Filomat 29:2 (2015), 361–370 Published by Faculty of Sciences and Mathematics, DOI 10.2298/FIL1502361C University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat Radial Growth and Hardy-Littlewood-Type Theorems on Hyperbolic Harmonic Functions Shaolin Chena, Zhenhua Sub In celebration of Matti Vuorinen’s 65-th birthday aDepartment of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan 421008, People’s Republic of China. bDepartment of Mathematics, Huaihua University, Huaihua, Hunan 418008, People’s Republic of China. Abstract. In this paper, we first show that a result of Girela et al. on analytic functions can be extended to hyperbolic-harmonic functions, and then we establish Hardy-Littlewood-type theorems on hyperbolic harmonic functions. 1. Introduction and main results For n 2, let Rn denote the usual real vector space of dimension n. Sometimes it is convenient to ≥ n identify each point x = (x ;:::; xn) R with an n 1 column matrix so that 1 2 × 0 1 B x1 C B C x B : C = B : C : @B AC xn n For a = (a ;:::; an) and x R , we define the Euclidean inner product ; by 1 2 h· ·i x; a = x a + + xnan h i 1 1 ··· so that the Euclidean length of x is defined by 1=2 2 2 1=2 x = x; x = ( x + + xn ) : j j h i j 1j ··· j j n Denote a ball in R with center x0 and radius r by Bn(x ; r) = x Rn : x x < r : 0 f 2 j − 0j g 2010 Mathematics Subject Classification. Primary 31C05, 46E15; Secondary 30J60, 30C62 Keywords. Hardy-Littlewood type theorem; majorant; hyperbolic harmonic function. Received: 9 November 2014; Accepted: 12 January 2015 Communicated by Miodrag Mateljevic´ This research was partly supported by the National Natural Science Foundation of China (No. 11401184 and No. 11326081), the Hunan Province Natural Science Foundation of China (No. 2015JJ3025), the Excellent Doctoral Dissertation of Special Foundation of Hunan Province (higher education 2050205), the Construct Program of the Key Discipline in Hunan Province. Email addresses: [email protected]. (Shaolin Chen), [email protected] (Zhenhua Su) Sh. Chen, Z. Su / Filomat 29:2 (2015), 361–370 362 In particular, Bn denotes the unit ball Bn(0; 1). Set D = B2, the open unit disk in the complex plane C. Let Ω be a proper subdomain of Rn. A function f 2(Ω) is called hyperbolic harmonic (briefly, h- harmonic, in the following) function in Ω if it satisfies the2hyperbolic C Laplace’s equation 2 2 2 ∆hu = (1 x ) ∆u + 2(n 2)(1 x ) u; x = 0; − j j − − j j hr i where ∆ denotes the ordinary Laplacian operator and denotes the gradient. Recall that hyperbolic harmonic functions are solutions of the Laplace-Beltrami equationr with respect to the Poincar´emetric Xn ds2 = (1 x 2)2 dx2 − j j k k=1 in the unit ball Bn. Obviously, when n = 2, all h-harmonic functions are harmonic functions. We refer to [2, 3, 6, 14, 18, 28, 29] for more details of h-harmonic functions. It turns out that if (@Bn), then the Dirichlet problem 2 C 8 > n <>∆h f = 0 in B > :> f = on @Bn n has an unique solution in (B ) and can be represented by C Z f (x) = Ph[ ](x) = Ph(x; ζ) (ζ)dσ(ζ); (1) @Bn n where dσ is the unique normalized surface measure on @B and Ph(x; ζ) is the hyperbolic Poisson kernel defined by !n 1 2 − 1 x n n Ph(x; ζ) = − j j (x B , ζ @B ). x ζ 2 2 2 j − j Throughout this paper, we use C to denote the various positive constants, whose value may change from one occurrence to the next. A continuous increasing function ! : [0; ) [0; ) with !(0) = 0 is called a majorant if !(t)=t is non-increasing for t > 0. Given a subset Ω of1Rn,! a function1 f : Ω Rm (m 1) is said to belong to the ! ≥ Lipschitz space Λ!(Ω) if there is a positive constant C such that f (x) f (y) C!( x y ) for all x; y Ω: (2) j − j ≤ j − j 2 For δ0 > 0, let Z δ !(t) dt C !(δ); 0 < δ < δ0 (3) 0 t ≤ · and Z 1 !(t) dt C δ 2 !(δ); 0 < δ < δ0; (4) δ t ≤ · where ! is a majorant. A majorant ! is said to be regular if it satisfies the conditions (3) and (4) (see [12, 13, 26]). n Let Ω be a proper subdomain of R . We use dΩ(x) to denote the Euclidean distance from x to the boundary @Ω of Ω. In particular, we always use d(x) to denote the Euclidean distance from x to the boundary of Bn: Sh. Chen, Z. Su / Filomat 29:2 (2015), 361–370 363 n A proper subdomain G of R is said to be Λ!-extension if Λ!(G) = locΛ!(G), where locΛ!(G) denotes the set of all functions f : G Rm (m 1) satisfying (2) with a fixed positive constant C, whenever x G !1 ≥ n 2 and y G such that x y < dG(x). Obviously, B is a Λ -extension domain. 2 j − j 2 ! In [22], the author proved that G is a Λ!-extension domain if and only if each pair of points x; y G can be joined by a rectifiable curve γ G satisfying 2 ⊂ Z !(d (τ)) G ds(τ) C!( x y ) (5) γ dG(τ) ≤ j − j with some fixed positive constant C = C(G;!), where ds stands for the arc length measure on γ. Furthermore, the author also proved that Λ!-extension domains exist only for majorants ! satisfying (3). See [13, 15, 22] for more details on Λ!-extension domains. For p (0; ], the Hardy class Hp(Bn) consists of those functions f : Bn R such that f is measurable, 2 1 ! Mp(r; f ) exists for all r (0; 1) and f p < , where 2 k k 1 8 > sup Mp(r; f ); if p (0; ); Z !1=p > 2 1 f = <0<r<1 and M (r; f ) = f (rζ) p dσ(ζ) : p > p k k >sup f (z) ; if p = @Bn j j :z Bn j j 1 2 A classical result of Hardy and Littlewood asserts that if p (0; ], α (1; ) and f is an analytic function in D, then 2 1 2 1 1 α Mp(r; f 0) = O as r 1 1 r ! − if and only if α 1 1 − Mp(r; f ) = O log as r 1; 1 r ! − Indeed the above result of Hardy and Littlewood provides a close relationship between the integral means of analytic functions and those of their derivatives [11, 19, 20]). In [16, Theorem 1(a)], Girela and Pelaez´ refined the above result for the case α = 1 as follows. Theorem 1.1. ([16, Theorem 1(a)]) Let p (2; ). For r (0; 1), if f is an analytic function in D such that 2 1 2 1 Mp(r; f 0) = O as r 1; 1 r ! − then for all β > 1=2, 1 β Mp(r; f ) = O log as r 1: (6) 1 r ! − In [16, P464, Equation (26)], Girela and Pelaez´ asked whether β in (6) can be substituted by 1=2. This problem was affirmatively settled by Girela, Pavlovic and Pelaez´ in [17] (see [17, Theorem 1.1]). We show that Theorem 1.6 can be extended to h-harmonic functions in Bn with β = 1=2. On the related topics, see [5, 7, 8, 10, 27]. Theorem 1.2. Let p [2; ) and ! be a majorant. For r (0; 1), if f is h-harmonic from Bn into R such that 2 1 2 1 Mp(r; f ) C! ; r ≤ 1 r − then 1 " n 2 2 Z 1 # 2 2 rp(p 1)(1 + r) − C !(1) 1 Mp(r; f ) f (0) + − ! dt : ≤ j j n 1 1 rt − 0 − Sh. Chen, Z. Su / Filomat 29:2 (2015), 361–370 364 Especially, if n = 2 and !(t) = t, then 1 ! 1 2 Mp(r; f ) = O log as r 1 (7) 1 r ! − and the estimate of (7) is sharp. Krantz [21] proved a Hardy-Littlewood-type theorem for harmonic functions in Bn with respect to the majorant !(t) = ! (t) = tα (0 < α 1) as follows. For the extended discussion on this topic, see [4, 9]. α ≤ Theorem 1.3. ([21, Theorem 15.8]) Let u be a harmonic function in Bn and 0 < α 1. Then u satisfies ≤ !α d(x) u(x) C for any x Bn jr j ≤ d(x) 2 if and only if u(x) u(y) C! ( x y ) for any x; y Bn: j − j ≤ α j − j 2 We generalize Theorem 1.3 to the following form. Theorem 1.4. Let ! be a majorant satisfying (3), Ω be a Λ!-extension domain and f be a h-harmonic function from Ω into R. Then f Λ!(Ω) if and only if 2 ! dΩ(x) f (x) C for any x Ω: jr j ≤ dΩ(x) 2 n B Let ! be a majorant and D be a bounded set of R . We use Λ!(D) to denote all the bounded continuous functions f in D with the norm ( ) f (x) f (y) f !;D = sup j − j < : k k x;y D;x,y !( x y ) 1 2 j − j Taking another majorant !0, we define the operator norm n Ph[ f ] !0;B Ph ! !0 = sup k k : k k ! B n f Bn f Λ (@B ); f Bn ,0 !;@ 2 ! k k!;@ k k For each a @Bn, we define 2 n ηa (ζ) = !( ζ a ) for ζ @B : ;! j − j 2 We refer to [1] for the similar definitions of harmonic functions.