Hyperbolic Green Function Estimates

Abstract

For a hyperbolic Brownian motion in the hyperbolic space \(\mathbb {H}^{n}, n\ge 3\), we prove a representation of a Green function and a Poisson kernel for bounded and smooth sets in terms of the corresponding objects for an ordinary Euclidean Brownian motion and a conditional gauge functional. Using this representation we prove bounds for the Green functions and Poisson kernels for smooth sets. In particular, we provide a two sided sharp estimate of the Green function of a hyperbolic ball of any radius. By usual isomorphism argument the same estimate holds in any other model of a real hyperbolic space.

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Acknowledgments

G. Serafin was supported by the National Science Centre, Poland, grant no. 2015/18/E/ST1/00239. M. Ryznar and T. Żak were partially supported by the National Science Centre, Poland, grant no. 2015/17/B/ST1/01043. We thank the referee for valuable comments and remarks which improved the presentation of the paper.

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Appendix

Appendix

Proposition A.1 1

Let \(a>\frac {n-1}2\), n ≥ 2, and let B be any Euclidean ball in \(\mathbb {R}^{n}\). Then we have

$${\int}_{\partial B}\frac{1}{|x-z|^{2a}}dz\approx \frac{1}{(\rho_{B}(x))^{2a-n+1}},\ x\in B,$$

where the comparability constant depends only on a and n, and ρB(x) stands for Euclidean distance between x and B.

Proof

Without loss of generality we may assume B = B(0, 1) and x = (0,..., 0,xn), where xn > 0. It is clear that for \(x_{n}<\frac 12\) the integral is comparable with a constant. For \(x_{n}\geq \frac 12\) we rewrite it as follows

$$ \begin{array}{@{}rcl@{}} {\int}_{\partial B}\frac{1}{|x-z|^{2a}}dz&={\int}_{\partial B}\frac{1}{\left( {z_{1}^{2}}+...+z_{n-1}^{2}+(x_{n}-z_{n})^{2}\right)^{a}}dz\\ &={\Omega}_{n-2}{\int}_{-1}^{1}\frac{(1-r^{2})^{(n-3)/2}}{\left( 1^{2}-r^{2}+(x_{n}-r)^{2}\right)^{a}}dr\\ &={\Omega}_{n-2}{{\int}_{0}^{2}}\frac{(u(2-u))^{(n-3)/2}}{\left( (1-x_{n})^{2}+2x_{n}u\right)^{a}}du\\ &{\approx{\int}_{0}^{1}}\frac{u^{(n-3)/2}}{\left( (1-x_{n})^{2}+2x_{n}u\right)^{a}}du, \end{array} $$

where Ωn− 2 is the surface area of the unit sphere in \(\mathbb {R}^{n-1}\). Next, we substitute \(u=\frac {(1-x_{n})^{2}}{2x_{n}}s\) and get

$$\frac{2x_{n}}{(1-x_{n})^{2}}^{a-(n-1)/2}{\int}_{0}^{2x_{n}/(1-x_{n})^{2}}\frac{s^{(n-3)/2}}{1+s^{a}}ds.$$

Since for \(x_{n}\geq \frac 12\) it holds xn ≈ 1 and 4xn/(1 − xn)2 > 8, we obtain the required estimate. □

Corollary A.1 1

Let γ > 0 and let BR, R > 1, be defined as in (18). If xBR and \(\delta _{B_{R}}(x)>1\), then

$${\int}_{\partial B_{R}}\frac{x_{n}^{2\gamma}}{|x-z|^{2\gamma+n-1}}dz\approx1, $$

where the comparability constant depends only on γ and n.

Proof

We fix R > 1 and denote B = BR. In view of the previous proposition, it is enough to show that ρB(x) ≈ xn whenever δB(x) > 1. The inequality ρB(x) ≤ xn is clear. On the other hand, we may write

$$\rho_{B}(x)=\sinh R-\sqrt{|\tilde x|^{2}+(x_{n}-\cosh R)^{2}}\approx \frac{\sinh^{2}R-|\tilde x|^{2}-(x_{n}-\cosh R)^{2}}{\sinh R}.$$

The assumption δB(x) > 1 implies xBR− 1, which gives us

$$ \begin{array}{@{}rcl@{}} &\sinh^{2}R-|\tilde x|^{2}-(x_{n}-\cosh R)^{2}\\ &>\sinh^{2}R-\sinh^{2}(R-1)+(x_{n}-\cosh (R-1))^{2}-(x_{n}-\cosh R)^{2}\\ &=2x_{n}(\cosh R-\cosh(R-1))\approx x_{n}\cosh R. \end{array} $$

Finally, we get

$$\rho_{B}(x)\ge C(n) x_{n}\frac{\cosh R}{\sinh R}\approx x_{n},$$

which ends the proof. □

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Ryznar, M., Serafin, G. & Żak, T. Hyperbolic Green Function Estimates. Potential Anal 54, 535–559 (2021). https://doi.org/10.1007/s11118-020-09837-5

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Keywords

  • Hitting times and distributions
  • Poisson kernels
  • Green functions
  • Hyperbolic spaces

Mathematics Subject Classification (2000)

  • Primary 60J45
  • Secondary 60J60