Monotonicity of Harnack Inequality for Positive Invariant Harmonic Functions

A monotonicity property and a reﬁned estimate of Harnack inequality are derived for positive solutions of the Weinstein equation.


Introduction
Let B n = {x ∈ R n : |x| < 1}, n ≥ 2, be the unit ball in R n , S n−1 = ∂B n . Consider the Weinstein equation in B n of the form Δ λ u = 1 − |x| 2 where u = u(x), x ∈ B n , λ ∈ R. In this paper, we prove a monotonicity property and a refined estimate of Harnack inequality for positive solutions of (1.1). The differential operator Δ λ in (1.1) is a natural extension of the Laplacian operator (λ = 0). If T is a Möbius transformation from B n onto B n and T (x) denotes the Jacobian matrix, then for every solution u of (1.1) in B n , the function detT (x) (n−2−2λ)/2n u T(x) (1.2) is also a solution of (1.1), as proved by Akın and Leutwiler [1]. More precisely, Δ λ detT (x) (n−2−2λ)/2n u T(x) = det T (x) ( [2]. Therefore, Δ λ is also called invariant Laplacian and the solutions of (1.1) are called invariant harmonic functions. The Dirichlet problem for (1.1) and its half-space counterpart are challenging and interesting, as summarized in Liu and Peng [2], where the authors also pointed out that invariant harmonic functions do not possess good boundary regularity in general. The classical Harnack inequality gives scale invariant bounds for nonnegative (nonpositive) harmonic functions in the plane. Harnack-type inequalities have been an important tool in the general theory of harmonic functions and partial differential equations, on which Kassmann [3] provided a through introduction and survey of the development and applications. In this paper, we give an estimate of Harnack inequality bounds for any two points in B n based on a monotonicity property of positive invariant harmonic functions.
In this section, we state the main results. The proofs are provided in the subsequent sections. For positive solutions of (1.1), Theorem 1.1 describes a monotonicity property, Theorem 1.2 gives bounds for a Harnack-type inequality for two points on the same ray, and Theorem 1.3 extends the estimates for the Harnack-type inequality to any two points in B n . Two interesting special cases-one on harmonic functions and another on the Laplace-Beltrami operator associated with the Poincaré metric-are stated as corollaries.
To study the properties of solutions of Δ λ u = 0, one often needs to distinguish the cases of λ ≥ − 1/2 and λ < −1/2. Throughout this paper, we denote Then for ζ ∈ S n−1 , the function is decreasing for 0 ≤ r < 1 and the function is increasing for 0 ≤ r < 1.

Proof of Theorem 1.1
Our proofs depend on an integral representation for positive solutions of Weinstein equations in B n derived by Leutwiler [4] based on earlier work of Huber [5] and Brelot-Collin and Brelot [6]. We state Leutwiler's representation as a theorem below (with a slightly different parametrization).
where μ is a positive measure on S n−1 .
According to the representation theorem, every positive solution of Δ λ u = 0 can be identified with its integral representation (2.1) and the corresponding positive measure μ. Notice that when λ ≥ − 1/2, the integrand of (2.1) is the Poisson kernel In the sequel, we will use the representation formula (2.1) in terms of the Poisson kernel for positive solutions of (1.1) for the case λ ≥ − 1/2. The solutions of (1.1) with λ < −1/2 is related to that of λ > −1/2 by a corresponding principle, also proved by Leutwiler [4].
We state a special case of the correspondence principle as a lemma.
We need the following two lemmas for the proof of Theorem 1.1. (2.6) To prove the right-side inequality in Lemma 2.3, it suffices to show which is equivalent to Since λ ≥ − 1/2, n + 2λ > 0, the above becomes which, after a simplification, is equivalent to The inequality is true since ζ,η ∈ S n−1 . To prove the left-side inequality in Lemma 2.3, it suffices to show that which is equivalent to Proof. By the representation theorem, u has the integral representation (2.1) of the Poisson kernel with a positive measure μ on S n−1 ,  Proof. First, consider the case λ ≥ − 1/2 (δ = 0). Define To prove Theorem 1.1 for λ ≥ − 1/2, we need to show that I(r,ζ) is decreasing and J(r,ζ) is increasing in r. By Lemma 4.1, (2.22) Therefore, log I(r,ζ) is decreasing in r, and so is I (r,ζ). Similarly, (2.23) Hence, J(r,ζ) is increasing in r. We have proved Theorem 1.1 for the case λ ≥ − 1/2.
by the correspondence principle (Lemma 2.2). From the above results for λ > −1/2, is decreasing in r, and is increasing in r. Recall that δ = 0 for λ ≥ − 1/2 and δ = 1 + 2λ for λ < −1/2, we have shown that is decreasing in r and is increasing in r for any λ ∈ R. This completes the proof of Theorem 1.1.

Proof of Theorem 1.2
The proof for Theorem 1.2 is based on Theorem 1.1 and the following lemma.

Proof of Theorem 1.3
The proof of Theorem 1.3 is based on the following three lemmas.
The proof of Theorem 1.3 follows immediately.