Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
Filomat 23:3 (2009), 199–202
DOI:10.2298/FIL0903199A
ON THE MODULUS OF CONTINUITY OF HARMONIC QUASIREGULAR MAPPINGS ON THE UNIT BALL IN Rn Miloˇ s Arsenovi´ c and Vesna Manojlovi´ c Abstract We show that, for a class of moduli functions ω(δ), 0 ≤ δ ≤ 2, the property |ϕ(ξ) − ϕ(η)| ≤ ω(|ξ − η|), ξ, η ∈ Sn−1 implies the corresponding property |u(x)−u(y)| ≤ Cω(|x−y|) x, y ∈ Bn , for u = P [ϕ], provided u is a quasiregular mapping. Our class of moduli functions includes ω(δ) = δ α (0 < α ≤ 1), so our result generalizes earlier results on H¨ older continuity (see [1]) and Lipschitz continuity (see [2]).
1
Introduction and notations
We set, for any n ≥ 2 Z P [ϕ](x) =
P (x, ξ)ϕ(ξ)dσ(ξ) Sn−1
1 − |x|2 is the Poisson kernel for Bn = {x ∈ R : |x| < 1}, dσ is the |x − ξ|n normalized surface measure on Sn−1 and ϕ : Sn−1 → Rn is a continuous mapping. We are going to work with moduli functions ω(δ), 0 ≤ δ ≤ 2, satisfying the following conditions: where P (x, ξ) =
1◦ ω(δ) is continuous, increasing and ω(0) = 0, 2◦ ω(δ)/δ is a decreasing function, Z δ ω(ρ) ◦ 3 dρ ≤ Cω(δ), ρ 0 We say that f is ω−continuous if |f (x) − f (y)| ≤ ω(|x − y|) for all x and y in the domain of f . We note that the following properties of ω follow from the conditions 1◦ and 2◦ : 2000 Mathematics Subject Classifications. 30C62. Key words and Phrases. quasihyperbolic metric, bilipschitz maps. Received: October 4, 2009 Communicated by Dragan S. Djordjevi´ c
200
Miloˇs Arsenovi´c and Vesna Manojlovi´c Z
2
◦
4
δ
Z
δ
5◦
ω(ρ) ω(δ) dρ ≤ C 2 , ρ3 δ ω(ρ)ρn−1 dρ ≤ Cδ n ω(δ).
0
2
The Main Result
Theorem 2.1 Assume ϕ : Sn−1 → Rn is ω−continuous mapping, where ω is a modulus function satisfying properties 1◦ − 3◦ . If the harmonic extension u = P [ϕ] of ϕ to Bn is K−quasiregular, then u is Cω−continuous, where C depends on n, K and ω only. In the case of Lipschitz continuity, i.e. ω(δ) = Lδ, this was proved in [2]. If ω(δ) = Lδ α , 0 < α < 1, then the conclusion holds without the assumption of quasiregularity of u = P [ϕ], see [3]. We use the same method of proof as in [2], adapted to deal with our class of moduli functions. Let us choose x0 = rξ0 ∈ B, r = |x0 |, ξ0 ∈ Sn−1 ; let T = Tx0 rSn−1 be the (n − 1)-dimensional tangent plane to the sphere rSn−1 at point x0 . The proof is based on the following estimate ||D(u|T )(x0 )|| ≤ C(ω, n) ·
ω(δ) , δ = 1 − |x0 |, δ
(∗)
which is of independent interest. Without loss of generality x0 = ren , where en = (0, 0, . . . , 0, 1) ∈ Sn−1 . We have, by a simple calculation, ∂ 2xj xj − ξj P (x, ξ) = − − n(1 − |x|2 ) . n ∂xj |x − ξ| |x − ξ|n+2 Hence, for 1 ≤ j < n and for x0 = ren we have ∂ ξj P (x0 , ξ) = n(1 − |x0 |2 ) . ∂xj |x0 − ξ|n+2 Note that this integral kernel is odd in ξ ∈ Sn−1 . We have, using this property of the kernel, ∂u (x0 ) = ∂xj =
Z
ξj dσ(ξ) |x0 − ξ|n+2 ZSn−1 ξj n(1 − |x0 |2 ) [ϕ(ξ) − ϕ(ξ0 )] dσ(ξ). |x0 − ξ|n+2 Sn−1 n(1 − |x0 |2 )
ϕ(ξ) ·
Of course, since x0 = ren , we have ξ0 = en . Now, using
On the modulus of continuity of harmonic quasiregular mappings on the unit ball in Rn 201
|ξj | ≤ |ξ − ξ0 |,
(1 ≤ j < n, ξ ∈ Sn−1 )
and ω continuity of ϕ we get, for 1 ≤ j < n, Z ¯ ∂u ¯ |ξ − ξ0 |ω(|ξ − ξ0 |) ¯ ¯ 2 (x0 )¯ ≤ n(1 − |x0 | ) · dσ(ξ). ¯ ∂xj |x0 − ξ|n+2 Sn−1 In order to estimate the last integral, we split Sn−1 into two disjoint subsets E = {ξ ∈ Sn−1 : |ξ − ξ0 | ≤ 1 − |x0 |} and F = {ξ ∈ Sn−1 : |ξ − ξ0 | > 1 − |x0 |}. Since |ξ − xo | ≥ 1 − |x0 | for ξ ∈ Sn−1 we have Z E
|ξ − ξ0 |ω(|ξ − ξ0 |) dσ(ξ) |x0 − ξ|n+2
Z ≤ (1 − |x0 |)
−n−2
· Z
≤ (1 − |x0 |)−n−2 ≤ C·
|ξ − ξ0 |ω(|ξ − ξ0 |)dσ(ξ) E δ
ρω(ρ)ρn−2 dρ
0
ω(δ) , δ2
where δ = 1 − |x0 |. Here we used property 5◦ of the modulus function ω. On the other hand, there is a constant Cn such that |ξ − ξ0 | ≤ Cn for ξ ∈ F ξ − x0 | and therefore, using property 4◦ of ω, we have Z F
Z |ξ − ξ0 |ω(|ξ − ξ0 |) ω(|ξ − ξ0 |)dσ(ξ) n+2 dσ(ξ) ≤ C n n+2 |x0 − ξ| |ξ − ξ0 |n+1 Z 2E ≤ C· ρ−n−1 ω(ρ)ρn−2 dρ δ
ω(δ) ≤ C· 2 , δ
δ = 1 − |x0 |.
Combining the above estimates for integrals over E and F we obtain, for 1 ≤ j < n, ¯ ∂u ¯ ω(δ) ¯ ¯ (x0 )¯ ≤ C(n, ω) · , δ = 1 − |x0 | ¯ ∂xj δ and this is precisely estimate (∗) in direction of coordinate axis xj (1 ≤ j < n). However, the same estimate is true for any tangential direction, by rotational symmetry and (∗) is proved. Now, K−quasiregularity gives the estimate of the derivative of u: ω(δ) , δ = 1 − |x|. δ Using property 3◦ of ω and a simple argument involving integration one concludes the proof. ||u0 (x)|| ≤ KC(n, ω)
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Miloˇs Arsenovi´c and Vesna Manojlovi´c
References [1] Craig A. Nodler and Daniel M. Oberlin, Moduli of continuity and a HardyLittlewood theorem. In Complex analysis, Joensuu 1987, volume 1351 of Lecture Notes in Math., pages 265-272. Springer, Berlin 1988. [2] Arsenovi´c M., Koji´c V. and Mateljevi´c M., On Lipschitz continuity of harmonic quasiregular maps on the unit ball in Rn , Ann. Acad. Sci. Fenn., Volumen 33, 2008, 315-318. [3] Antoni Zygmund, Trigonometrical Series, 2nd ed. Vol I, II, Cambridge university Press, New York, 1959. [4] Sheldon Axler, Paul Bourdon and Wade Ramey, Harmonic Function Theory., Springer Verlag New York 1992. [5] M. Vuorinen, Conformal Geometry and quasiregular mappings, Lecture Notes in Math., 1319, Springer, Berlin, 1988 [6] Pavlovi´c M., Remarks on Lp -oscillation of the modulus of a holomorphic function, J. Math. Anal. Appl. 326 (2007), 1-11. [7] Pavlovi´c M., Lipschitz conditions on the modulus of a harmonic function, Revista Mat. Iberoamericana, 23 (2007), no. 3, 831-845. Miloˇs Arsenovi´c Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia E-mail:
[email protected] Vesna Manojlovi´c University of Belgrade, Faculty of Organizational Sciences, Jove Ilica 154, Belgrade, Serbia E-mail:
[email protected]
