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Gauge functions, eikonal equations and Bôcher’s theorem on stratified Lie groups. (English) Zbl 1137.31003
Let $$\mathcal{L}=\sum_1^mX_i^2$$ be a sub-Laplacian on a Carnot group $$G$$ (i.e., $$G=\mathbb{R}^N$$ equipped with a family of dilations $$\{\delta_\lambda\}_{\lambda>0}$$ of the form $$\delta_\lambda(x_1,\ldots,x_N)=(\lambda^{\alpha_1}x_1,\ldots,\lambda^{\alpha_N}x_N),$$ with integers $$1=\alpha_1\leq\ldots\leq\alpha_N$$) and let $$\nabla_{\mathcal{L}}=(X_1,\ldots,X_m)$$ be the intrinsic gradient related to $$\mathcal{L}.$$ Let $$d$$ be a homogeneous norm on $$G.$$ A homogeneous norm $$d$$ is said to be $$\mathcal{L}$$-gauge if there exists a constant $$\gamma\not=0$$ such that $$d^\gamma$$ is $$\mathcal{L}$$-harmonic in $$G\setminus\{0\}.$$
The first main result (Theorem 1.1) says that if $$d$$ is a $$\mathcal{L}$$-gauge satisfying the $$\mathcal{L}$$-eikonal equation $$| \nabla_{\mathcal{L}}d| =1$$ on $$G\setminus\{0\}$$ then $$G$$ is the Euclidean group $$(\mathbb{R}^N,+).$$
Let $$d$$ be a $$\mathcal{L}$$-gauge and define the kernels: $$K=| \nabla_{\mathcal{L}}d| ^2$$ and $$\mathcal{K}=K/| \nabla d| ,$$ where $$\nabla$$ is the standard gradient vector. Put $$m_r(u)=\int_{\partial B_d(0,r)}u\mathcal{K}\,dH^{N-1},$$ where $$H^{N-1}$$ is the $$(N-1)$$-Hausdorff measure in $$\mathbb{R}^N$$ and $$B_d(x,r)$$ is the $$d$$-ball of center $$x$$ and radius $$r,$$ $$B_d(x,r)=\{y\in G:d(x^{-1}y)<r\}.$$ With this notation the authors prove the following result (Theorem 1.5): Let $$d$$ be a $$\mathcal{L}$$-gauge and let $$\gamma$$ be the related exponent. If $$u$$ is $$\mathcal{L}$$-harmonic in the unit punctured ball $$B_d(0,1)\setminus\{0\},$$ then there exist real constants $$a,b$$ such that $$r^{\gamma-1}m_r(u)=ar^\gamma+b$$ for every $$r\in(0,1).$$
As a corollary, if $$d$$ is a $$\mathcal{L}$$-gauge with exponent $$\gamma,$$ then the function $$\Gamma=\beta_dd^{2-Q},$$ where $$Q$$ is a homogeneous dimension of $$G,$$ is the fundamental solution (with pole at $$0$$) for $$\mathcal{L}.$$ The constant $$\beta_d=1/(Q(Q-2)\int_{B_d(0,1)}K).$$
The next main result is Theorem 1.8: Let $$d$$ be a $$\mathcal{L}$$-gauge and let $$w$$ be a nonnegative $$\mathcal{L}$$-harmonic function in $$B_d(0,1)\setminus\{0\}.$$ Assume that $$w$$ is continuous up to the boundary of $$B_d(0,1)$$ and $$w(x)=0$$ for every $$x\in\partial B_d(0,1).$$ Then $$w$$ is radially symmetric. More precisely, there is a real constant $$a>0$$ such that $$w(x)=a(d^{2-Q}(x)-1)$$ for every $$x\in B_d(0,1)\setminus\{0\}.$$
As a corollary, from the above theorem the authors obtain, using the left-invariance of $$\mathcal{L},$$ the Bôcher-type theorem for nonnegative $$\mathcal{L}$$-harmonic functions in the punctured open sets $$\Omega\setminus\{x_0\}\subset G,$$ where $$x_0\in\Omega$$ is fixed.

##### MSC:
 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 35J70 Degenerate elliptic equations 35H20 Subelliptic equations
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##### References:
 [1] Axler S., Bourdon P. and Ramey W. (1992). Bôcher’s Theorem. Am. Math. Mon. 99: 51–55 · Zbl 0758.31002 · doi:10.2307/2324549 [2] Bonfiglioli A. and Lanconelli E. (2001). Liouville-type theorems for real sub-Laplacians. Manuscripta Math. 105: 111–124 · Zbl 1016.35014 · doi:10.1007/PL00005872 [3] Bonfiglioli A. and Lanconelli E. (2003). Subharmonic functions on Carnot groups. Math. Ann. 325: 97–122 · Zbl 1017.31003 · doi:10.1007/s00208-002-0371-z [4] Bonfiglioli A., Lanconelli E. and Uguzzoni F. (2002). Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups. Adv. Differ. Equ. 7: 1153–1192 · Zbl 1036.35061 [5] Constantinescu C. and Cornea A. (1972). Potential Theory on Harmonic Spaces, Die Grundlehren der mathematischen Wissenschaften, Band 158. Springer, New York · Zbl 0248.31011 [6] Folland G.B. (1975). Subelliptic estimates and function spaces on nilpotent groups. Arkiv för Mat. 13: 161–207 · Zbl 0312.35026 · doi:10.1007/BF02386204 [7] Gallardo, L.: Capacités, mouvement brownien et problème de l’épine de Lebesgue sur les groupes de Lie nilpotents. In: Probability measures on groups (Oberwolfach, 1981), pp.$$\sim$$96–120, Lecture Notes in Math., 928, Springer, Berlin (1982) · Zbl 0501.60017 [8] Hörmander L. (1968). Hypoelliptic second-order differential equations. Acta Math. 121: 147–171 · Zbl 0156.10701 [9] Monti R. and Serra Cassano F. (2001). Surface measures in Carnot-Carathéodory spaces. Calc. Var. Partial Differ. Equ. 13: 339–376 · Zbl 1032.49045 · doi:10.1007/s005260000076 [10] Trudinger N.S. and Wang X.-J. (2002). On the weak continuity of elliptic operators and applications to potential theory. Amer. J. Math. 124: 369–410 · Zbl 1067.35023 · doi:10.1353/ajm.2002.0012 [11] Zalcman L. (1973). Mean values and differential equations. Isr. J. Math. 14: 339–352 · Zbl 0263.35013 · doi:10.1007/BF02764713
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