On Harnack inequality and harmonic Schwarz lemma

In this paper, we study the $(s, C(s))$-Harnack inequality in a domain $G\subset \mathbb{R}^n$ for $s\in(0,1)$ and $C(s)\geq1$ and present a series of inequalities related to $(s, C(s))$-Harnack functions and the Harnack metric. We also investigate the behavior of the Harnack metric under $K$-quasiconformal and $K$-quasiregular mappings, where $K\geq 1$. Finally, we provide a type of harmonic Schwarz lemma and improve the Schwarz-Pick estimate for a real-valued harmonic function.


Introduction
Harnack's inequality is a fundamental result in the study of partial differential equations (PDEs), with applications across various branches of mathematics, particularly in the theory of elliptic and parabolic equations.The Harnack inequality typically concerns positive solutions to elliptic or parabolic equations in divergence form.In the case of elliptic equations, which describe steady-state problems such as heat conduction or electrostatics, the inequality establishes bounds on the solutions by comparing the maximum and minimum values within a domain.Moreover, the German mathematician Axel Harnack developed the original formulation of this inequality for harmonic functions in the plane, see [18] for more details.It should be noted that this inequality was first published in 1887 in the book [11].
In the context of the theory of partial differential equations, the current formulation of the Harnack inequality for harmonic functions is expressed as follows: Harnack inequality.Let B n (x, r) be a Euclidean ball centered at x with the radius r ∈ (0, 1) such that the concentric ball B n (x, 2r) is contained in a domain G ⊂ R n , n ≥ 2. Then there exists a positive constant C depending on n such that (1.1) sup holds for all nonnegative harmonic functions u : G → R.
We recall that a real-valued function u : G ⊂ R n → R is called harmonic in a domain G ⊂ R n if it is twice continuously differentiable and satisfies the Laplace equation n i=1 ∂ 2 u/∂x 2 i = 0.The progress of potential analysis linked to the Laplace equation hinges on the key role of Harnack's inequality (1.1), see [12].
Subsequently, we revisit a definition presented in [24].Define R + as the set {x ∈ R : x > 0}.
Definition 1.1.Consider a proper subdomain G of R n , and let u : G → R + ∪ {0} be a continuous function.We say that u satisfies the Harnack inequality in G if there exist numbers s ∈ (0, 1) and C(s) ≥ 1 such that Here are some examples: ii) Let G be a domain and d(x, ∂G) be the minimum distance from x to the boundary of , where s = 1/2; see [10, Exercise 6.33 (1)].
In this paper, we study the (s, C(s))-Harnack inequality, which is defined as follows, where s ∈ (0, 1) and C(s) ≥ 1.This paper is organized as follows: Section 2 provides the essential notations and definitions required for the discussions in this paper.In Section 3, we investigate the behavior of the (s, C(s))-Harnack functions and the Harnack metric.Lastly, Section 4 presents a version of the harmonic Schwarz lemma and improves the Schwarz-Pick estimate for a real-valued harmonic function.

Preliminaries
This section establishes a foundation for our subsequent discussions by introducing essential notations and definitions.
Let sh, ch, th, and arth denote the hyperbolic functions sinh, cosh, tanh, and arctanh respectively.Consider the Euclidean space R n with n ≥ 2 and define H n = {x = (x 1 , . . ., x n ) ∈ R n : x n > 0} as the Poincaré half-space or the upper half-plane.The ball with center x in R n and radius r > 0 is denoted as B n (x, r), defined as the set {y ∈ R n : |y − x| < r}.Correspondingly, the sphere sharing the same center and radius is and The quasihyperbolic distance, denoted as k G (x, y), between points x and y in the domain G, is formally defined as the infimum of the integral along rectifiable curves γ ⊂ G containing both x and y.This integral is calculated as the quotient of the absolute value of the differential element dx by the distance function d G (x), as given by the expression: Gehring and Palka introduced the metric k G (x, y) in [8, p. 173] and provided a proof for the sharp inequalities ([8, Lemma 2.1]).These inequalities are expressed as follows: (2.1) For a detailed discussion, we refer to [10, p. 68].It is well-known that (see [8, p. 174]) For any open set Ω in R n , where Ω is not equal to the entire space R n , the distance ratio metric is defined by When Ω ∈ {B n , H n } as per [10, Lemma 4.9], the following double-inequality holds: (2.4) j Ω (x, y) ≤ ρ Ω (x, y) ≤ 2j Ω (x, y).
Modulus of a curve family.Let Γ be a family of curves in R n .Also, let F (Γ) denote the family of all non-negative Borel-measurable functions σ : γ σdτ ≥ 1 for each locally rectifiable curve γ ∈ Γ.The modulus of a curve family Γ ⊂ R n is defined by (see [10, p. 104]) where m stands for the n-dimensional Lebesgue measure.
We denote by ∆(E, F ; G) the family of all closed non-constant curves joining two nonempty sets E and F in a domain G, where E, F , and G are subsets of R n .
Modulus metric.Let G be a proper subdomain of R n .The modulus metric is defined by µ G (x, y) = inf For an open set Ω in R n , an ACL mapping φ : Ω → R is said to be ACL n , n ≥ 1, if φ is locally L n -integrable in Ω and if the partial derivatives ∂ j φ (which exist a.e. and are measurable) of φ are locally L n -integrable as well; see Ref. [20, p. 22].

Quasiregular mappings. Consider a domain
almost everywhere in G. Here, f ′ (x) and J f (x) represent the formal derivative and the Jacobian determinant of f at the point x, respectively.Quasiconformal mappings.Let G, G ′ be domains in R n = R n ∪ {∞}, K ≥ 1 and let f : G → G ′ be a homeomorphism.Then, f is K-quasiconformal if and only if the following conditions are satisfied: where L(λ) = min |φ|=1 |λφ|.The Harnack inequality provides a basis for defining a Harnack (pseudo) metric.Consider H + (G) as the class of all positive harmonic functions u in G. Harnack metric.For arbitrary x, y ∈ G, the Harnack metric is defined by where the supremum is taken over all u ∈ H + (G).This metric has been investigated in various contexts, including studies in [3,5,14,15,19,22].

(s, C(s))-Harnack functions and Harnack metric
In this section, we present our results on (s, C(s))-Harnack functions and the Harnack metric under K-quasiconformal and K-quasiregular mappings.We start with the following: Proof.Let u be any positive harmonic function on B n (x, r) and 0 < δ < r.Then, by [12, Theorem 3.2.1]we have 2) since rs < r for all s ∈ (0, 1).(ii) If u is a positive harmonic function, x ∈ B n , s ∈ (0, 1) and y ∈ S n−1 (x, s(1 − |x|)), then Proof.(i) The proof follows from Definition 1.2 and [10, Lemma 6.23].
We continue with the following result on quasiregular mappings; in fact, we show that if f : G → R n is a quasiregular mapping, and if ∂f G satisfies some additional conditions, then the function u(x) = d f G (f (x)), (x ∈ G), satisfies the (s, C(s))-Harnack inequality.
Remark 3.1.It is important to clarify that the theorem presented herein diverges from Theorem 5.2 in [23].Specifically, our theorem assumes that f G is a A-uniform domain with a connected boundary, while Sugawa et al. [23] regarded ∂f G as uniformly perfect.The connectedness of ∂f G is decisive in the following theorem, as demonstrated in Remark 3.2 below.Conversely, in the proof of Theorem 5.2, Sugawa et al. [23] employ the definition of the modulus metric µ G to establish an upper bound, whereas we utilize a general upper bound derived from Lemma 10.6(2) of [10] for y ∈ B n (x, sd G (x)).Moreover, the constant C(s) obtained here is more generality than the constant obtained by Sugawa et al. in [23].
Theorem 3.2.Let G be a proper subdomain of R n , and f : G → R n be a K-quasiregular mapping such that f G ⊂ R n is a A-uniform domain.Also, let ∂f G be connected such that it consists of at least two points.Then, the function u(x) = d f G (f (x)), (x ∈ G), satisfies the (s, C(s))-Harnack inequality with the constant is the inner dilatation of f , and c n is a constant number depending only on n.
Proof.Since ∂f G is a connected domain and f G is a A-uniform domain, by [10, Lemma 10.8 (1)] and by definition, we have where A ≥ 1, and c n is a constant number depending on n.Also, by [10,Theorem 15.36(1)] the following inequality This establishes the desired inequality (3.3), and thus concludes the proof.
Additionally, employing a straightforward calculation, we can infer from (2.2) that Moreover, due to k B 2 (x, y) ≤ 2j B 2 (x, y), the preceding inequality leads to Finally, by applying Theorem 3.2 and utilizing (2.4), we derive As ρ B 2 (x p , x p+1 ) = 1, the right-hand side of the last inequality remains bounded.However, the left-hand side of the same inequality diverges to infinity as p approaches infinity.Consequently, we can infer that the assertion in Theorem 3.2 loses validity when ∂f G includes isolated points.
In the following, we shall study the Harnack metric h G (x, y), where G is a proper subdomain of R n .Theorem 3.3.Let s ∈ (0, 1) and C(s Proof.(i) Let u : G → (0, ∞) be a Harnack function.By [10,Lemma 6.23] we have where t = k G (x, y)/(2 log(1 + s)).The claim is now a direct consequence of the Harnack metric definition.
(ii) According to [10,Lemma 6.23], the proof closely resembles that of part (i), so we skip the details.
To prove the next results, the following two lemmas will be helpful.
holds, where α = K 1/(1−n) and b is a constant depending on K and n.Here, b tends to 1 as K tends to 1.
Proof.(i) By [10, Theorem 16.2 (2)] we have for all x, y ∈ B n , where f : B n → f B n ⊂ B n is a K-quasiregular mapping.It follows also from Lemma 3.7 that, for x, y ∈ B n (3.10) Now, combining (3.10) and (3.9) with Lemma 3.7 gives the desired result.
(ii) Let f : B n → f B n = B n be a K-quasiconformal mapping and x, y ∈ B n .Then, by Corollary 18.5 in [10] we have: where α = K 1/(1−n) and b is a constant depending on K and n.Now, by (3.11), and using Lemma 3.7, the conclusion is obtained.
where K ≥ 1.Also, by Lemma 3.8, for all x, y ∈ f H n ⊂ H n , we have: The result now follows from (3.12)-(3.13),and Lemma 3.8.The proof is now complete.

Harmonic Schwarz lemma
This section first generalizes the Schwarz lemma for harmonic functions in the complex plane utilizing the Poisson integral formula.Then, it improves the Schwarz-Pick estimate for a real-valued harmonic function.First, we recall that the classical Schwarz lemma states that if u : B 2 → B 2 is a holomorphic function with u(0) = 0, then • |u(z)| ≤ |z| for all z ∈ B 2 ; • |u ′ (0)| ≤ 1. Heinz (see [13]) has obtained an improvement of the classical Schwarz lemma for a complex-valued harmonic function, see Lemma 4.1 below.A complex-valued function f : G → C, where f = u + iv is said to be harmonic if both u : G → R and v : G → R are harmonic in the sense defined above.The inequality is sharp for each point z ∈ B 2 .
The following Theorem 4.1 is known as the Poisson integral formula (see, for example, [7]).Theorem 4.1.Let u be a complex-valued function continuous on B 2 (a, R), (R > 0), and harmonic on B 2 (a, R).Then for r ∈ [0, R) and t ∈ R the following formulas hold: Motivated by Lemma 4.1 and applying Theorem 4.1, we derive the following Theorem 4.2 which is an extension of the above Schwarz lemma: The result is sharp.
By the last equality and the assumption |u| ≤ M, we obtain dθ.
Now, we calculate the integral It is easy to check that, Thus, from (4.4) follows that which implies the desired result.It is easy to see that the result is sharp for the function or one of its rotations, where 0 < r < R and M > 0, completing the proof.
In 1989 (see [6]), Colonna proved the following Schwarz-Pick estimate for complex-valued harmonic functions u from the unit disk B 2 to itself: If u is a real-valued function, Kalaj and Vuorinen established the above Schwarz-Pick estimate as the following theorem; refer to [16,Theorem 1.8] for details.Theorem 4.3.Let u be a real harmonic function of the unit disk into (−1, 1).Then the following sharp inequality holds: In accordance with the findings of Chen [4, Theorem 1.2], the subsequent result has been derived: holds for z ∈ B 2 .The inequality is sharp for any z ∈ B 2 and any value of u(z), and the equality occurs for some point in B 2 if and only if u(z) = (4Re{arctan f (z)})/π, z ∈ B 2 with a Möbius transformation f of B 2 onto itself.
In the subsequent discussion, we aim to expand upon Theorem 4.3 in the following manner: Furthermore, it is worth noting that our extension encompasses the findings presented in Theorem 6.26 of [1].The result is sharp.
Proof.Define v(z) as where u : B 2 → (α, β) is a real valued harmonic function, α and β are real numbers such that α < β.Then it is clear that v is a harmonic function of the unit disk B 2 into (−1, 1).
Therefore, v satisfies the assumption of Theorem 4.3.Moreover, we have where α and β are real numbers such that α < β.The result is sharp.
Proof.The proof is the same as the proof of Theorem 4.5, therefore, we omit the details.
We conclude this paper by presenting the following open question: Open question.What is the connection between the Harnack metric h and the hyperbolic metric ρ in a simply connected Jordan domain in the complex plane C?
B n (x, r) ⊂ G and B x = B n (x, sr).A function that meets this condition is referred to as a Harnack function.

Definition 1 . 2 .
Under the assumptions of Definition 1.1, for s ∈ (0, 1) and C s ≥ 1 we say that u satisfies the (s, C(s))-Harnack inequality in a domain G ⊂ R n , if the inequality (1.2) holds.A function satisfying (1.2) for all s ∈ (0, 1) is called the (s, C(s))-Harnack function.

Remark 4 . 1 .
It should be noted that Theorem 4.2 is also an extension of [21, Theorem 3.6.1].Indeed, Pavlović proved that if f : B 2 → B 2 is a complex-valued harmonic function, then the following sharp inequality holds: