\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 84, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/84\hfil Grushin fundamental solutions] {Fundamental solutions to the $p$-Laplace equation in a class of Grushin vector fields} \author[T. Bieske \hfil EJDE-2011/84\hfilneg] {Thomas Bieske} \address{Thomas Bieske \newline Department of Mathematics\\ University of South Florida\\ Tampa, FL 33620, USA} \email{tbieske@math.usf.edu} \thanks{Submitted December 10, 2010. Published June 29, 2011.} \subjclass[2000]{35H20, 53C17, 17B70} \keywords{Grushin-type spaces; $p$-Laplacian} \begin{abstract} We find the fundamental solution to the $p$-Laplace equation in a class of Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \newcommand{\ip}[2]{\langle #1,#2\rangle} \section{Motivation} The $p$-Laplace equation is the model equation for nonlinear potential theory. The Euclidean results of \cite{HKM:NPT} can be extended into a class of sub-Riemannian spaces possessing an algebraic group law, called Carnot groups \cite{HH:QR}. Fundamental solutions to the $p$-Laplace equation in a subclass of Carnot groups called groups of Heisenberg-type have been found in \cite{CDG, HH:QR}. The exploration of the $p$-Laplace equation in sub-Riemannian spaces without an algebraic group law is currently a topic of interest. In this paper, we will find the fundamental solution to the $p$-Laplace equation for $1< p < \infty$ in a class of Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. \section{Grushin-type spaces} Before presenting the main theorem, we recall the construction of such spaces and their main properties. We begin with $\mathbb{R}^n$, possessing coordinates $(x_1,x_2,\dots,x_n)$ and vector fields $$X_i = \rho_i (x_1,x_2,\dots,x_{i-1}) \frac{\partial}{\partial x_i}$$ for $i=2,3,\dots, n$ where $\rho_i (x_1,x_2,\dots,x_{i-1})$ is a (possibly constant) real-valued function. We decree that $\rho_1 \equiv 1$ so that $$X_1=\frac{\partial}{\partial x_1}\,.$$ A quick calculation shows that when $i < j$ and $\rho_j (x_1,x_2,\dots,x_{j-1})$ is differentiable, the Lie bracket is given by $$\label{bracket} X_{ij} \equiv [X_i,X_j]= \rho_i (x_1,x_2,\dots,x_{i-1}) \frac{\partial \rho_j (x_1,x_2,\dots,x_{j-1})}{\partial x_i} \frac{\partial }{\partial x_j}.$$ If the $\rho_i$'s are polynomials, at each point there is a finite number of iterations of the Lie bracket so that $\frac{\partial}{\partial x_i}$ has a non-zero coefficient. This is easily seen for $X_1$ and $X_2$, and the result is obtained inductively for $X_i$. (It is noted that the number of iterations necessary is a function of the point.) It follows that H\"{o}rmander's condition is satisfied by such vector fields. We may further endow $\mathbb{R}^n$ with an inner product (singular where the polynomials vanish) so that the collection $\{X_i\}$ forms an orthonormal basis. This produces a sub-Riemannian manifold that we shall call $g_n$, which is also the tangent space to a generalized Grushin-type space $G_n$. Points in $G_n$ will also be denoted by $x=(x_1,x_2,\dots, x_n)$. Though $G_n$ is not a Lie group, it is a metric space with the natural metric being the Carnot-Carath\'{e}odory distance, which is defined for points $x$ and $y$ as follows: \begin{equation*} d_C(x,y)= \inf_{\Gamma} \int_{0}^{1} \| \gamma '(t) \|\,dt\,. \end{equation*} Here $\Gamma$ is the set of all curves $\gamma$ such that $\gamma (0) = x, \gamma (1) = y$ and $$\gamma '(t) \in \operatorname{span} \{\{X_i(\gamma(t))\}_{i=1}^n\} .$$ In the case of polynomials, Chow's theorem (see, for example, \cite{BR:SRG}) states any two points can be joined by such a curve. In the non-polynomial case, one can explicitly construct horizontal curves of finite length connecting any two points. This means $d_C(x,y)$ is an honest metric. Using this metric, we can define a Carnot-Carath\'{e}odory ball of radius $r$ centered at a point $x_0$ by $$B_C(x_0,r)=\{p\in G_n : d_C(x,x_0) < r\}$$ and similarly, we shall denote a bounded domain in $G_n$ by $\Omega$. The Carnot-Carath\'{e}odory metric behaves differently at points where the functions $\rho_i$ vanish. Fixing a point $x_0$ where the $\rho_i$ are sufficiently differentiable, consider the $n$-tuple $r_{x_0}=(r^1_{x_0},r^2_{x_0},\dots,r^n_{x_0})$ where $r^i_{x_0}$ is the minimal number of Lie bracket iterations required to produce $$[X_{j_1},[X_{j_2},[\cdots[X_{j_{r^i_{x_0}}},X_i]\cdots](x_0) \neq 0\,.$$ Note that though the minimal length is unique, the iteration used to obtain that minimum is not. Note also that $$\rho_i(x_0) \neq 0 \leftrightarrow r^i_{x_0}=0\,.$$ Using \cite[Theorem 7.34]{BR:SRG} we obtain the local estimate at $x_0$ $$\label{distest} d_C(x_0,x) \sim \sum _{i=1}^n |x_i-x_i^0|^\frac{1}{1+r^i_{x_0}} .$$ Given a smooth function $f$ on $G_n$, we define the horizontal gradient of $f$ as $$\nabla_0f(x) = (X_1f(x),X_2f(x),\dots, X_nf(x))$$ and the symmetrized second order (horizontal) derivative matrix by $$((D^2f(x))^{\star})_{ij} = \frac{1}{2} (X_iX_jf(x)+X_jX_if(x))$$ for $i,j=1,2,\dots n$. \begin{definition} \label{def1} \rm The function $f: G_n \to \mathbb{R}$ is said to be $C_{\rm sub}^1$ if $X_if$ is continuous for all $i=1,2,\dots,n$. Similarly, the function $f$ is $C_{\rm sub}^2$ if $X_iX_jf$ is continuous for all $i,j=1,2,\dots,n$. \end{definition} \begin{remark} \label{rmk1} \rm We note that Euclidean $C^1$ functions are $C_{\rm sub}^1$ functions, but the class of $C_{\rm sub}^1$ functions is larger than the class of Euclidean $C^1$ functions. For example, when $n=2$ and $\rho_2(x_1)=x_1$, we have $\sqrt{x_2}$ is $C_{\rm sub}^1$ at the origin, but is clearly not Euclidean $C^1$ at the origin. The interested reader is directed to \cite{BR:SRG} for a more complete discussion. \end{remark} Using these derivatives, we consider two main operators on $C_{\rm sub}^2$ functions called the $p$-Laplacian $$\Delta_{p}f=\operatorname{div}(\|\nabla_0f\|^{p-2}\nabla_0f)=\sum_{i=1}^nX_i (\|\nabla_0f\|^{p-2}X_if)$$ defined for $1 < p < \infty$ and the infinite Laplacian $\Delta_{\infty}f = \sum_{i,j=1}^n X_ifX_jfX_iX_jf = \ip{\nabla_0 f}{(D^2f)^{\star}\nabla_0 f}.$ For a more in-depth study of Grushin-type spaces, the reader is directed to \cite{BR:SRG,B:GG,B:RV} and the references therein. \section{The co-area formula and measure theory} We begin by fixing $m,n \in \mathbb{N}$ and $k,c \in \mathbb{R}$ so that $m1$, we may apply the Co-Area Formula to the well-defined function $g = \begin{cases} (\|\nabla_0 \psi\|^p/ \|\nabla \psi\|)\cdot \|\nabla \psi\| & \|\nabla \psi\| \neq 0 \\ 0 & \|\nabla \psi\| = 0 \end{cases}$ to obtain the following result. \begin{proposition} \label{prop3.3} With the hypotheses as above, let $\mathcal{V}$ be an absolutely continuous measure to $\mathcal{L}_n$ with Radon-Nikodym derivative $\|\nabla_0 \psi\|^p = [d\mathcal{V} / d\mathcal{L}_n]$. Then for sufficiently small $R > 0$, $$\label{coarea2} \mathcal{V}(\mathcal{B}_R) = \int_{\mathcal{B}_R} d\mathcal{V} = \int_0^R \int_{\partial \mathcal{B}_r} \frac{\|\nabla_0 \psi\|^p}{\|\nabla \psi\|} \, d\mathcal{H} dr$$ \end{proposition} In light of the equality in \eqref{coarea2}, we see that the measure space $(G_n, \mathcal{V})$ is globally Ahlfors $Q$-regular with respect to balls centered at $x_0$. In particular, for $R > 0$, $$\label{ahlfors} \mathcal{V}(\mathcal{B}_R) = \sigma_p R^Q$$ where $Q = m+(k+1)(n-m)=k(n-m)+n$ and $\sigma_p = \mathcal{V}(\mathcal{B}_1)$ is a fixed positive constant. For technical purposes we proceed to study the boundary behavior of precompact domains $\Omega$. This now motivates the following definition. \begin{definition} \label{def2} \rm For small values $R>0$, define a measure $\mathcal{S}$ on $\partial \mathcal{B}_R$ as $$\mathcal{S}(\partial \mathcal{B}_R) = \int_{\partial \mathcal{B}_R} d\mathcal{S} = \int_{\partial \mathcal{B}_R} \frac{\|\nabla_0 \psi\|^p}{\|\nabla \psi\|} \,d\mathcal{H}.$$ \end{definition} In particular, $S$ is absolutely continuous with respect to the Hausdorff $(n-1)$-measure $\mathcal{H}$. Using previous results, in particular, the fact that $\psi$ is smooth away from $x_0$, we now conclude: \begin{corollary} \label{coro3.4} \begin{enumerate} \item $S$ is locally Ahlfors $(Q-1)$-regular and $$\label{ahlfors2} \mathcal{S}(\partial \mathcal{B}_R) = Q \sigma_p R^{Q-1}.$$ \item Let $\varphi$ be a continuous and integrable function on $\mathcal{B}_R$. Then as $R \to 0$, $$\label{density} \frac{R^{1-Q}}{Q\sigma_p} \int_{\partial \mathcal{B}_R} \varphi \,d\mathcal{S} \; \to \; \varphi(x_0)$$ \end{enumerate} \end{corollary} \begin{proof}[Sketch of Proof] Equation \eqref{ahlfors2} follows immediately from differentiating both Equations \eqref{coarea2} and \eqref{ahlfors}. Since $\mathcal{S}$ is absolutely continuous with respect to Hausdorff $(n-1)$-measure $\mathcal{H}$, it follows that $\mathcal{S}$ is Borel regular. As a result, Equation \eqref{density} is the analogue of the Lebesgue Density Theorem. \end{proof} \section{The $p$-Laplace equation} In this section, we compute the fundamental solution of the $p$-Laplacian for the previously-defined vector fields \eqref{vfds} and for $10$ and letting $\varepsilon \to 0$. However, we shall proceed formally. Suppressing the variables $(x_1, x_2, \dots , x_n)$, and letting $$\Sigma=\sum_{i=1}^m(x_i-a_i)^{2}$$ we compute for $p \neq Q$: \begin{gather*} X_if = \alpha h^{w-1}c^2(x_i-a_i)\Sigma^{k}\ \textmd{for $i=1,2,\dots, m$}\\ X_{j}f = c \alpha h^{w-1}\Sigma^{\frac{k}{2}}(k+1)(x_{j}-b_j)\ \textmd{for $j=m+1,\dots, n$}\\ \|\nabla_0f\|^2 = c^2 \alpha^2h^{2w-1}\Sigma^{k}\\ \|\nabla_0f\|^{p-2} = |c\alpha|^{p-2}h^{w(p-2)-\frac{p-2}{2}}\Sigma^{\frac{k(p-2)}{2}}. \end{gather*} We then are able to compute, for $i=1,2,\dots, m$, $\|\nabla_0f\|^{p-2}X_if = \alpha|\alpha|^{p-2}|c|^p h^{w(p-1)-\frac{p}{2}} \Sigma^{\frac{kp}{2}}(x_i-a_i)$ and for $j=m+1,m+2,\dots, n$, \begin{equation*} \|\nabla_0f\|^{p-2}X_{j}f = \alpha|\alpha|^{p-2}c|c|^{p-2} h^{w(p-1)-\frac{p}{2}} \Sigma^{\frac{kp}{2}-\frac{k}{2}}(k+1)(x_{j}-b_j). \end{equation*} Setting $$D_p \equiv \frac{\Delta_pf}{\alpha |\alpha|^{p-2}|c|^p}\quad\text{and}\quad \Upsilon=w(p-1)-\frac{p}{2}$$ we can then compute \begin{align*} D_p &= \sum_{i=1}^m h^{\Upsilon}\Sigma^{\frac{kp}{2}}+ \sum_{i=1}^m h^{\Upsilon}(kp)\Sigma^{\frac{kp-2}{2}}(x_i-a_i)^2\\ &\quad + \sum_{i=1}^m \Upsilon h^{\Upsilon-1}2c^2(k+1) \Sigma^{\frac{kp}{2}+k}(x_i-a_i)^2\\ &\quad+\sum_{j=m+1}^n h^{\Upsilon}(k+1)\Sigma^{\frac{kp}{2}}+ \sum_{j=m+1}^n 2\Upsilon h^{\Upsilon-1}\Sigma^{\frac{kp}{2}}(k+1)^3 (x_{j}-b_j)^2\\ &= h^{\Upsilon-1}\Sigma^{\frac{kp}{2}} \bigg( mh+(kp)h+(k+1)(n-m)h\\ &\quad+ \big(\alpha(p-1)-p(k+1)\big) \Big(c^2\Sigma^{k+1}+(k+1)^2 \sum_{j=m+1}^n(x_{j}-b_j)^2\Big)\bigg)\\ &= h^{\Upsilon}\Sigma^{\frac{kp}{2}}\Big( \alpha(p-1)-p(k+1)+m+(kp)+(k+1)(n-m)\Big)\\ &= h^{\Upsilon}\Sigma^{\frac{kp}{2}}((p-Q)-p+Q)=0. \end{align*} Note that these computations are valid wherever the function $f$ is smooth and in particular, these are valid away from the point $x_0$. We next note that $$\|\nabla_0f\|^{p-1} \sim \psi^{1-Q}$$ and so we conclude that $\|\nabla_0f\|^{p-1}$ is locally integrable on $G_n$ with respect to Lebegue measure. We then consider $\phi \in C^{\infty}_0$ with compact support in the ball $$\mathcal{B}_R = \{x: \psi(x) < R \}.$$ Let $0 < r < R$ be given so that $\mathcal{B}_r \subset \mathcal{B}_R$. In the annulus $\mathcal{A} := \mathcal{B}_R \setminus \overline{\mathcal{B}_r}$ we have, via the Leibniz rule, \begin{align*} \operatorname{div}(\phi \|\nabla_0f\|^{p-2}\nabla_0f) &= \phi \operatorname{div}(\|\nabla_0f\|^{p-2}\nabla_0f) + \|\nabla_0f\|^{p-2}\ip{\nabla_0f}{\nabla_0\phi} \\ &= 0 + \|\nabla_0f\|^{p-2}\ip{\nabla_0f}{\nabla_0\phi}. \end{align*} Let $\mathcal{L}_n$ and $\mathcal{H}$ be the measures from \eqref{coarea} and recall $$\Sigma\equiv \sum_{j=1}^m(x_i - a_i)^2.$$ Applying Stokes' Theorem, \begin{align*} & \int_{\mathcal{A}}\|\nabla_0f\|^{p-2} \ip{\nabla_0f}{\nabla_0\phi}d\mathcal{L}_n \\ &= \int_{\mathcal{A}}\operatorname{div}(\phi \|\nabla_0f\|^{p-2}\nabla_0f)d\mathcal{L}_n \\ &=\int_{\mathcal{A}} \Big( \sum_{i=1}^m X_i[\phi \|\nabla_0f\|^{p-2}X_if] + c \Sigma^{\frac{k}{2}} \sum_{j=m+1}^n\frac{\partial}{\partial x_{j}}\Big(\phi \|\nabla_0f\|^{p-2}X_{j}f\Big) \Big) d\mathcal{L}_n \\ &=\int_{\mathcal{A}} \Big( \sum_{i=1}^m X_i[\phi \|\nabla_0f\|^{p-2}X_if] + \sum_{j=m+1}^n \frac{\partial}{\partial x_{j}}\Big( c \Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2}X_{j}f\Big) \Big) d\mathcal{L}_n \\ &=\int_{\mathcal{A}} \operatorname{div}_{\operatorname{eucl}} \begin{bmatrix} \phi \|\nabla_0f\|^{p-2}X_1f \\ \vdots \\ \phi \|\nabla_0f\|^{p-2}X_mf \\ c \Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2}X_{m+1}f \\ \vdots \\ c \Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2}X_n f \end{bmatrix} d\mathcal{L}_n \\ &=\int_{\partial\mathcal{A}} \frac{1}{\|\nu\|} \Big( \phi \|\nabla_0f\|^{p-2}\sum_{i=1}^mX_if \nu_i + c\Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2} \sum_{j=m+1}^nX_{j}f \nu_{j} \Big) d\mathcal{H} \\ &=-\int_{\partial \mathcal{B}_r} \frac{1}{\|\nu\|} \Big( \phi \|\nabla_0f\|^{p-2}\sum_{i=1}^mX_if \nu_i + c\Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2} \sum_{j=m+1}^nX_{j}f \nu_{j} \Big) d\mathcal{H} \end{align*} where $\nu$ is the outward Euclidean normal of $\mathcal{A}$. Recalling that $$\psi(x_1, x_2, \dots, x_n) = [h(x_1, x_2, \dots, x_n)]^{1/(2k+2)},$$ and that $\nu_j = \frac{\partial\psi}{\partial x_j}$, we may proceed with the computation, \begin{align*} & \int_{\mathcal{A}}\|\nabla_0f\|^{p-2} \ip{\nabla_0f}{\nabla_0\phi}d\mathcal{L}_n \\ &= -\int_{\partial \mathcal{B}_r} \frac{\alpha \psi^{\alpha-1}}{\|\nu\|} \phi \|\nabla_0\psi^\alpha\|^{p-2} \Big(\sum_{i=1}^m(\frac{\partial\psi}{\partial x_i})^2 + c^2 \Sigma^k \sum_{j=m+1}^n(\frac{\partial\psi}{\partial x_{j}})^2 \Big) d\mathcal{H} \\ &= -\int_{\partial \mathcal{B}_r} \frac{\alpha \psi^{\alpha-1}}{\|\nu\|} \phi \|\nabla_0\psi\|^{p-2}|\alpha|^{p-2} \psi^{(p-2)(\alpha-1)} \Big( \| \nabla_0\psi\|^2 \Big) d\mathcal{H} \\ &= -\int_{\partial \mathcal{B}_r} \frac{|\alpha|^{p-2}\alpha \psi^{(p-1)(\alpha-1)}}{\|\nu\|} \phi \|\nabla_0\psi\|^p d\mathcal{H}. \end{align*} Recall that by definition, $\psi \equiv r$ on $\partial \mathcal{B}_r$. We then have $\int_{\mathcal{A}}\|\nabla_0f\|^{p-2} \ip{\nabla_0f}{\nabla_0\phi} d\mathcal{L}_n = - |\alpha|^{p-2} \alpha r^{1-Q} \int_{\partial \mathcal{B}_r} \frac{\phi \|\nabla_0\psi\|^p}{\|\nu\|} \, d\mathcal{H}.$ Letting $r \to 0$, we apply \eqref{density} and obtain $$\label{scaling} \int_{\mathcal{A}}\|\nabla_0f\|^{p-2} \ip{\nabla_0f}{\nabla_0\phi} d\mathcal{L}_n \; \to \; - |\alpha|^{p-2} \alpha (Q\sigma_p) \phi(x_0).$$ We then obtain the case for $p \neq Q$. The case of $p=Q$ is similar and left to the reader. \end{proof} It was shown in \cite{B:GG} and \cite{B:RV} that in Grushin-type spaces, viscosity infinite harmonic functions are limits of weak $p$-harmonic functions as $p$ tends to infinity. This motivates the following corollary. \begin{corollary} \label{coro4.2} The function $\psi$, as defined above, is infinite harmonic in the space $G_n \setminus \{ x_0 \}$. \end{corollary} \begin{proof} We use the formula that for a smooth function $u$, $$\Delta_{\infty}u=\frac{1}{2}\nabla_0u \cdot \nabla_0\|\nabla_0u\|^2.$$ Computing as in the proof of the Theorem, we have $$\|\nabla_0\psi\|^2 = c^2\Sigma^k h^\frac{-2k}{2k+2}.$$ Thus we obtain for $i=1,2,\dots, m$, $X_i\|\nabla_0\psi\|^2 = 2kc^2h^{\frac{-2k}{2k+2}-1}\Sigma^{k-1}(x_i-a_i) \big(h-c^2 \Sigma^{k+1}\big)$ and for $j=m+1,m+2,\dots, n$, \begin{align*} X_{j}\|\nabla_0\psi\|^2 = -2kc^3 h^{\frac{-2k}{2k+2}-1}\Sigma^{\frac{3k}{2}}(k+1)(x_{j}-b_j) \end{align*} so that using the derivatives as in the proof of the Theorem, \begin{align*} \Delta_{\infty}\psi &= \sum_{i=1}^m 2kc^4 h^{\frac{-4k-1}{2k+2}-1}\Sigma^{2k-1}(x_i-a_i)^2 \big(h-c^2 \Sigma^{k+1}\big)\\ & \quad - 2kc^4h^{\frac{-4k-1}{2k+2}-1}(k+1)^2\Sigma^{2k}\sum_{j=m+1}^n(x_{j}-b_j)^2\\ &= 2kc^4h^{\frac{-4k-1}{2k+2}-1}\Sigma^{2k} \Big(h-c^2 \Sigma^{k+1}-(k+1)^2\sum_{j=m+1}^n(x_{j}-b_j)^2\Big)\\ &= 2kc^4h^{\frac{-4k-1}{2k+2}-1}\Sigma^{2k}\times (0). \end{align*} \end{proof} Using the existence-uniqueness of viscosity infinite harmonic functions \cite{B:GG,B:RV} and the fact that absolute minimizers in Grushin spaces are viscosity infinite harmonic functions and enjoy comparison with cones \cite{BDM:BDM}, we conclude the following corollary. \begin{corollary} \label{coro4.3} Let \$0 Q. \end{cases} \] \end{theorem} \subsection*{Acknowledgements} The author wishes to thank the anonymous referee for his/her careful reading of this manuscript and for the helpful suggestions to improve its readability and clarity. \begin{thebibliography}{99} \bibitem{BR:SRG}Bella\" {\i}che, Andr\' {e}. The Tangent Space in Sub-Riemannian Geometry. In \emph{Sub-Riemannian Geometry}; Bella\" {\i}che, Andr\' {e}., Risler, Jean-Jacques., Eds.; Progress in Mathematics; Birkh\" {a}user: Basel, Switzerland. 1996; Vol. 144, 1--78. \bibitem{B:GG}Bieske, Thomas. Lipschitz Extensions on Generalized Grushin Spaces. Mich. Math. J., \textbf{2005}, 53 (1), 3--31. \bibitem{B:RV}Bieske, Thomas. Properties of Infinite Harmonic Functions on Grushin-type Spaces. Rocky Mtn J. of Math. \textbf{2009}, 39 (3), 729--756. \bibitem{BG} Bieske, Thomas.; Gong, Jasun. The p-Laplace Equation on a class of Grushin-type Spaces. Proc. Amer. Math. Soc. 2006, 134:12, 3585--3594. \bibitem {BDM:BDM} Bieske, Thomas.; Dragoni, Federica.; Manfredi, Juan. The Carnot-Carath\'{e}odory distance and the infinite Laplacian J. of Geo. Anal. 2009, \emph{19} (4), 737--754. \bibitem{CDG} Capogna, Luca.; Danielli, Donatella.; Garofalo, Nicola. Capacitary Estimates and Subelliptic Equations. Amer. J. of Math. 1996, 118:6, 1153--1196. \bibitem{C:C} Chavel, Isaac. \emph{Eigenvalues in Riemannian Geometry}; Academic Press: Orlando, 1984. \bibitem{HH:QR}Heinonen, Juha.; Holopainen, Ilkka. Quasiregular Maps on Carnot Groups. J. of Geo. Anal. 1997, 7:1, 109--148. \bibitem{HKM:NPT} Heinonen, Juha.; Kilpel\"{a}inen, Tero.; Martio, Olli. \emph{Nonlinear Potential Theory of Degenerate Elliptic Equations}; Oxford Mathematical Monographs; Oxford University Press: New York, 1993. \bibitem{MSC:co}Monti, Roberto.; Serra Cassano, Francesco. Surfaces Measures in Carnot-Carath\'{e}odory Spaces. Calc. Var. PDE, 2001, 13 (3), 339--376. \end{thebibliography} \end{document}