# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Axler S., Bourdon P. and Ramey W. (1992). Bôcher’s Theorem. Am. Math. Mon. 99: 51–55 · Zbl 0758.31002 · doi:10.2307/2324549  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  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  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  Constantinescu C. and Cornea A. (1972). Potential Theory on Harmonic Spaces, Die Grundlehren der mathematischen Wissenschaften, Band 158. Springer, New York · Zbl 0248.31011  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  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  Hörmander L. (1968). Hypoelliptic second-order differential equations. Acta Math. 121: 147–171 · Zbl 0156.10701  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  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  Zalcman L. (1973). Mean values and differential equations. Isr. J. Math. 14: 339–352 · Zbl 0263.35013 · doi:10.1007/BF02764713
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.