The Harnack distance on space  and its conformal invariance were constructed and studied by Herron. In this paper, we obtain the Harnack distance on domains  in . Then, we use this concept to investigate some properties of the positive harmonic function class. These results are obtained in the complex plane, so it is advantageous to take some tools of the complex analysis. The main result of this paper is the property of the local uniform convergence of the positive harmonic sequences on a domain in the complex plane.The Harnack distance on space  and its conformal invariance were constructed and studied by Herron. In this paper, we obtain the Harnack distance on domains  in . Then, we use this concept to investigate some properties of the positive harmonic function class. These results are obtained in the complex plane, so it is advantageous to take some tools of the complex analysis. The main result of this paper is the property of the local uniform convergence of the positive harmonic sequences on a domain in the complex plane

Do, Dang Thinh

Hoang, Nhat Quy

Tran, Le Dieu Linh

English

Hue University Journal of Science (HU JOS)

Hue University Journal of Science: Natural Science Vol. 131, No. 1D, 61–65, 2022 pISSN 1859-1388 eISSN 2615-9678  DOI: 10.26459/hueunijns.v131i1D.6663 61   The local uniform convergence of positive harmonic function sequence  Do Dang Thinh*, Tran Le Dieu Linh, Hoang Nhat Quy Faculty of Mathematics, University of Science and Education, The University of Da Nang, 459 Ton Duc Thang St., Lien Chieu District, Da Nang, Vietnam * Correspondence to Do Dang Thinh <dodangthinh45@gmail.com> (Received: 15 March 2022; Accepted: 02 June 2022) Abstract. The Harnack distance on space ℝ𝑛  and its conformal invariance were constructed and studied by Herron. In this paper, we obtain the Harnack distance on domains 𝐷 in ℂ. Then, we use this concept to investigate some properties of the positive harmonic function class. These results are obtained in the complex plane, so it is advantageous to take some tools of the complex analysis. The main result of this paper is the property of the local uniform convergence of the positive harmonic sequences on a domain in the complex plane. Keywords: harmonic function, Harnack distance, local uniform convergence 1 Introduction Harmonic functions and subharmonic functions are the main research subject of potential theory. These function classes are usually presented in space ℝ𝑛 [1-3]. In [4], the author constructed the Harnack distance on space ℝ𝑛  and studied its conformal invariance. This is convenient to use the differential operators and integral operators that are common in the theory of multivariable functions. However, it does not take advantage of the powerful tools of complex analysis. Moreover, it is hard to expand many results to the pluripotential theory [3]. In this paper, we investigate some results of the positive harmonic function class in ℂ. Then, we use these results to prove the property of the local uniform convergence of the positive harmonic sequences.  From this research, we also pose an open question of whether it is possible to extend this result to the new function classes [5-7]. 2 Preliminaries We denote ℂ be a complex plane. A subset 𝐷 ⊂ ℂ is called a domain if it is open, connected, and non-empty. A disc centred at 𝑤 with radius 𝛿 is denoted and defined as ∆(𝑤, 𝛿) ≔ {𝑧 ∈ ℂ: |𝑧 − 𝑤| < 𝛿}. Let 𝑈 be an open subset in ℂ. The function ℎ: 𝑈 → ℝ is harmonic if it is in 𝐶2(𝑈) and satisfies the Laplace equation as follows: ∆ℎ(𝑧) ≔𝜕2ℎ𝜕𝑥2(𝑧) +𝜕2ℎ𝜕𝑦2(𝑧) = 0, for all 𝑧 = 𝑥 + 𝑖𝑦 ∈ 𝑈 . A positive harmonic function ℎ on 𝑈 is a harmonic function and ℎ ≥0 on 𝑈.  We need the identity principle for the harmonic class to prove the main result. We cite here for convenient reading. Theorem 2.1 Let ℎ  and 𝑘  be harmonic functions on domain 𝐷  in ℂ . If ℎ = 𝑘  on non-empty open subset 𝑈  of 𝐷 ; then ℎ = 𝑘  on the whole 𝐷. Do Dang Thinh et al.  62   Proof: If ℎ, 𝑘 are harmonic functions on 𝐷, then ℎ − 𝑘 is also a harmonic function on 𝐷. So, we can suppose, without loss of generality, that 𝑘 = 0. Set 𝑔 =𝜕ℎ𝜕𝑥− 𝑖𝜕ℎ𝜕𝑦. Then, from the proof of Theorem 1 (page 111 [8]), 𝑔 is holomorphic on 𝐷, and also 𝑔 = 0  on 𝑈  since ℎ = 0 . From the identity principle for holomorphic functions, it follows that 𝑔 = 0  throughout 𝐷 , and, hence, that 𝜕ℎ𝜕𝑥=𝜕ℎ𝜕𝑦= 0  on 𝐷 . Therefore, ℎ  is constant on 𝐷; and since ℎ = 0 on 𝑈, this implies that the constant must be zero. □ One of the most important results for the positive harmonic function class is the Harnack’s inequality Theorem 2.2 Let ℎ be a positive harmonic function on disc ∆(𝑤, 𝛿). Then, for all 𝑟 < 𝛿 and 0 ≤ 𝑡 < 2𝜋, 𝛿 − 𝑟𝛿 + 𝑟ℎ(𝑤) ≤ ℎ(𝑤 + 𝑟𝑒𝑖𝑡) ≤𝛿 + 𝑟𝛿 − 𝑟ℎ(𝑤). Proof: see Theorem 2.14 [4].          □ Prompted by the Harnack’s inequality, the Harnack distance is defined as follow: Definition 2.3 Let 𝐷  be a domain in ℂ . Given 𝑧, 𝑤 ∈ 𝐷, the Harnack distance between 𝑧 and 𝑤, denoted by 𝜏𝐷(𝑧, 𝑤), and defined as 𝜏𝐷(𝑧, 𝜔) = min{𝜏 ∈ ℝ: 𝜏−1ℎ(𝜔) ≤ ℎ(𝑧) ≤ 𝜏ℎ(𝜔)}, where 𝑚𝑖𝑛 is taken over all the positive harmonic functions ℎ on 𝐷.  The following result is a formulation of the Harnack distance from a point in a disc to its centre. Theorem 2.4 If ∆= ∆(𝑤, 𝛿), then 𝜏∆(𝑧, 𝑤) =𝛿 + |𝑧 − 𝑤|𝛿 − |𝑧 − 𝑤|  (𝑧 ∈ ∆). Proof: Let 𝑧 ∈ ∆. From Harnack’s inequality (Theorem 2.2), we have 𝜌 − |𝑧 − 𝜔|𝜌 + |𝑧 − 𝜔|ℎ(𝜔) ≤ ℎ(𝑧) ≤𝜌 + |𝑧 − 𝜔|𝜌 − |𝑧 − 𝜔|ℎ(𝜔) for all positive harmonic functions ℎ on ∆. From the definition of the Harnack distance, we infer 𝜏∆(𝑧, 𝜔) ≤𝜌 + |𝑧 − 𝜔|𝜌 − |𝑧 − 𝜔|.   (1) On the other hand, for each |𝜉| = 1, let ℎ𝜉  be a function definite on ∆ as follows: ℎ𝜉(𝑧) = 𝑃 (𝑧 − 𝜔𝜌, 𝜉) = 𝑅𝑒 (𝜌𝜉 + (𝑧 − 𝜔)𝜌𝜉 − (𝑧 − 𝜔)), where 𝑃  is the Poisson kernel [3]. From the property of the Poisson kernel (Lemma 2.2.1 [3]), we infer that ℎ𝜉  is a positive harmonic function on ∆ . We have ℎ𝜉(𝜔) = 1 . Set 𝑧 = 𝜔 + 𝑟𝑒𝑖𝑡 , where 𝑟 = |𝑧 − 𝜔| and 𝜉 = 𝑒𝑖𝜃, then  ℎ𝜉(𝑧) = 𝑅𝑒 (𝜌𝑒𝑖𝜃+𝑟𝑒𝑖𝑡𝜌𝑒𝑖𝜃−𝑟𝑒𝑖𝑡) =𝜌2−𝑟2𝜌2−2𝑟𝜌 cos(𝑡−𝜃)+𝑟2.  From the definition of 𝜏Δ(𝑧, 𝜔), we have 𝜏Δ−1(𝑧, 𝜔)ℎ𝜉(𝜔) ≤ ℎ𝜉(𝑧) ≤ 𝜏∆(𝑧, 𝜔)ℎ𝜉(𝜔). This is equivalent to 𝜏Δ−1(𝑧, 𝜔)ℎ𝜉(𝜔) ≤𝜌2 − 𝑟2𝜌2 − 2𝑟𝜌 cos(𝑡 − 𝜃) + 𝑟2≤ 𝜏∆(𝑧, 𝜔)ℎ𝜉(𝜔), for all 0 ≤ 𝜃 < 2𝜋. Choose 𝜃 = 𝑡, we have 𝜏Δ−1(𝑧, 𝜔)ℎ𝜉(𝜔) ≤𝜌+𝑟𝜌−𝑟≤ 𝜏∆(𝑧, 𝜔)ℎ𝜉(𝜔).  Sine ℎ𝜉(𝜔) = 1, so 𝜏∆(𝑧, 𝜔) ≥𝜌 + 𝑟𝜌 − 𝑟=𝜌 + |𝑧 − 𝜔|𝜌 − |𝑧 − 𝜔|.   (2) From (1) and (2), we infer that the item is proved. □ In the main result of this paper, we use the Harnack distance to prove a result of the local uniform convergence of the positive harmonic sequence. Let (ℎ𝑛)  be a sequence of functions that are defined on open set 𝑈. The sequence (ℎ𝑛) is said to be point-wise convergence to function ℎ if for each 𝑧 ∈ 𝑈  and > 0 , there exists 𝑁 =𝑁(𝑧, ) > 0 such that |ℎ𝑛(𝑧) − ℎ(𝑧)| <    (∀ 𝑛 ≥ 𝑁). Hue University Journal of Science: Natural Science Vol. 131, No. 1D, 61–65, 2022 pISSN 1859-1388 eISSN 2615-9678  DOI: 10.26459/hueunijns.v131i1D.6663 63   Let 𝐸 ⊂ 𝑈. The sequence (ℎ𝑛) is said to be a uniform convergence to ℎ on 𝐸 if for each >0, there exists 𝑁 = 𝑁( ) > 0, such that |ℎ𝑛(𝑧) − ℎ(𝑧)| <    (∀𝑧 ∈ 𝐸, ∀ 𝑛 ≥ 𝑁). The sequence (ℎ𝑛)  is said to be a local uniform convergence to ℎ on 𝑈 if for each 𝑧 ∈ 𝑈, there exists a neighborhood 𝐸𝑥  of 𝑥  such that ℎ𝑛 → ℎ uniformly on 𝐸𝑥. We also need the following for the proof of the main theorem. Theorem 2.5 Let (ℎ𝑛) be positive harmonic functions on domain 𝐷  in ℂ . Then, either    ℎ𝑛 → +∞  locally uniform, or else some subsequence ℎ𝑛𝑗 → ℎ locally uniform, where ℎ is harmonic on 𝐷. Proof: Fix 𝑤 ∈ 𝐷. For all 𝑧 ∈ 𝐷 and 𝑛 ≥ 1, from the inequalities 𝜏𝐷−1(𝑧, 𝑤)ℎ𝑛(𝑤) ≤ ℎ𝑛(𝑧) ≤ 𝜏𝐷(𝑧, 𝑤)ℎ𝑛(𝑤), (∗) it follows that if ℎ𝑛(𝑤) → +∞, then, also ℎ𝑛 → +∞ locally uniform on 𝐷, and if ℎ𝑛(𝑤) → 0, then, also ℎ𝑛 → 0 locally uniform on 𝐷. Therefore, replacing (ℎ𝑛)  by a subsequence if necessary, we can reduce to the case where the sequence (log ℎ𝑛(𝑤))𝑛≥1  is bounded. The inequality (∗) then implies that (log ℎ𝑛)𝑛≥1 is locally uniformly bounded on 𝐷 , and so it suffices to prove that there is a subsequence (ℎ𝑛𝑗)  such that (log ℎ𝑛𝑗)𝑗≥1 is locally uniformly convergent on 𝐷.  Let 𝑆  be a countable dense subset of 𝐷 . The sequence (log ℎ𝑛(𝜉))𝑛≥1 is bounded for each 𝜉 ∈ 𝑆, so with a “diagonal argument”, we may find a subsequence (ℎ𝑛𝑗) such that (log ℎ𝑛𝑗(𝜉))𝑗≥1 is convergent for each 𝜉 ∈ 𝑆. We shall show that, for this subsequence, (log ℎ𝑛𝑗)𝑗≥1  is locally uninformly convergent on 𝐷. Let 𝐾  be a compact subset of 𝐷 , and let > 0. For each 𝑧 ∈ 𝐾, let  𝑉𝑧 = {𝑧′ ∈ 𝐷: log 𝜏𝐷(𝑧, 𝑧′) < }, And let 𝑉𝑧1 , … , 𝑉𝑧𝑚  be a finite subcover of 𝐾. As 𝑆 is dense in 𝐷, for each 𝑙, we can pick a point 𝜉𝑙 ∈ 𝑉𝑧𝑙 ∩ 𝑆 . Then, there exists 𝑁 ≥ 1 such that  |log ℎ𝑛𝑗(𝜉𝑙) − log ℎ𝑛𝑘(𝜉𝑙)| ≤ , for all 𝑛𝑗 , 𝑛𝑘 ≥ 𝑁, 𝑙 = 1,2, . . , 𝑚 . According to the definition of the Harnack distance,  |log ℎ𝑛𝑗(𝑧) − log ℎ𝑛𝑘(𝜉𝑙)| ≤ log 𝜏𝐷(𝑧, 𝜉𝑙) < 2 , for all 𝑧 ∈ 𝑉𝑧𝑙. Then, we have a similar inequality for ℎ𝑛𝑘. Hence, |log ℎ𝑛𝑗(𝑧) − log ℎ𝑛𝑘(𝑧)| ≤ 5  (𝑛𝑗, 𝑛𝑘 ≥ 𝑁, 𝑧 ∈ 𝐾). Thus, (log ℎ𝑛𝑗)𝑗≥1 is a uniform Cauchy on 𝐾, and so uniformly convergent there.        □ 3 Main results Theorem 3.1 Let (ℎ𝑛)𝑛≥1  be positive harmonic functions on domain 𝐷 ⊂ ℂ. If (ℎ𝑛)𝑛≥1 converge point-wise for each 𝑧 in a non-empty open subset of 𝐷, then (ℎ𝑛)𝑛≥1 converge locally uniformly on the whole of 𝐷. Proof: Following the hypothesis, there exists ∅ ≠ 𝜔 ⊂ 𝐷, and 𝜔 is open such that the sequence (ℎ𝑛)𝑛≥1 converges for every 𝑧 ∈ 𝜔. Then, we have ℎ𝑛 ↛ +∞, on 𝐷 . According to Theorem 2.5, there exists a subsequence (ℎ𝑛𝑗) such that  ℎ𝑛𝑗 → ℎ, 𝑗 → +∞ is local uniform on 𝐷  and ℎ  is a harmonic function on 𝐷. First, we shall prove that the sequence (ℎ𝑛) converges point-wise to ℎ on 𝐷: we suppose that (ℎ𝑛)  does not converge point-wise to ℎ  on 𝐷 . Then, there exists a point 𝑧0 ∈ 𝐷, a number > 0 and a subsequence (ℎ𝑚𝑗) such that  |ℎ𝑚𝑗(𝑧0) − ℎ(𝑧0)| > , (∀𝑗 = 1,2, … ). Do Dang Thinh et al.  64   By applying Theorem 2.5 for the subsequence (ℎ𝑚𝑗), we have two cases that can occur as follows: • Either the sequence ℎ𝑚𝑗 → +∞ when 𝑗 → +∞ is locally uniform on 𝐷 . Then, ℎ = +∞ on open subset 𝜔  of 𝐷 . So, according to the identity principle (Theorem 2.1), ℎ = +∞ on 𝐷 . This is contrary to the hypothesis. • Or, there exists a subsequence (ℎ𝑚𝑗𝑘) such that  ℎ𝑚𝑗𝑘 → ℎ̂, 𝑘 → +∞, Is locally uniform on 𝐷 . Then, ℎ = ℎ̂  on open subset 𝜔  of 𝐷 . Again, according to the identity principle, ℎ = ℎ̂ on 𝐷. According to this and the contrary hypothesis, |ℎ𝑚𝑗𝑘(𝑧0) − ℎ̂(𝑧0)| > , (∀𝑗 = 1,2, … ). This is contrary to the convergence of the sequence (ℎ𝑚𝑗𝑘) of the function ℎ̂ on 𝐷.  Next, we prove the sequence (ℎ𝑛) converges locally uniformly to ℎ on 𝐷: Let 𝑧1 ∈𝐷. Choose 𝛿 > 0 such that ∆(𝑧1, 𝛿) ⊂ 𝐷. We will prove (ℎ𝑛) converges locally uniformly to ℎ on ∆(𝑧1,𝛿2) . Indeed, given > 0 . Then, for every 𝑧, 𝑤 ∈ ∆(𝑧1,𝛿2)  and every positive harmonic function ℎ on 𝐷, we have 𝜏∆(𝑤,𝛿)−1 (𝑧, 𝑤)ℎ(𝑤) ≤ ℎ(𝑧) ≤ 𝜏∆(𝑤,𝛿)(𝑧, 𝑤)ℎ(𝑤)(∗∗) where 𝜏∆(𝑤,𝛿)(𝑧, 𝑤) =𝛿 + |𝑧 − 𝑤|𝛿 − |𝑧 − 𝑤|<𝛿 +𝛿2𝛿 −𝛿2= 3. – Applying (∗∗) for 𝑤 = 𝑧1, we infer each point 𝑧 ∈ ∆(𝑧1,𝛿2), there exists a number 𝑀𝑧 > 0 such that  |ℎ(𝑧)| < 𝑀𝑧 ,   (1) for all positive harmonic function ℎ on 𝐷. – Since lim|𝑧−𝑤|→0𝛿 + |𝑧 − 𝑤|𝛿 − |𝑧 − 𝑤|= 1, So, there exists 𝛿1 > 0 such that for every |𝑧 − 𝑤| < 𝛿1, we have 𝛿 + |𝑧 − 𝑤|𝛿 − |𝑧 − 𝑤|− 1 <4𝑀𝑧. equivalent to 𝜏∆(𝑤,𝛿)(𝑧, 𝑤) − 1 < 4𝑀𝑧.     (2) We consider 𝛿1 – a finite net {𝑧1, 𝑧2, … , 𝑧𝑚} ⊂ ∆(𝑧1,𝛿2) such that ⋃ ∆(𝑧𝑗 , 𝛿1)𝑚𝑗=1⊃ ∆ (𝑧1,𝛿2). – Since the sequence (ℎ𝑛)  converges at points 𝑧1, 𝑧2, … , 𝑧𝑚  to ℎ , so there exists 𝑛0 ≥ 1 large enough such that for every 𝑛 > 𝑛0, we have |ℎ𝑛(𝑧𝑗) − ℎ(𝑧𝑗)| < 2   (∀𝑗 = 1,2, … ). Each 𝑤 ∈ ∆(𝑧1,𝛿2)  is arbitrary. Then, there exists 𝑧𝑗 such that |𝑤 − 𝑧𝑗| < 𝛿1. We have |ℎ𝑛(𝑤) − ℎ(𝑤)| ≤ |ℎ𝑛(𝑤) − ℎ𝑛(𝑧𝑗)|+ |ℎ𝑛(𝑧𝑗) − ℎ(𝑧𝑗)|+ |ℎ(𝑧𝑗) − ℎ(𝑤)|.  (3) For all positive harmonic functions 𝑢, we have (𝜏∆(𝑤,𝛿)−1 (𝑧𝑗 , 𝑤) − 1)𝑢(𝑧𝑗) ≤ 𝑢(𝑤) − 𝑢(𝑧𝑗)≤ (𝜏∆(𝑤,𝛿)(𝑧𝑗 , 𝑤) − 1)𝑢(𝑧𝑗). From this, we infer |𝑢(𝑤) − 𝑢(𝑧𝑗)| ≤ (𝜏∆(𝑤,𝛿)(𝑧𝑗 , 𝑤) − 1)|𝑢(𝑧𝑗)|. From (1) and (2), we imply that |𝑢(𝑤) − 𝑢(𝑧𝑗)| < 4𝑀𝑧𝑗𝑀𝑧𝑗 = 4.    (4) Applying (4) for functions ℎ𝑛 and ℎ, then input them to (3), we imply that |ℎ𝑛(𝑤) − ℎ(𝑤)| < 4+2+4= , Hue University Journal of Science: Natural Science Vol. 131, No. 1D, 61–65, 2022 pISSN 1859-1388 eISSN 2615-9678  DOI: 10.26459/hueunijns.v131i1D.6663 65   for all 𝑛 > 𝑛0 and all 𝑤 ∈ ∆(𝑧1,𝛿2).       □ 4 Conclusions In this note, we prove a property of the local uniform convergence of the positive harmonic sequence with the main result, and it is Theorem 3.1. All the results in this paper are presented in space ℝ2  equivalent to complex plane ℂ . This allows us to use the powerful tools of complex analysis to demonstrate and prove the results. By the way, we propose an open question of whether this property is valid for new function classes. References 1. Axler S, Bourdon P, Ramey W. Harmonic function theory. New York: Springer – Verlag; 2001. 264 p.  2. Klimek M. Pluripotential Theory. Oxford: Clarendon Press; 1991. 3. Helms LL. Introduction to potential theory. New York: John Wiley and Sons; 1969. 4. Herron DA. The harnack and other conformally invariant metrics. Kodai Mathematical Journal. 1987;10(1):9-19. 5. Hiep PH. Singularities of plurisubharmonic functions. Pub Hou Sci and Tec; 2016. 6. Quy HN. The topology on the space 𝛿ℰ𝜒 . Univ Iagel Acta Math. 2014;51:61-73. 7. Hung VV, Quy HN. The m-Hessian Operator on Some Weighted Energy Classes of Delta m-Subharmonic Functions. Results in Mathematics. 2020;75(3):112.  8. Khue NV, Hai LM. Functions of One complex variable. Hanoi: Hanoi Vietnam National University Press; 1997. 

The local uniform convergence of positive harmonic function sequence

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The local uniform convergence of positive harmonic function sequence

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