithmic Potential Theory and Approximation Theory spread among Markov, Bernstein and Nikolski type polynomial inequalities, asymptotic of optimal polynomial interpolation arrays and Fekete points, overconvergence phenomena (i.e. uniformly convergent sequence of polynomials defining a holomorphic function in a larger open set) and its quantitative version, the Bernstein Walsh Theorem , asymptotic of orthogonal polynomials, random polynomials and random matrices. Moreover, most of such relations extend to the more general case of weighted polynomials and Logarithmic Potential Theory in presence of an external field. We refer to [1, 2, 3, 4, 5] and references therein for extensive treatments of this subjects.

More recently a non linear potential theory in multi dimensional complex spaces has been introduced and many analogies with the linear case have been shown, provided a suitable ”translation” of the quantities that come into the play. Pluripotential Theory (see for instance [6, 7]) is the study of plurisubharmonic functions (i.e., functions which are subharmonic along each complex line) and the complex Monge Ampere operator; [8].

Though the lack of linearity makes this new theory much more difficult and requires to work with different tools, many connections with polynomial approximation has been extended to this multi dimensional framework; see [9, 10]. Indeed, polynomial inequalities in $\mathbb{C}^{n}$ are usually obtained by means of Pluripotential Theory, see for instance [11, 12], the Bernstein Walsh Theorem has been extended by Siciak to $\mathbb{C}^{n}$ [13] and more general complex spaces by Zeriahi [14]. In his seminal work [15, 16, 17], Zaharjuta extended the equivalence between (a suitably re-defined version of) the Chebyshev Constant (i.e., the asymptotic of the uniform norms of monic polynomials) and the Transfinite Diameter (i.e., the asymptotic of the maximum of the Vandermonde determinant). Very recently, Berman Boucksom and Nystrom [18, 19] showed that Fekete points converge weak∗ to the pluripotential equilibrium measure of the considered set in $\mathbb{C}^{n}$ and in much more general settings, this is a deep extension of the one dimensional case which can be obtained only by means of the weighted theory. The work of Berman and Boucksom stimulated different lines of research as $L^{2}$ theory and general orthogonal polynomials [20], the study of multi-variate random polynomials [21, 22, 23], and the theory of sampling and interpolation arrays [24, 25, 26]. More importantly from our point of view, the widely used heuristic that the ”best” measure for producing uniform polynomial approximations by $L^{2}$ projection is the equilibrium measure has been fully motivated and theoretically explained in [19] also in its multivariate setting.

The present work concerns, on one hand, to (partially) extend to the $\mathbb{C}^{n}$ case another connection between polynomials and Potential Theory, on the other hand, to highlight how the polynomial $L^{2}$ approximation with respect to the equilibrium measure may be regarded as Fourier Analysis on a suitable Riemaniann manifold. These ideas rest upon the relation between the Laplace Beltrami operator relative to the Baran metric and the orthogonal polynomials with respect to the pluripotential equilibrium measure.

We would like to introduce such relations starting by some examples that treat the instances of the interval $[-1,1]$ and the unit sphere.

1.2. Two motivational examples

1.2.1. Chebyshev polynomials

Chebyshev polynomials $T_{n}(x):=\arccos(n\cos x)$ are the orthogonal polynomials with respect to $\frac{1}{\pi\sqrt{1-x^{2}}}dx$, the equilibrium measure of the interval $[-1,1]$ as a subset of $\mathbb{C},$ i.e., the unique minimizer of the logarithmic potential $-\int\log|z-w|d\mu(z)d\mu(w)$ among all Borel probability measures $\mu$ on the interval $[-1,1]$. Another classical characterization of Chebyshev polynomials is given by the eigen-functions of the Sturm-Liouville eigenvalue problem

[TABLE]

The set of eigenvalues turns out to be $\{n^{2}:n\in\mathbb{N}\}$ and $\mathscr{S}[T_{n}]=n^{2}T_{n}.$

Instead, we re-write such an eigenvalue problem as

[TABLE]

This apparently useless manipulation actually enlightens another property of Chebyshev polynomials. To explain this property, we first recall that the Laplace Beltrami operator relative to a metric $g$ can be written in local coordinates as

[TABLE]

where $g^{i,j}$ are the components of the inverse of the matrix representing $g.$

Let us endow $]-1,1[$ with the Riemaniann metric $g(x):=\frac{1}{1-x^{2}},$ we canonically obtain the Riemannian distance $d(x_{0},x_{1})=\int_{x_{0}}^{x_{1}}\frac{1}{\sqrt{1-x^{2}}}dx.$ Note that, up to a re-normalization, the resulting volume form is precisely the equilibrium measure of $[-1,1].$ If we plug $g(x):=\frac{1}{1-x^{2}}$ in the expression (3) of the Laplace Beltrami operator, we obtain precisely the left hand side of (2). In other words, we observe that:

• •

Chebyshev polynomials are eigenfunctions of the Laplace Beltrami operator with respect to the equilibrium measure of the interval.

It is relevant to notice that the density of the equilibrium measure on $[-1,1]$ at $x$ is obtained as the normal (i.e., purely complex) derivative of the Green function of $\mathbb{C}\setminus[-1,1]$ with pole at infinity; see [1, Ch II.1]. This operation has a multidimensional counterpart (see [12]) that, under some assumptions, leads to the so called Baran metric ([27, 28, 29]), see equation (8) below.

1.2.2. Spherical harmonics

We mention another relevant example of this relation between eigenfunctions of the Laplace Beltrami operator with respect to the metric defined by (pluri-)potential theory and the (pluripotential) equilibrium measure. In contrast with the case of Chebyshev polynomials, now we work in a multi dimensional setting and the flat euclidean space $\mathbb{C}^{n}$ is replaced by a complex manifold. A more detailed account of this example requires some preliminary notions in addition to the ones of Subsection 2.1, thus we decided to present the explicit computations in Appendix A, together with the needed recalls from pluripotential theory on algebraic varieties. At this stage we only sketch the results to underline the analogy with the case of Chebyshev polynomials.

Let us consider the unit sphere $\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$ endowed it with the round metric $g$ induced by the flat metric on $\mathbb{R}^{n}$ and denote by $\Delta$ the Laplace Beltrami operator on $\mathbb{S}^{n-1}.$ It is well known that spherical harmonics are a dense orthogonal system of $L^{2}(\mathbb{S}^{n-1})$ which consists of polynomials that are eigenfunctions of $\Delta.$

Let us look at $\mathbb{S}^{n-1}$ as a compact subset of the complexified sphere $\mathcal{S}^{n-1}:=\{z\in\mathbb{C}^{n}:\sum z_{i}^{2}=1\}.$ By a fundamental result due to Sadullaev [30], since $\mathcal{S}^{n-1}$ is a irreducible algebraic variety, one can relate ( see Appendix A) the traces of polynomials on $\mathcal{S}^{n-1}$ to the pluripotential theory on the complex manifold of $\mathcal{S}^{n-1}.$ On the other hand, due to Lemma A.1 below, we can define a smooth Riemaniann metric $g_{\mathbb{S}^{n-1}}$ on $\mathbb{S}^{n-1}$ suitably modifying the construction (see eq. (8)) of the Baran metric of convex real bodies. In particular such a definition is given by the generalization of the case of the real interval $[-1,1]$. Indeed, it turns out that $g_{\mathbb{S}^{n-1}}=g$ and its volume form is, up to a constant scaling factor, the pluripotential equilibrium measure (see equation (31) below) of $\mathbb{S}^{n-1},$ as compact subset of $\mathcal{S}^{n-1}.$ In other words

• •

the eigenfunctions of the Laplace Beltrami operator of $(\mathbb{S}^{n-1},g)$ are the orthogonal polynomials with respect to the pluripotential equilibrium measure of $\mathbb{S}^{n-1},$ seen as compact subset of $\mathcal{S}^{n-1};$ see Corollary A.1.

1.3. Our results and conjecture

The aim of the present paper is to present a conjecture on the extension to the $\mathbb{C}^{n}$ case of the relation between potential theory and certain Riemaniann structure that holds in the examples above. We support it by full proofs of all the few known instances fulfilling the required hypothesis, see Theorems 1 and 2 below.

Conjecture 1**.**

Let $\mathcal{C}$ denote either $\mathbb{C}^{n}$ or any irreducible algebraic sub-variety of it. Let $E\subset\mathcal{C}$ be a fat111This should be intended as the closure in $\mathcal{C}$ of the interior in $\mathbb{R}^{n}\cap\mathcal{C}$ of $E$ equals to $E$ itself. real compact set. Assume that the Baran metric $\delta_{E}$ of $E$ is a Riemannian metric on $\operatorname{int}_{\mathbb{R}^{n}\cap\mathcal{C}}E,$ then the orthonormal polynomials with respect to the pluripotential equilibrium measure $\mu_{E,\mathcal{C}}$ of $E$ in $\mathcal{C}$ are eigenfunctions of the Laplace Beltrami operator relative to the metric $\delta_{E}.$

Remark 1**.**

We stress that the orthogonal bases used in our proofs as well as most of their properties are already known in the framework of orthogonal polynomials (see [31, 32] and references therein). Moreover, our differential operators (i.e., Laplace Beltrami operators with respect to the metrics arising from Pluripotential Theory) turn out to be already studied in relation with certain symmetry groups [32, Ch. 8], but they have not been related to any potential theoretic aspects before. More precisely, the Laplace Beltrami operator on the ball endowed with its Baran metric turns out to be the operator $\mathscr{D}_{\mu}$ in [31, pg. 142] with the parameter choice $\mu=0$. Instead, in the simplex case, $\Delta$ is precisely the operator defined in [31, eq. 5.3.4] (see equation 24 and Theorem 5 below) if we set (in the authors notation) $\kappa=(0,\dots,0)\in\mathbb{R}^{n+1}.$

*Our goal is precisely to relate such families of functions and their properties to the Riemaniann structure that comes from Pluripotential Theory. ***

Theorem 1** (Laplace Beltrami on the Baran Ball).**

Let us denote by $\Delta$ the Laplace Beltrami operator of the Riemannian manifold $(B^{n},g_{B^{n}})$ acting on

[TABLE]

where $B^{n}:=\{x\in\mathbb{R}^{n}:|x|<1\}$ and $g_{B^{n}}(x)$ is represented by the matrix

[TABLE]

The operator $\Delta$ is symmetric and unbounded, it has discrete spectrum

[TABLE]

and the eigen-space of $\lambda_{s}$ is $\operatorname{span}\{\varphi_{\alpha},|\alpha|=s\},$ where $\varphi_{\alpha}$ (see Proposition 3.1) are orthonormal polynomials with respect to the pluripotential equilibrium measure

[TABLE]

Moreover, $\Delta$ can be closed to a self-adjoint operator $\mathcal{D}(\Delta)\rightarrow L^{2}(B^{n},g_{B^{n}})$ (having the same spectrum), where

[TABLE]

and $\hat{u}_{\alpha}$ is the Fourier coefficient $\int_{B^{n}}u\frac{\varphi_{\alpha}}{\|\varphi_{\alpha}\|^{2}_{L^{2}(\mu_{B^{n}})}}d\mu_{B^{n}}.$

The operator $\Delta^{1/2}$ has domain

[TABLE]

For a precise definition of the Sobolev space $H^{1}(B^{n},g_{B^{n}})$ see Subsection 2.2.2 below.

Theorem 2** (Laplace Beltrami on the Baran Simplex).**

Let us denote by $\Delta$ the Laplace Beltrami operator on the Riemannian manifold $(S^{n},g_{S^{n}})$, acting on

[TABLE]

where $S^{n}:=\{x\in\mathbb{R}^{n}:\sum_{i=1}^{n}x_{i}<1,x_{i}>0\,\forall i=1,2,\dots,n\}$ and $g_{S^{n}}(x)$ is represented by the matrix

[TABLE]

The operator $\Delta$ is symmetric and unbounded, it has discrete spectrum

[TABLE]

and the eigen-space of $\lambda_{s}$ is $\operatorname{span}\{\psi_{\alpha},|\alpha|=s\},$ where $\psi_{\alpha}$ (see Proposition 3.2) are orthonormal polynomials with respect to the pluripotential equilibrium measure of the simplex

[TABLE]

Moreover, $\Delta$ can be closed to a self-adjoint operator (still denoted by $\Delta$) $\mathcal{D}(\Delta)\rightarrow L^{2}(S^{n},g_{S^{n}})$ (having the same spectrum) where

[TABLE]

and $\hat{u}_{\alpha}$ is the Fourier coefficient $\int_{B^{n}}u\frac{\psi_{\alpha}}{\|\psi_{\alpha}^{2}|_{L^{2}(\mu_{S^{n}})}}d\mu_{S^{n}}.$

The operator $\Delta^{1/2}$ has domain

[TABLE]

Remark 2** (Refinement of Conjecture 1).**

The computations performed in [33], [34] and [27] allow us to prove that Conjecture 1 does hold for $E$ being the real ball and the real simplex. We remark that in this cases $\partial E$ is a real algebraic (possibly reduceble) set, thus one may add such an assumption to Conjecture 1.

Remark 3**.**

In order to better understand how the Baran metrics of the ball and the simplex look like, it is worth to recall their special relation with certain portion of the sphere.

Let us denote by $(\mathbb{H}^{n}_{+},g_{\mathbb{H}^{n}_{+}})$ the upper unit hemisphere, i.e., the Riemaniann manifold which can be obtained by intersecting the unit sphere $\mathbb{S}^{n}$ (thought as a sub-manifold of $\mathbb{\mathbb{R}}^{n+1}$ endowed with the euclidean metric) with the positive half space $\{\xi\in\mathbb{R}^{n+1}:\xi_{n+1}>0\}.$ The map $\pi:\mathbb{H}^{n}_{+}\rightarrow B^{n}$, $\pi(\xi):=(\xi_{1},\dots,\xi_{n})$ clearly is a one-to-one $\mathscr{C}^{\infty}$ map of manifolds. Therefore we can define a metric $g$ on $B^{n}$ by means of the pull-back operator with respect to $F:=\pi^{-1}$:

[TABLE]

One can verify by direct computations that indeed $g\equiv g_{B^{n}}.$

Similarly, we can define the map $\operatorname{Sqrt}:S^{n}\rightarrow B^{n}\cap\{x\in\mathbb{R}^{n}:x_{i}>0,\forall i=1,\dots,n\}$, $\operatorname{Sqrt}(x):=(\sqrt{x_{1}},\sqrt{x_{2}},\dots,\sqrt{x_{n}}),$ and pull back by $\operatorname{Sqrt}$ on $S^{n}$ the Baran metric of the ball. Again this new metric indeed coincide with the Baran metric of the simplex.

In other words the maps $F$ and $sqrt$ are isometries of Riemaniann manifolds.

Note that, since the manifolds $(B^{n},g_{B^{n}})$ and $(S^{n},g_{S^{n}})$ are isometric to certain portions of $\mathbb{S}^{n}$, the local differential and metric properties of this manifolds are the same of $\mathbb{S}^{n}.$ We recall that a Riemaniann manifold $(M,g)$ is termed Einstein when its metric tensor is a solution of the Einstein vacuum field equation

[TABLE]

Here

[TABLE]

is the Ricci tensor (written by means of the Christoffel symbols $\Gamma^{i}_{j,k}$) and $k>0.$ Since it is a well known fact that $(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$ is Einstein, we get the following proposition as a consequence of Remark 3.

Proposition 1.1**.**

The unit ball and the unit simplex, endowed with their Baran metric respectively, are Einstein Manifolds.

Since for all cases where the Baran metric is known to be Riemaniann it happens that it solves Equation (4), the following question naturally arises.

Question 1**.**

Assume that $E$ is a Baran body in the sense of Definition 3 below. Is it necessary for its Baran metric tensor to solve the Einstein vacuum field equation (4)?

Remark 4**.**

Recently, Zelditch [35] studied the spectral theory of the Laplace Beltrami operator on a real analytic Riemaniann manifold $M$ in connection with the Pluripotential Theory of the so called Bruhat-Whitney complexification $M_{\mathbb{C}}$ of $M.$ In particular, working under the assumption of ergodicity of the geodesic flow, [36, 37] present asymptotic results on the zero distribution of the eigenfunctions and series of functions with random Fourier coefficients. These results closely resemble the relation between the behaviour of zeros of orthogonal polynomials (or random polynomials) and the pluripotential equilibrium measure.

Even though our study is far to be as general as the context of the above references, in the author’s opinion our result may be casted within this framework and offer concrete examples where explicit computations are performed. Indeed our Appendix A exactly fits in the framework of [35].

The paper is structured as follows. In Section 2 we furnish all the required definitions from Pluripotential Theory, Operator Theory and Differential Geometry. In Section 3 we prove Theorems 1 and 2, giving a precise spectral characterization of the involved Sobolev spaces. Finally, in the Appendix A it is shown how to define the Baran metric on the sphere and its equivalence with the standard round metric.

Acknowledgements

The ideas of the present paper surfaced during the open problems session of the workshop Dolomites Research Week on Approximation (DRWA16), held in Canazei (TN) Italy in September 2016. However, the content of the present paper has been deeply influenced by the note [28] told by its second author during his visit at University of Padova in 2012. Therefore we would like to thank Norm Levenberg and the organizers of the conference and the Doctoral School of Mathematics of the University of Padova. Also, we would like to thank Prof. P.D. Lamberti for his helpfulness, Prof. M. Putti and M. Vianello for the useful discussions and the support they offered.

2. Preliminaries and tools

2.1. The Pluripotential Theory framework

Pluripotential Theory is the study of plurisubharmonic functions, i.e., any upper-semicontinuous function $u:\Omega\rightarrow[-\infty,+\infty[$ being subharmonic along each one complex dimension affine variety in $\Omega\subseteq_{open}\mathbb{C}^{n}.$ We use the operators $d:=\partial+\bar{\partial}$ and $d^{c}:=i(-\partial+\bar{\partial})$, where

[TABLE]

The operator $\operatorname*{dd^{c}}$ is sometimes referred as complex Laplacian and correspond with the usual Laplacian (up to a scaling factor) when $n=1.$

Since $\operatorname*{dd^{c}}$ is a linear operator, one can consider $\operatorname*{dd^{c}}u$ for a $L^{1}_{\text{loc}}$ function in the sense of currents (distribution on the space of differential forms) and it turns out that, for an upper-semicontinuous function $u$, $\operatorname*{dd^{c}}u\geq 0$ if and only iff $u$ is plurisubharmonic.

The complex Monge Ampere operator $(\operatorname*{dd^{c}}{})^{n}$ is defined for $\mathscr{C}^{2}$ functions as

[TABLE]

Clearly trying to define wedge products of factors of the type $\operatorname*{dd^{c}}u$ for any plurisubharmonic function $u$ leads to serious difficulties due to the lack of linearity. Bedford and Taylor [8] showed that the definition of equation (5) can be extended to any locally bounded plurisubharmonic function, being $(\operatorname*{dd^{c}}u)^{n}$ a positive Borel measure.

One may think to plurisubharmonic functions in $\mathbb{C}^{n}$ as ”the correct counterpart” (see [6, Preface]) of subharmonic functions on $\mathbb{C}$, while harmonic functions should be replaced in this multi dimensional setting by maximal plurisubharmonic functions, i.e., functions $u$ dominating on any subdomain $\Omega^{\prime}$ any plurisubharmonic function $v$ such that $u\geq v$ on $\partial\Omega^{\prime}.$ Locally bounded maximal plurisubharmonic functions satisfy $(\operatorname*{dd^{c}}u)^{n}=0.$

The multi dimensional counterpart of the Green function for the complement of a compact set $E$ is the pluricomplex Green function (also known as Siciak-Zaharjuta extremal function) $V_{E}^{*}.$ Let $E\subset\mathbb{C}^{n}$ be a compact set, then we set

[TABLE]

Here $\mathcal{L}(\mathbb{C}^{n})$ is the Lelong class of plurisubharmonic functions of logarithmic growth, i.e., $u(z)-\log|z|$ is bounded at infinity.

It is worth to recall that, as in the one dimensional case, due to [13] (see also [6]) we can express $V_{E}^{*}$ by means of polynomials $\mathscr{P}(\mathbb{C}^{n}).$ That is

[TABLE]

The function $V_{E}^{*}$ is either identically $+\infty$ or a locally bounded plurisubharmonic function on $\mathbb{C}^{n}$, maximal on $\mathbb{C}^{n}\setminus E$ (i.e., $\left(\operatorname*{dd^{c}}V_{E}^{*}\right)^{n}$ is a positive Borel measure with support in $E$) having logarithmic growth at $\infty;$ if the latter case occurs we say that $E$ is non pluripolar. In principle $V_{E}^{*}$ is only a upper semi-continuous function. When $V_{E}^{*}$ is continuous the compact set $E$ is said regular. It is worth to recall that it turns out that $V_{E}^{*}$ is continuous if and only if $V_{E}^{*}$ identically vanishes on $E.$ We will treat only such a case in what follows.

For any non pluripolar compact set $E\subset\mathbb{C}^{n}$ the pluripotential equilibrium measure of $E$ is defined as

[TABLE]

this is a Borel probability measure supported on $E.$ We stress that, since $\mu_{E}(E)=1$ for any non pluripolar set [8], the total mass of the measures (and volume forms) that we are going to deal with is not important. We avoid to introduce normalizing constant in the metrics to keep the notation simple.

Let $E$ be a real convex body, Baran showed that in such a case

[TABLE]

exists for any $x\in\operatorname{int}E,$ $v\in\mathbb{R}^{n}.$ We refer to $\delta_{E}(x,v)$ as the Baran metric of $E.$ We refer the reader to [34] for a study on the connections among this metric, polynomials inequalities and polynomial sampling. The Baran metric defines in general a Finsler distance on $E$

[TABLE]

however it may happen that $\delta_{E}(x,v)$ is indeed Riemaniann, i.e.

[TABLE]

for a positive definite matrix $G_{E}(x).$ Note that $G_{E}(x)$ is then well defined by the parallelogram law. More precisely we have

[TABLE]

One of the possible motivation for the interest on the Baran metric comes from Approximation Theory. Indeed the Baran Inequality (see [29, 38] and [28])

[TABLE]

can be understood as a generalization of the classical Bernstein Inequality. For instance such inequality may be used to construct good sampling sets for polynomials, namely admissible meshes; see [39, 40, 41, 42, 43]

We believe that the following definition is worth to be introduced.

Definition 3** (Baran body).**

Let $\mathcal{C}$ denote either $\mathbb{C}^{n}$ or a irreducible algebraic variety of pure dimension $n,$ and let $\mathcal{C}_{\mathbb{R}}$ denote the real points of $\mathcal{C}.$ Let $E\subset\mathcal{C}_{\mathbb{R}}$ a compact fat222This mean that the closure in $\mathcal{C}_{\mathbb{R}}$ of the interior of $E$ in $\mathcal{C}_{\mathbb{R}}$ coincides with $E.$ non pluripolar set. If the Baran metric of $E$ is Riemaniann, then we term $E$ a Baran body.

In [28, 34], the Baran metrics of the real ball, real simplex are computed (see Theorem 1 and Theorem 2 above), showing in particular that they are Baran bodies. To the best author’s knowledge, these are all the known examples of Baran compact sets in $\mathbb{C}^{n}.$ We offer a further instance of a Baran compact in Appendix A: the real sphere as subset of the complexified sphere.

2.2. Differential operators and Sobolev spaces on a Riemaniann manifold

2.2.1. Differential operators

We recall that a liner connnection on a vector bundle $\pi:E\rightarrow M$ (built on the differentiable manifold $M$) is an application (here $\mathcal{E}(M)$ is the space of smooth sections of the vector bundle $E$ and $\mathcal{T}(M)$ is the tangent bundle)

[TABLE]

such that it is $\mathscr{C}^{\infty}$-linear in $X$, $\mathbb{R}$-linear in $V,$ and for which holds the Liebnitz Rule $\nabla_{X}(fV)=VX(f)+f\nabla_{X}(V),$ for any $f\in\mathscr{C}^{\infty}(M).$ In particular we have $\nabla_{X}f=X(f).$

Let $(M,g)$ be a (possibly non compact) Riemaniann manifold. It is well known that there exists a unique torsion-free linear connection on $\mathcal{T}(M)$ that is compatible with the metric $g$; namely the Levi-Civita connection. Since we will deal only with such a connection we will still denote it by $\nabla.$ Indeed, the proof of the Levi Civita Theorem is fully constructive: the desired connection is expanded over a canonical basis and its coefficients, the Christoffel symbols usually denoted by $\Gamma_{i,j}^{k},$ are computed in terms the metric and its partial derivatives.

Note that, for a given $u\in\mathscr{C}^{\infty}(M),$ $\nabla u$ is a $(1,0)$ tensor field (i.e., point-wise it is a linear form) having the property that $(X,\nabla u)_{g}=\nabla_{X}u=X(u)$ and thus it can be written in local coordinates

[TABLE]

Here $(\cdot,\cdot)_{g}$ is the canonical duality induced by $g$ and $g^{ij}$ are the components of the matrix representing $g^{-1}.$ Hence it is convenient to define the tangent vector

[TABLE]

namely the covariant gradient of $u$, having the property that $(X,\nabla u)_{g}=\langle X,\operatorname*{grad}u\rangle_{g}.$

The divergence operator acting on $X\in\mathcal{T}(M)$ is defined by

[TABLE]

Finally we can recall the definition of the Laplace Beltrami operator $\Delta.$

[TABLE]

2.2.2. Sobolev Spaces

Let $(M,g)$ be a Riemaniann manifold. Let us introduce on $\mathscr{C}^{\infty}(M)$ the norm

[TABLE]

where $|\operatorname*{grad}u|^{2}=\langle\operatorname*{grad}u,\operatorname*{grad}u\rangle_{g}.$ Let us denote by $\mathscr{C}_{1,2}^{\infty}(M)$ the space $\{u\in\mathscr{C}^{\infty}(M),\|u\|_{1,2}<\infty\}.$

The Sobolev space $H^{1}(M,g)$ is defined as the closure of $\mathscr{C}_{1,2}^{\infty}(M)$ with respect to $\|\cdot\|_{1,2}$ in the space of square integrable functions, also we introduce the space $H^{1}_{0}(M,g)$ defined as the closure of $\mathscr{C}^{\infty}_{c}(M)$ in the same norm. Note that in principle $H^{1}_{0}(M,g)\subseteq H^{1}(M,g).$

An important fact about Sobolev spaces and manifold is that the above two spaces may coincide, that is

[TABLE]

Our interest on this phenomena is mainly due to the fact that the Laplace operator does not need to be complemented with boundary conditions in such a case.

Indeed, $H^{1}_{0}(M,g)\equiv H^{1}(M,g)$ for any complete Riemaniann manifold $M$; see [44, Th. 3.1]. We recall for the reader’s convenience that a Riemaniann manifold $(M,g)$ is said to be complete if the metric space $(M,d_{g})$ is complete, where

[TABLE]

The Hopf-Rinow Theorem asserts that the completeness of $(M,g)$ is equivalent to the fact that any closed bounded subset of $M$ is compact.

We denote by $\mathscr{C}^{\infty}_{b}(M)$ the set uniformly bounded functions that have uniformly bounded partial derivatives of any order. Since for a complete manifold $\mathscr{C}^{\infty}_{c}(M)\subseteq\mathscr{C}^{\infty}_{b}(M)\subset H^{1}(M,g)$, it follows that for any complete manifold $(M,g)$, $\mathscr{C}^{\infty}_{b}(M)$ is dense in $H^{1}(M,g).$

Unfortunately, both $(B^{n},g_{B^{n}})$ and $(S^{n},g_{S^{n}})$ fail to be complete: it is very easy to construct a Cauchy sequence in $B^{n}$ not converging in $B^{n}.$ For instance take $\{x_{k}\}:=\cos{(2^{-k})}u$ for any unit vector $u\in\mathbb{R}^{n}$. Since $d(x_{k},x_{l})\leq 2^{-\min(k,l)},$ this is a Cauchy sequence, however $x_{k}\to u\notin B^{n}.$ Nevertheless, one may wonder weather equation (11) holds true in this instances. This fact indeed depends on finer properties of the manifolds than completeness. Namely, Masamune [45, 46] showed that equality (11) holds if and only if the metric completion of $M$ lies in the category of manifolds with almost polar boundary.

We recall that the Riemaniann manifold $(M\cup\Gamma,g)$ with boundary $\Gamma$ is said to have almost polar boundary if the outer capacity $\operatorname{cap}(\Gamma)$ of $\Gamma$ vanishes. Here we use the notation $\operatorname{cap}(A)$ for the Sobolev (outer) capacity of the Borel subset $A$ of $M\cup\Gamma,$ where for any open subset $O$ of $M\cup\Gamma$ we set

[TABLE]

and for for any Borel subset $S$ we set

[TABLE]

It is clear that one can replace $\mathscr{C}^{\infty}_{c}(M\cup\Gamma)$ by $H^{1}_{0}(M\cup\Gamma,g)$ in the definition of $\operatorname{cap}(O)$ obtaining an equivalent definition.

At this stage we can observe that $\partial B^{n}$ fails the sufficient condition (see [45, Th. 7]) to be polar

[TABLE]

Here equality case is considered since $\partial B^{n}$ itself is a manifold (see [45, Th. 7]).

Let us denote by $N_{\varepsilon}$ the set $\{x\in B^{n}:d(x,\partial B^{n})<\varepsilon\}$, we have $N_{\varepsilon}=B^{n}\setminus(\cos\varepsilon)\cdot B^{n}$, moreover

[TABLE]

Here $\beta(a,b,z)$ denotes the Incomplete Beta Function $\int_{0}^{z}t^{a-1}(1-t)^{b-1}dt.$ Hence

[TABLE]

Note that

[TABLE]

Thus we have $\liminf_{\varepsilon\to 0^{+}}\frac{\operatorname{Vol}N_{\varepsilon}}{\varepsilon^{2}}=+\infty$ that in particular implies $\frac{\log\operatorname{Vol}N_{\varepsilon}}{\log\varepsilon}<2$ for any $\varepsilon<\varepsilon_{0}.$

Since the condition (12) is not fulfilled by $\partial B^{n}$ nor $\partial S^{n}$ we wonder if the ball and the simplex, endowed with their Baran metrics, are not manifold with almost polar boundary. Indeed this is the case, as stated in the following proposition. However, since these conclusions are obtained as a consequence of Theorem 1 and Theorem 2 respectively, we cannot use them in the proof of such theorems.

Proposition 2.1**.**

The manifolds $(B^{n},g_{B^{n}})$ and $(S^{n},g_{S^{n}})$ are not manifold with almost polar boundary and

[TABLE]

Remark 5**.**

We warn the reader that $H^{1}(B^{n},g_{B^{n}})\neq H^{1}_{0}(B^{n},g_{B^{n}})$ does not imply in general that the eigenvalue problem $\Delta u=\lambda u$ is not well posed when we do not impose any boundary condition. The motivation depends on the following proposition which allows us to write the weak formulations (21) and (27) of the Laplace Beltrami operator used in the proofs of Theorem 1 and 2 which is based on $\mathscr{C}^{\infty}_{b}$ functions (for which the boundary terms appearing in the integration by parts formulas we use vanish).

Proposition 2.2**.**

Let $(M,g)$ be $(B,g_{B})$ or $(S,g_{S})$. The space $\mathscr{C}^{\infty}_{b}(M)$ is dense in $\mathscr{C}^{\infty}_{1,2}(M)$ with respect to the norm $\|\cdot\|_{1,2}.$ Thus $\mathscr{C}^{\infty}_{b}(M)$ is dense in $H^{1}(M,g).$

Before proving Proposition 2.2 we need this two technical Lemmata whose proofs are omitted since it is sufficient to check the statements by easy direct computations.

Lemma 2.1** (The inverse Baran metric of the ball).**

Let us denote by $G_{B^{n}}^{-1}$ the inverse of the matrix $G_{B^{n}}$ which represents the Baran metric of the $n$-dimensional ball. Then we have

[TABLE]

The matrix $G_{B^{n}}^{-1}(x)$ has eigenvalues $\{1,1-|x|^{2}\}$, where the eigen-space of $1$ is the tangent space at $x$ to the sphere of radius $|x|$ and centred at zero, while the eigen-space of $1-|x|^{2}$ is the Euclidean normal to such a sphere at $x.$

Lemma 2.2** (The inverse Baran metric of the simplex).**

Let us denote by $G_{S^{n}}^{-1}$ the inverse of the matrix $G_{S^{n}}$ which represents the Baran metric of the $n$-dimensional simplex. Then we have

[TABLE]

Moreover we have

[TABLE]

Proof of Proposition 2.2.

Let us start by considering the case $M=B^{n}\subset\mathbb{R}^{n}.$ We denote by $\mathbb{S}^{n}$ the $n$ dimensional unit real sphere endowed with the standard round metric $g_{\mathbb{S}^{n}}$ and we introduce the embedding map

[TABLE]

where

[TABLE]

and

[TABLE]

We claim that $E$ is an isometry of Hilbert spaces.

Before proving such a claim we stress that this would conclude the proof for the case of the ball. For, by standard mollification we can construct a sequence $\{\hat{f}_{k}\}$ of function in $\mathscr{C}^{\infty}(\mathbb{S}^{n})$ converging to $E[f]$ in $H^{1}(\mathbb{S}^{n},g_{\mathbb{S}^{n}}).$ To ensure that $\tilde{f}_{k}\in H_{\text{even}}^{1}(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$ we set $\tilde{f}_{k}:=(\hat{f}_{k}(x_{1},\dots,x_{n+1})+\hat{f}_{k}(x_{1},\dots,-x_{n+1}))/2.$ Finally define $\{f_{k}\}:=\{E^{-1}[\tilde{f}_{k}]\}$ and note that the claim above implies that $f_{k}\to f$ in $H^{1}(B^{n},g_{B^{n}}).$

We stress that, while the injectivity of $E$ is trivial, one needs to notice that the global boundedness of $\tilde{f}_{k}$ together with its derivatives ensure that $E^{-1}[\tilde{f}_{k}]$ is a well defined element of $\mathscr{C}^{\infty}_{1,2}(M)$ which in particular is in $\mathscr{C}^{\infty}_{b}(M).$

Let us go back to prove that $E$ is an isometric embedding. For simplicity we work in the easy case of $n=2$, the general case can be proved in a completely equivalent way. Consider the spherical coordinates

[TABLE]

We recall that the round metric represented in this coordinates is

[TABLE]

and the corresponding volume form can be written $d\operatorname{Vol}_{\mathbb{S}^{2}}=\cos\theta d\theta d\varphi.$ It follows that, for any $h\in H^{1}(\mathbb{S}^{2},g_{\mathbb{S}^{2}})$ we have

[TABLE]

To compute $\|E[f]\|_{H^{1}(\mathbb{S}^{2})}$ we perform the change of variables suggested by the first two components of the spherical coordinates, i.e.,

[TABLE]

It is easy to verify by a direct computation that

[TABLE]

Let us now consider the case $M=S^{n}.$ We introduce the embedding map

[TABLE]

where

[TABLE]

and

[TABLE]

Again if the closure of $F$ to $H^{1}(S^{n},g_{S^{n}})$ is an isometric embedding we are done, since, for any given target function $h\in H^{1}(S^{n},g_{S^{n}})$ we can pull back to $\mathscr{C}^{\infty}_{b}(S^{n})$ any sequence of $\mathscr{C}^{\infty}_{b}(B^{n})$ approximations to $F[h].$

To this aim, we introduce the partition $Q_{1},\dots,Q_{2^{n}}$ of $[-1,1]^{n}$ given by the coordinates hyperplanes, we denote by $T:S^{n}\rightarrow B^{n}$ the map $(\xi_{1},\dots,\xi_{n})\mapsto(\xi_{1}^{2},\dots,\xi_{n}^{2})=x$ and we notice that, for any $f\in\mathscr{C}^{\infty}(S^{n})$, we have

[TABLE]

Finally we compute

[TABLE]

Since, due to equation (16),

[TABLE]

we conclude that $\|F[h]\|_{H^{1}(B^{n},g_{B^{n}})}=\|h\|_{H^{1}(S^{n},g_{S^{n}})}.$ In view of the above reasoning this concludes the proof. ∎

2.3. Unbounded linear operators on Hilbert spaces, some tools.

We need to recall some concepts from Operator Theory that allow a more precise and compact formulation of our results. A linear operator on a Banach space $\mathscr{B}$ is a couple $(\mathcal{D}_{\mathscr{B}}(T),T)$, where $\mathcal{D}_{\mathscr{B}}(T)$ is a dense linear subspace of $\mathscr{B}$ and $T$ is a linear map $\mathcal{D}_{\mathscr{B}}(T)\rightarrow\mathscr{B}.$

Let $(\mathcal{D}_{\mathscr{B}}(T),T)$ be a linear operator. If for any sequence $\{f_{n}\}$ in $\mathcal{D}_{\mathscr{B}}(T)$ such that

• •

$\|f_{n}\to f\|_{\mathscr{B}}\to 0$ for some $f\in\mathscr{B},$

• •

there exists $g\in\mathscr{B}$ with $\|Tf_{n}-g\|_{\mathscr{B}}\to 0$

it follows that $f\in\mathcal{D}_{\mathscr{B}}(T)$ and $Tf=g$, then the operator $T$ is said to be closed. If $\mathscr{B}$ is not finite dimensional, the notion of spectrum and set of eigenvalues are not coinciding. More precisely, we denote by $\sigma(T)$ the spectrum of $T$

[TABLE]

Instead, $\lambda$ is an eigenvalue of $T$ if there exists an element $f\in\mathscr{B}$ such that $Tf=\lambda f.$

If an operator $T$ is not closed we may try to find and extension of it, i.e., $(\tilde{T},\mathcal{D}_{\mathscr{B}}(\tilde{T}))$ such that $\mathcal{D}_{\mathscr{B}}(\tilde{T})\supset\mathcal{D}_{\mathscr{B}}(T)$ and $\tilde{T}f=Tf$ for any $f\in\mathcal{D}_{\mathscr{B}}(T).$ If we can find such an extension in the category of closed operators, then $T$ is said to be closable and its minimal closed extension $\overline{T}$ is termed the closure of $T.$

Now we replace the Banach space $\mathscr{B}$ by an Hilbert space $\mathscr{H}$, clearly the above terminologies are still well defined, since any Hilbert space is in particular Banach.

If for any $f,g\in\mathcal{D}_{\mathscr{H}}(T)$ we have $\langle Tf,g\rangle_{\mathscr{H}}=\langle f,Tg\rangle_{\mathscr{H}}$, then the operator $T$ is said to be symmetric. It is a very useful fact that any symmetric operator is closable to a symmetric operator. Again, if $\mathscr{H}$ is infinite dimensional, one must pay attention to the difference among symmetric and self-adjoint operators.

The adjoint $T^{*}$ of the operator $T$ is defined by the relation

[TABLE]

where

[TABLE]

Clearly, we term $T$ self-adjoint when the two domains indeed coincide.

The proofs of our results, besides the explicit computations, rely on the following theorem which collects some classical results of Operator Theory; see for instance [47, Ch. 1 and Ch. 4].

Theorem 4**.**

Let $T$ be a linear non negative unbounded operator on the Hilbert space $(\mathscr{H},\|\cdot\|)$ with domain $\mathcal{D}(T)$. Assume that

1. a)

$T$* is symmetric,* 2. b)

It has discrete real spectrum $\sigma(T)=\{\lambda_{j}\}_{j\in\mathbb{N}}$ diverging to $+\infty.$

Then

1. i)

the closure $\bar{T}$ of $T$ is a self-adjoint unbounded operator (i.e., $T$ is essentially self-adjoint), 2. ii)

$\sigma(\bar{T})=\sigma(T)$, 3. iii)

the domain of $\bar{T}$ is

[TABLE] 4. iv)

the quadratic form

[TABLE]

has domain

[TABLE]

which is complete in the norm

[TABLE]

3. Proofs

The strategy of the proofs of Theorems 1 and 2 is to show that the conditions $a)$ and $b)$ of Theorem 4 holds for $T$ being the Laplace Beltrami operator (with respect to the considered metric), then to conclude applying Theorem 4. This will be done by considering the weak formulation of the Laplace Beltrami operator and performing explicit computations on a suitable orthogonal system.

3.1. Orthogonal polynomials in $L^{2}_{\mu_{B^{n}}}$

The following family of orthogonal functions on the unit ball has been first introduced in the Approximation Theory framework, indeed the formula we will use is a special case of orthogonal polynomials for certain radial weight functions; see [32, Ch. 5].

Proposition 3.1** ([32]).**

Let us set for any $\alpha\in\mathbb{N}^{n}$

[TABLE]

where $T_{k}$ is the Chebyshev polynomial of degree $k$, $\lambda_{j}:=\frac{n-j}{2}+\sum_{k=j+1}^{n}\alpha_{k}$ and $C^{s}_{t}$ denote the monic Gegenbauer polynomials of degree $t$ (i.e., $C^{s}_{t}:=J^{s-1/2,s-1/2}_{t}$ and $J_{t}^{\alpha,\beta}$ is the monic Jacobi polynomial).

The set $\{\varphi_{\alpha}:\alpha\in\mathbb{N}^{n}\}$ is a dense orthogonal system in $L^{2}({B^{n}},g_{B^{n}})$ and

[TABLE]

where $\|f\|_{a,b}:=\left(\int_{-1}^{1}|f(t)|^{2}(1-t)^{a}(1+t)^{b}dt\right)^{1/2}.$

Note that the density of the linear subspace $\operatorname{span}\{\varphi_{\alpha}:\alpha\in\mathbb{N}^{n}\}$ in $H^{1}({B^{n}},g_{B^{n}})$ follows by Proposition 2.2.

3.2. Proof of Theorem 1

we warn the reader that we will denote throughout this section by $Df$ the *Euclidean gradient of *$f.$

Proof of Theorem 1.

We start showing that $\Delta$ acting on $\mathscr{C}^{2}_{b}(B^{n})$ is a symmetric operator. Namely, for any $u,v\in\mathscr{C}^{2}_{b}(B^{n}),$ we have

[TABLE]

In order to prove this formula we perform two integrations by parts.

[TABLE]

Here $\nu$ is the (euclidean) unit outward normal to $\partial{B^{n}}_{r}:=\{x\in\mathbb{R}^{n}:|x|\leq r\}.$

The proof of (21) is concluded if we show that

[TABLE]

for any $u,v\in\mathscr{C}^{2}_{b}({B^{n}}).$ For, simply observe (see Lemma 2.1) that $\nu$ is an eigenvector of $G_{B^{n}}^{-1}$ of eigenvalue $(\det g)^{-1}|_{|x|=r}=(1-|x|^{2})$, thus we have

[TABLE]

This shows that condition $a)$ of Theorem 4 holds for $\Delta.$ To conclude the proof we need to show that $b)$ holds as well, i.e., there exists a $L^{2}(B^{n},g_{B^{n}})$-orthogonal system in $\mathscr{C}^{2}_{b}(B^{n})$ dense in $L^{2}(B^{n},g_{B^{n}})$ made of eigenfunctions of $\Delta$ such that the corresponding eigenvalues are a positive diverging sequence. We claim that such an orthogonal system is, indeed $\{\varphi_{\alpha},\alpha\in\mathbb{N}^{n}\},$ see Proposition 3.1.

For the sake of readability we present here the case $n=2$, which leads to slightly easier notation and computations with respect to the general one. However, all the elements of the proof of the general case are presented in such a simplified exposition. To easy the notation we denote $B^{2}$ by $B.$

The orthogonal basis of Proposition 3.1 reads as

[TABLE]

where we denoted by $J_{m}^{\alpha,\beta}$ the $m$-th Jacobi orthogonal polynomial with respect to $(1-x)^{\alpha}(1+x)^{\beta}.$ We need to verify

[TABLE]

Since $\varphi_{s,k}$ are elements of $\mathscr{C}^{\infty}_{b}(B)$ we can use the above weak formulation (21) to get

[TABLE]

Let us introduce a change of variables

[TABLE]

We denote by $J\psi$ the Jacobian matrix of $\Psi$ so we get

[TABLE]

Note that not only $\Psi$ is a change of variables that diagonalizes $G_{B}^{-1}$, also it has the property of giving to the basis functions $\varphi_{s,k}$ a tensor product structure. Indeed we have $\varphi_{s,k}\circ\Psi(x,z)=(1-x^{2})^{k/2}J_{s-k}^{k,k}(x)T_{k}(z)$, thus

[TABLE]

It is well known that

[TABLE]

Also one has $T_{k}^{\prime}=kU_{k-1},$ where $U_{k}$ are orthogonal Chebyshev ppolynomials of the second kind, i.e.,

[TABLE]

Using such orthogonality and differentiation relations in the above computation we get

[TABLE]

Now we note that

[TABLE]

integration by parts in the second term leads to

[TABLE]

We plug this last identity in (22) so we get

[TABLE]

The last term in the sum vanishes for any $m\neq s$, this follows from the orthogonality of Jacobi polynomials. When instead $m=s$ we have (see for instance [31])

[TABLE]

For the first term, we recall that $\frac{d}{dx}J_{s-k}^{k,k}=\frac{s+k+1}{2}J_{s-k-1}^{k+1,k+1}$, Hence, using again the orthogonality, we get

[TABLE]

We finally computed

[TABLE]

Here the last line is due to Proposition 3.1. ∎

3.3. Orthogonal polynomials in $L^{2}_{\mu_{S^{n}}}$

Proposition 3.2** ([32]).**

Let us set for any $\alpha\in\mathbb{N}^{n}$ and $x\in S^{n}$

[TABLE]

where $J_{m}^{a,b}$ is the $m$-th Jacobi polynomial of parameters $a,b$ and

[TABLE]

*The set $\{\psi_{\alpha}:\alpha\in\mathbb{N}^{n}\}$ is a dense orthogonal system in $L^{2}(S^{n},g_{S^{n}}).$ *

This result (see Th. 8.2.2 in [32]) plays a key role in our proof.

Theorem 5** ([32]).**

Let us introduce the differential operator

[TABLE]

Then we have

[TABLE]

3.4. Proof of Theorem 2

Proof of Theorem 2.

Let us introduce, see Figure 2, the following notations for $\varepsilon>0$

[TABLE]

Also let $\nu_{i}$ be the Euclidean unit normal to $T^{n,i}_{\varepsilon}$ (for any $\varepsilon>0$). We note that $\partial S^{n}_{\varepsilon}=\cup_{j=0}^{n}T^{n,i}_{\varepsilon}.$

Following the first part of the proof of Theorem 1, we show that $\Delta$ is a symmetric operator on the space $\mathscr{C}^{\infty}_{b}(S^{n})$ which is dense (see Proposition 2.2) in $H^{1}(S^{n},g_{S^{n}}).$

To this aim we perform integration by parts twice. Let $u,v\in\mathscr{C}^{\infty}_{b}(S^{n}),$ then

[TABLE]

Thus we need to prove that for any $u,v\in\mathscr{C}^{\infty}_{b}(S^{n})$ and any $i\in\{0,1,\dots,n\}$ we have

[TABLE]

For, it is sufficient to notice (using Lemma 2.2) that for any $x\in T^{n,0}_{\varepsilon}$

[TABLE]

and for any $x\in T^{n,i}_{\varepsilon}$, $i=1,2,\dots,n$

[TABLE]

Therefore we have

[TABLE]

and, for any $i=1,2,\dots,n$

[TABLE]

and thus (26) holds true. This shows that $\Delta$ is a symmetric operator on $\mathscr{C}^{\infty}_{b}(S^{n}),$ i.e., for any such $u$ and $v$

[TABLE]

Now we want to show that $\Delta$ has discrete spectrum $\sigma(\Delta_{S})=\{\lambda_{s}:=s(s+\frac{n-1}{2}):s\in\mathbb{N}\}$ and the eigen-space of $\lambda_{s}$ is $\operatorname{span}\{\psi_{\alpha},|\alpha|=s\}$ (see Proposition 3.2).

Instead of proving this directly, we rely on the known properties of the basis $\{\psi_{\alpha}\}$, namely (24), and we simply show that for smooth functions

[TABLE]

this allows us to characterize $\sigma(\Delta)$ due to Theorem 5. Then we apply Theorem 4 and the thesis follows.

We introduce the notation $h(x):=(1-\sum_{k=1}^{n}x_{k})\prod_{k=1}^{n}x_{k}.$ It is worth to note that

[TABLE]

For any smooth $f$ we have

[TABLE]

∎

3.5. Proof of Proposition 2.1

Let us first recall a result of Masamune [46, Th. 3] which the proof of Proposition 2.1 relies on. Assume $(M,g)$ to be a compact Riemaniann manifold and let $\Sigma$ be a submanifold of $M,$ let us define $\Delta_{M}$ as the standard Laplace Beltrami operator acting on $\mathscr{C}^{\infty}_{c}(M\setminus\Sigma).$ Then

[TABLE]

Proof of Proposition 2.1.

Let $M:=S^{n}\subset\mathbb{R}^{n+1}$ and $\Sigma:=\{x\in M:x_{n+1}=0\}.$ Also introduce the notation $(x_{1},x_{2},\dots,x_{n},x_{n+1})=(\xi,x_{n+1}).$

Let us assume by contradiction that $\mathscr{C}^{\infty}_{c}(B^{n})$ is dense in $H^{1}(B^{n},g_{B^{n}}).$ In view of the proof of Proposition 2.2 we have

[TABLE]

Here $\mathscr{C}^{\infty}_{c,\text{odd}}(M\setminus\Sigma)$ denotes the subspace

[TABLE]

and $\mathscr{C}^{\infty}_{c,\text{even}}(M\setminus\Sigma)$ is defined similarly. Note that, given $u\in\mathscr{C}^{\infty}_{c}(M\setminus\Sigma)$ we can define $u_{\text{even}}:=1/2(u(\xi,x_{n+1})+u(\xi,-x_{n+1}))\in\mathscr{E}_{1}$ and $u_{\text{odd}}:=1/2(u(\xi,x_{n+1})-u(\xi,-x_{n+1}))\in\mathscr{E}_{2}$ such that $u=u_{\text{even}}+u_{\text{odd}}$.

The assumption $\mathscr{C}^{\infty}_{c}(B^{n})$ dense in $H^{1}(B^{n},g_{B^{n}})$ together with Theorem 1 and the isometry property of the map $E$ in the proof of Proposition 2.2 implies that the Laplace Beltrami operator $\Delta_{1}$ acting on $\mathscr{E}_{1}$ and $\Delta_{2}$ acting on $\mathscr{E}_{2}$ are essentially self-adjoint. Moreover, since $\Delta_{M}u=\Delta_{1}u_{\text{even}}+\Delta_{2}u_{\text{odd}}$ for any $u\in\mathscr{C}^{\infty}_{c}(M\setminus\Sigma),$ it follows that $\Delta_{M}$ itself is essentially self-adjoint.

On the other hand, $\dim\Sigma=n-1$ and $\dim M=n$, this is in contrast with Masamune’s result (29) and thus $\mathscr{C}^{\infty}_{c}(B^{n})$ can not be dense in $H^{1}(B^{n},g_{B^{n}})$ and thus $H^{1}(B^{n},g_{B^{n}})\neq H^{1}_{0}(B^{n},g_{B^{n}}).$ Note that, in view of [45, Th. 1], this is equivalent to the fact that $B^{n}$ is not a manifold with almost polar boundary.

The proof for the simplex can be done in a equivalent way but using the map $F$ defined in the proof of Proposition 2.2 instead of the map $E.$ ∎

Appendix A Pluripotential Theory on the complexified sphere and spherical harmonics

In this section we consider $\mathbb{S}^{n-1}$ as a compact subset of the complexified sphere $\mathcal{S}^{n-1}:=\{z\in\mathbb{C}^{n}:\sum z_{i}^{2}=1\}.$ We can consider the space $\operatorname*{psh}(\mathcal{S}^{n-1})$ of plurisubharmonic functions on the complex manifold $\mathcal{S}^{n-1}$ and form the usual upper envelope

[TABLE]

defining the extremal plurisubharmonic function; compare this definition with equation (6). This is a locally bounded plurisubharmonic function which is maximal on $\mathcal{S}^{n-1}\setminus\mathbb{S}^{n-1};$ [14, 48].

On the other hand, it is clear that $\mathcal{S}^{n-1}$ is a irreducible algebraic sub-variety of $\mathbb{C}^{n}$ of pure dimension $n-1,$, hence we can use the result of Sadullaev [30] to get

[TABLE]

Here $\mathscr{P}$ denotes the space of algebraic polynomials with complex coefficients. It is worth to stress that here $\deg$ denotes the degree of a polynomial on $\mathbb{C}^{n}$, not the degree over the coordinate ring of $\mathcal{S}^{n-1}.$

The operator $(\operatorname*{dd^{c}})^{n-1}$ maps the space of locally bounded plurisubharmonic functions on positive Borel measures [8, 48] and

[TABLE]

is a probability measure, termed the pluripotential equilibrium measure of $\mathbb{S}^{n-1}.$

The invariance of the couple $\mathbb{S}^{n-1},\mathcal{S}^{n-1}$ and the operator $(\operatorname*{dd^{c}})^{n-1}$ under complex rotations can be used to show that $\mu_{\mathbb{S}^{n-1},\mathcal{S}^{n-1}}$ is indeed absolutely continuous with respect to the standard volume measure of $\mathbb{S}^{n-1}$ with constant density. Since $\mu_{\mathbb{S}^{n-1},\mathcal{S}^{n-1}}$ by definition has total mass $1$ its density with respect to the standard volume form is $c_{n-1}:=(2\pi^{n/2}/\Gamma(n/2))^{-1}.$

In [49, Prop. 4.1] authors prove the formula

[TABLE]

we note that this function can be used to define the Baran metric on the sphere, due to the following differentiability property.

Lemma A.1**.**

Let $x\in\mathbb{S}^{n-1},$ the function $V_{\mathbb{S}^{n-1}}(\cdot,\mathcal{S}^{n-1})$ has right tangent directional derivative at $x$ in any direction $i\cdot v$, for any $v\in T_{x}\mathbb{S}^{n-1},$ that is

[TABLE]

where $\gamma:[0,1]\mapsto\mathcal{S}^{n-1}$ is any differentiable arc with $\gamma(0)=x$, $\gamma^{\prime}(0^{+})=i\cdot v.$

Moreover we have $\partial_{i\cdot v}V_{\mathbb{S}^{n-1}}(x,\mathcal{S}^{n-1})=|v|.$

Proof.

The problem is clearly rotation independent. We can thus assume $x=(1,0,\dots,0)=e_{1}$ and $v=|v|(0,1,0,\dots,0)=|v|e_{2}$ without loss of generality.

Let us introduce the arc

[TABLE]

It is easy to verify that $z$ enjoys the properties

[TABLE]

Thus we are left to show that, setting $u(t):=V_{\mathbb{S}^{n-1}}^{*}(z(t),\mathcal{S}^{n-1}),$ we have $\frac{d}{dt}u(0^{+})=|v|.$

Let us note first that $|z(t)|^{2}=1+2|v|^{2}\log^{2}(1+t)$, then we can compute

[TABLE]

Therefore

[TABLE]

∎

Due to Lemma A.1 we can define the Baran metric on the real unit sphere by setting

[TABLE]

note the analogy of the partial derivative taken in the Lemma with the definition of the Baran metric in the standard ”flat” case.

Using the Parallelogram Identity we can define for any $x\in\mathbb{S}^{n-1}$ and any $u,v\in T_{x}\mathbb{S}^{n-1}$ the scalar product related to the Baran metric as

[TABLE]

that turns out to coincide with the standard (round) metric.

It is very well known that the Laplace Beltrami operator on the real unit sphere (endowed with the round metric) has a discrete diverging set eigenvalues and its eigenfunctions are polynomials: the spherical harmonics.

These observations lead automatically to the desired conclusion that we state as a corollary.

Corollary A.1**.**

The eigenfunctions of the Laplace Beltrami operator with respect to the Baran metric on the real unit sphere are the orthogonal polynomials with respect to the pluripotential equilibrium measure $\mu_{\mathbb{S}^{n-1},\mathcal{S}^{n-1}}$ of the real unit sphere $\mathbb{S}^{n-1}$ in the complexified sphere $\mathcal{S}^{n-1}.$