International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article
Special Issue

New Trends in Geometric Function Theory

View this Special Issue

Research Article | Open Access

Volume 2010 |Article ID 382179 | 8 pages | https://doi.org/10.1155/2010/382179

Bi-Lipschitz Mappings and Quasinearly Subharmonic Functions

Academic Editor: Stanisława R. Kanas
Received30 Nov 2009
Accepted25 Dec 2009
Published27 Jan 2010

Abstract

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojić has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under conformal mappings. We give partial generalizations to her results by showing that in ℝ𝑛, 𝑛≥2, these both classes are invariant under bi-Lipschitz mappings.

1. Introduction

Notation. Our notation is rather standard; see, for example, [1–3] and the references therein. We recall here only the following. The Lebesgue measure in ℝ𝑛, 𝑛≥2, is denoted by 𝑚𝑛. We write 𝐵𝑛(𝑥,𝑟) for the ball in ℝ𝑛, with center 𝑥 and radius 𝑟. Recall that 𝑚𝑛(𝐵𝑛(𝑥,𝑟))=𝜈𝑛𝑟𝑛, where 𝜈𝑛∶=𝑚𝑛(𝐵𝑛(0,1)). If 𝐷 is an open set in ℝ𝑛, and 𝑥∈𝐷, then we write 𝛿𝐷(𝑥) for the distance between the point 𝑥 and the boundary 𝜕𝐷 of 𝐷. Our constants 𝐶 are nonnegative, mostly ≥1, and may vary from line to line.

1.1. Subharmonic Functions and Generalizations

Let Ω be an open set in ℝ𝑛, 𝑛≥2. Let 𝑢∶Ω→[−∞,+∞) be a Lebesgue measurable function. We adopt the following definitions.

(i)𝑢 is subharmonic if 𝑢 is upper semicontinuous and if 1𝑢(𝑥)≤𝜈𝑛𝑟𝑛𝐵𝑛(𝑥,𝑟)𝑢(𝑦)𝑑𝑚𝑛(𝑦)(1.1) for all balls 𝐵𝑛(𝑥,𝑟)⊂Ω. A subharmonic function may be ≡−∞ on any component of Ω; see [3, page 9] and [4, page 60]. (ii)𝑢 is nearly subharmonic if 𝑢+∈ℒ1loc(Ω) and 1𝑢(𝑥)≤𝜈𝑛𝑟𝑛𝐵𝑛(𝑥,𝑟)𝑢(𝑦)𝑑𝑚𝑛(𝑦)(1.2) for all balls 𝐵𝑛(𝑥,𝑟)⊂Ω. Observe that this definition, see [5, page 51], is slightly more general than the standard one [3, page 14]. (iii)Let 𝐾≥1. Then 𝑢 is 𝐾-quasinearly subharmonic if 𝑢+∈ℒ1loc(Ω) and 𝑢𝐿(𝐾𝑥)≤𝜈𝑛𝑟𝑛𝐵𝑛(𝑥,𝑟)𝑢𝐿(𝑦)𝑑𝑚𝑛(𝑦)(1.3) for all 𝐿≥0 and for all balls 𝐵𝑛(𝑥,𝑟)⊂Ω. Here 𝑢𝐿∶=max{𝑢,−𝐿}+𝐿.

The function 𝑢 is quasinearly subharmonic if 𝑢 is 𝐾-quasinearly subharmonic for some 𝐾≥1. For the definition and properties of quasinearly subharmonic functions, see, for example, [1, 4–7] and the references therein.

Proposition 1.1 (cf. [5, Proposition 2.1, pages 54-55]). The following holds. (i)A subharmonic function is nearly subharmonic but not conversely. (ii)A function is nearly subharmonic if and only if it is 1-quasinearly subharmonic. (iii)A nearly subharmonic function is quasinearly subharmonic but not conversely. (iv)If 𝑢∶Ω→[0,+∞) is Lebesgue measurable, then 𝑢 is 𝐾-quasinearly subharmonic if and only if 𝑢∈ℒ1loc(Ω) and 𝐾𝑢(𝑥)≤𝜈𝑛𝑟𝑛𝐵𝑛(𝑥,𝑟)𝑢(𝑦)𝑑𝑚𝑛(𝑦)(1.4) for all balls 𝐵𝑛(𝑥,𝑟)⊂Ω.

1.2. Bi-Lipschitz Mappings

Let 𝐷 be an open set in ℝ𝑛, 𝑛≥2. Let 𝑀≥1 be arbitrary. A function 𝑓∶𝐷→ℝ𝑛 is 𝑀-bi-Lipschitz if

||||𝑦−𝑥𝑀≤||||||||𝑓(𝑦)−𝑓(𝑥)≤𝑀𝑦−𝑥(1.5) for all 𝑥,𝑦∈𝐷. A function is bi-Lipschitz if it is 𝑀-bi-Lipschitz for some 𝑀≥1. It is easy to see that if 𝑓∶𝐷→ℝ𝑛 is 𝑀-bi-Lipschitz, then also 𝑓−1âˆ¶ğ·î…žâ†’â„ğ‘› is 𝑀-bi-Lipschitz, where ğ·î…žâˆ¶=𝑓(𝐷).

Let Ω be an open subset of ℝ𝑛. Let 𝑝𝐷∈𝐷 and 𝑥Ω∈Ω. We write

𝑝𝑀-BiLip𝐷,𝑥Ω,𝐷,Ω∶=â„Žâˆ¶ğ·â†’â„ğ‘›î€·ğ‘âˆ¶â„Ži𝑠𝑀-bi-Lipschitz,â„Žğ·î€¸=𝑥Ω,ℎ(𝐷)⊂Ω.(1.6)

2. On the Generalized Mean Value Inequality

Lemma 2.1. Let 𝐷 be a bounded open set in ℝ𝑛, 𝑛≥2. Fix a point 𝑝𝐷∈𝐷. Let Ω be a domain in ℝ𝑛. Let 𝑢∶Ω→[0,+∞) be a 𝐾-quasinearly subharmonic function. Then there is 𝐶=𝐶(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)≥1 such that 𝑢𝑥Ω≤𝐶𝑚𝑛(ℎ(𝐷))ℎ(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦)(2.1) for every point 𝑥Ω∈Ω and all â„Žâˆˆğ‘€-BiLip(𝑝𝐷,𝑥Ω,𝐷,Ω), 𝑀≥1.

Proof. Take 𝑥Ω∈Ω and â„Žâˆˆğ‘€-BiLip(𝑝𝐷,𝑥Ω,𝐷,Ω), 𝑀≥1, arbitrarily. (Observe that the set of bi-Lipschitz mappings is (in general) nonempty.) Write 𝑅𝐷∶=sup𝑦∈𝐷||𝑝𝐷||−𝑦,𝑟𝐷∶=𝛿𝐷𝑝𝐷.(2.2) Using the fact that â„Žâˆ£ğµğ‘›(𝑝𝐷,𝑟𝐷)∶𝐵𝑛(𝑝𝐷,𝑟𝐷)→ℎ(𝐵𝑛(𝑝𝐷,𝑟𝐷)) is a homeomorphism, one sees easily that 𝐵𝑛(𝑥Ω,𝑟𝐷/𝑀)⊂ℎ(𝐷). Since ℎ is 𝑀-bi-Lipschitz, it follows from a result of Radó-Reichelderfer, see, for example, [8, Theorem 2.2, page 99], that 𝑚𝑛(ℎ(𝐷))≤𝑛!𝑀𝑛𝑚𝑛(𝐷). (Observe that bi-Lipschitz mappings satisfy the property 𝑁 and are differentiable almost everywhere, see, for example, [9, Theorem 33.2, page 112, Theorem 32.1, page 109].) Therefore, 𝑢𝑥Ω≤𝐾𝜈𝑛(𝑟𝐷/𝑀)𝑛𝐵𝑛(𝑥Ω,𝑟𝐷/𝑀)𝑢(𝑦)𝑑𝑚𝑛(≤𝑦)𝐾𝑀𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛𝐵𝑛𝑝𝐷,ğ‘…ğ·î€œî€¸î€¸â„Ž(𝐷)𝑢(𝑦)𝑑𝑚𝑛≤(𝑦)𝐾𝑀𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛(𝐷)ℎ(𝐷)𝑢(𝑦)𝑑𝑚𝑛≤(𝑦)𝐾𝑀𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛(ℎ(𝐷))/𝑛!ğ‘€ğ‘›î€œâ„Ž(𝐷)𝑢(𝑦)𝑑𝑚𝑛≤(𝑦)𝑛!𝐾𝑀2𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛(ℎ(𝐷))ℎ(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦).(2.3) Thus (2.1) holds with 𝐶=𝐶(𝐾,𝑛,𝑀,𝐷,𝑝𝐷).

Theorem 2.2. Let 𝐷 be an open set in ℝ𝑛, 𝑛≥2, with 𝑚𝑛(𝐷)<+∞. Fix a point 𝑝𝐷∈𝐷. Let Ω be an open set in ℝ𝑛. Let 𝑢∶Ω→[0,+∞) be a 𝐾-quasinearly subharmonic function. Then there is a constant 𝐶=𝐶(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)≥1 such that (2.1) holds for every point 𝑥Ω∈Ω and all â„Žâˆˆğ‘€-BiLip(𝑝𝐷,𝑥Ω,𝐷,Ω), 𝑀≥1.

Proof. Let 𝑡>1 be arbitrary. It is easy to see that 𝑡𝑚𝑛(𝐷∩𝐵𝑛(𝑝𝐷,𝑟𝑡))≥𝑚𝑛(𝐷) for some 𝑟𝑡>0. Write 𝐷𝑡∶=𝐷∩𝐵𝑛(𝑝𝐷,𝑟𝑡) and 𝑝𝐷𝑡=𝑝𝐷. One sees easily that 𝐷𝑡 satisfies the assumptions of Lemma 2.1; that is, 𝐷𝑡 is a bounded domain, ℎ(𝐷𝑡)⊂ℎ(𝐷)⊂Ω and ℎ(𝑝𝐷𝑡)=ℎ(𝑝𝐷)=𝑥Ω. Hence there is a constant 𝐶1=𝐶1(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)≥1 such that 𝑢𝑥Ω≤𝐶1ğ‘šğ‘›î€·â„Žî€·ğ·ğ‘¡î€œî€¸î€¸â„Ž(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)(2.4) for every point 𝑥Ω∈Ω and all â„Žâˆˆğ‘€-BiLip(𝑝𝐷𝑡,𝑥Ω,𝐷𝑡,Ω). Since ℎ and ℎ−1 are 𝑀-bi-Lipschitz, it follows that 𝑚𝑛(ℎ(𝐷))≤𝑛!𝑀𝑛𝑚𝑛(𝐷) and 𝑚𝑛(𝐷𝑡)≤𝑛!𝑀𝑛𝑚𝑛(ℎ(𝐷𝑡)); see again [8, Theorem 2.2, page 99]. Thus for 𝐶2=𝐶2(𝑛,𝑀)=(𝑛!)2𝑀2𝑛, 𝑚𝑛𝐷𝑡𝑚𝑛(𝐷)≤𝐶2â‹…ğ‘šğ‘›î€·â„Žî€·ğ·ğ‘¡î€¸î€¸ğ‘šğ‘›.(ℎ(𝐷))(2.5) Proceed then as follows: 1ğ‘šğ‘›î€·â„Žî€·ğ·ğ‘¡î€œî€¸î€¸â„Ž(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)≤𝐶2⋅𝑚𝑛(𝐷)𝑚𝑛𝐷𝑡⋅1𝑚𝑛(ℎ(𝐷))ℎ(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)≤𝐶2⋅𝑡𝑚𝑛(ℎ(𝐷))ℎ(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)≤𝐶2⋅𝑡𝑚𝑛(ℎ(𝐷))ℎ(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦).(2.6) Therefore 𝑢𝑥Ω≤𝐶1𝐶2𝑡𝑚𝑛(ℎ(𝐷))ℎ(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦),(2.7) concluding the proof.

3. An Invariance of the Class of Quasinearly Subharmonic Functions

Suppose that 𝐺 and 𝑈 are open sets in the complex plane ℂ. If 𝑓∶𝑈→𝐺 is analytic and 𝑢∶𝐺→[−∞,+∞) is subharmonic, then 𝑢∘𝑓 is subharmonic; see, for example, [3, page 37] and [4, Corollary 3.3.4, page 70]. Using Koebe’s one-quarter and distortion theorems, Kojić proved the following partial generalization.

Theorem 3.1 (see [6, Theorem 1, page 245]). Let Ω and 𝐺 be open sets in ℂ. Let 𝑢∶Ω→[0,+∞) be a 𝐾-quasinearly subharmonic function. If 𝜑∶𝐺→Ω is conformal, then the composition mapping 𝑢∘𝜑∶𝐺→[0,+∞) is 𝐶-quasinearly subharmonic for some 𝐶=𝐶(𝐾).

For the definition and properties of conformal mappings, see, for example, [9, pages 13–15] and [8, pages 171-172].

Below we give a partial generalization to Kojić’s result. Our result gives also a partial generalization to the standard result according to which in ℝ𝑛, 𝑛≥2, the class of subharmonic functions is invariant under orthogonal transformations; see [10, page 55].

Theorem 3.2. Let Ω and 𝑈 be open sets in ℝ𝑛,𝑛≥2. Let 𝑢∶Ω→[0,+∞) be a 𝐾-quasinearly subharmonic function. If 𝑓∶𝑈→Ω is 𝑀-bi-Lipschitz, then the composition mapping 𝑢∘𝑓∶𝑈→[0,+∞) is 𝐶-quasinearly subharmonic for some 𝐶=𝐶(𝐾,𝑛,𝑀).

Proof. It is sufficient to show that there exists a constant 𝐶=𝐶(𝐾,𝑛,𝑓)>0 such that (𝑥𝑢∘𝑓)0≤𝐶𝑚𝑛𝐵𝑛𝑥0,𝑟0𝐵𝑛(𝑥0,𝑟0)(𝑢∘𝑓)(𝑥)𝑑𝑚𝑛(𝑥)(3.1) for all 𝐵𝑛(𝑥0,𝑟0)⊂𝑈. To see this, observe first that ğµğ‘›î‚€ğ‘¥î…ž0,𝑟0𝑀𝐵⊂𝑓𝑛𝑥0,𝑟0î€¸î€¸âŠ‚ğµğ‘›î€·ğ‘¥î…ž0,𝑀𝑟0,(3.2) where ğ‘¥î…ž0=𝑓(𝑥0).
Then (𝑥𝑢∘𝑓)0𝑥=ğ‘¢î…ž0î€¸â‰¤ğ¾ğ‘šğ‘›î€·ğµğ‘›î€·ğ‘¥î…ž0,𝑟0/𝑀𝐵𝑛(𝑥′0,𝑟0/𝑀)𝑢(𝑦)𝑑𝑚𝑛(≤𝐾𝑦)𝜈𝑛(𝑟0/𝑀)𝑛𝐵𝑛(𝑥′0,𝑟0/𝑀)𝑓(𝑢∘𝑓)−1(𝑦)𝑑𝑚𝑛≤(𝑦)𝐾𝑀2𝑛𝜈𝑛(𝑀𝑟0)𝑛𝐵𝑛(𝑥′0,𝑟0/𝑀)𝑓(𝑢∘𝑓)−1(𝑦)𝑑𝑚𝑛≤(𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢∘𝑓)−1(𝑦)𝑑𝑚𝑛≤(𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢∘𝑓)−1||𝐽(𝑦)𝑓−1||⋅1(𝑦)||𝐽𝑓−1||(𝑦)𝑑𝑚𝑛≤(𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢∘𝑓)−1||𝐽(𝑦)𝑓−1||⋅||𝐽(𝑦)𝑓𝑓−1(||𝑦)𝑑𝑚𝑛(≤𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢∘𝑓)−1||𝐽(𝑦)𝑓−1||(𝑦)⋅𝑛!𝑀𝑛𝑑𝑚𝑛≤(𝑦)𝑛!𝐾𝑀3𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢∘𝑓)−1||𝐽(𝑦)𝑓−1||(𝑦)𝑑𝑚𝑛≤(𝑦)𝑛!𝐾𝑀3𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝐵𝑛(𝑥0,𝑟0)(𝑢∘𝑓)(𝑥)𝑑𝑚𝑛≤(𝑥)𝑛!𝐾𝑀3ğ‘›ğ‘šğ‘›î€·ğµğ‘›î€·ğ‘¥î…ž0,𝑟0/𝑀𝐵𝑛(𝑥0,𝑟0)(𝑢∘𝑓)(𝑥)𝑑𝑚𝑛≤(𝑥)𝑛!𝐾𝑀4𝑛𝑚𝑛𝐵𝑛𝑥0,𝑟0𝐵𝑛(𝑥0,𝑟0)(𝑢∘𝑓)(𝑥)𝑑𝑚𝑛(𝑥).(3.3) Above we have used the routineous fact that for 𝑀-bi-Lipschitz mappings, ||𝐽𝑓𝑓−1||(𝑦)≤𝑛!𝑀𝑛,(3.4) and the already cited change of variable result of Radó-Reichelderfer; see [8, Theorem 2.2, page 99]. (Recall again that bi-Lipschitz mappings satisfy the property 𝑁 and are differentiable almost everywhere.)

4. An Invariance of Regularly Oscillating Functions

Let Ω be an open set in ℝ𝑛, 𝑛≥2. Let 𝑓∶Ω→ℝ𝑚 be continuous. Write

𝐿(𝑥,𝑓)∶=limsup𝑦→𝑥||||𝑓(𝑦)−𝑓(𝑥)||||.𝑦−𝑥(4.1) The function 𝑥↦𝐿(𝑥,𝑓) is a Borel function in Ω. If 𝑓 is differentiable at 𝑥, then 𝐿(𝑥,𝑓)=|ğ‘“î…ž(𝑥)|; see [9, page 11], [11, page 19], and [12, page 93].

A function 𝑓∶Ω→ℝ is regularly oscillating, if there is 𝐾≥1 such that

𝐿(𝑥,𝑓)≤𝐾𝑟−1sup𝑦∈𝐵𝑛(𝑥,𝑟)||||,𝑓(𝑦)−𝑓(𝑥)𝐵𝑛(𝑥,𝑟)⊂Ω.(4.2) The class of such functions is denoted by OC1𝐾(Ω). The class of all regularly oscillating functions is denoted by RO(Ω); see [11, page 19], [13, page 17], [14], [6, page 245], and [12, page 96].

Using again Koebe’s results, Kojić proved also the following result.

Theorem 4.1 (see [6, Theorem 2, page 245]). Let Ω and 𝐺 be open sets in ℂ. Let 𝑢∈OC1𝐾(Ω). If 𝑓∶𝐺→Ω is conformal, then 𝑢∘𝑓∈OC1𝐶(𝐺), where 𝐶 depends only on 𝐾.

Below we give a partial generalization to Kojić’s above result.

Theorem 4.2. Let Ω and 𝑈 be open sets in ℝ𝑛,𝑛≥2. Let 𝑢∈OC1𝐾(Ω). If 𝜑∶𝑈→Ω is 𝑀-bi-Lipschitz, 𝑀≥1, then 𝑢∘𝜑∈OC1𝐾𝑀2(𝑈).

Proof. Let 𝜑∶𝑈→Ω be 𝑀-bi-Lipschitz. Take 𝑥0∈𝑈 and 𝑟0>0 arbitrarily such that 𝐵𝑛(𝑥0,𝑟0)⊂𝑈. Write ğ‘¥î…ž0=𝜑(𝑥0) and ğ‘¥î…ž=𝜑(𝑥) for 𝑥∈𝑈. Then 𝐿𝑥0,𝑢∘𝜑=limsup𝑥→𝑥0||𝑢𝜑𝑥(𝜑(𝑥))−𝑢0||||𝑥−𝑥0||=limsup𝑥→𝑥0||𝜑𝑥𝑢(𝜑(𝑥))−𝑢0||||𝜑𝑥(𝑥)−𝜑0||⋅||𝑥𝜑(𝑥)−𝜑0||||𝑥−𝑥0||≤limsup𝑥′→𝑥′0||ğ‘¢î€·ğ‘¥î…žî€¸î€·ğ‘¥âˆ’ğ‘¢î…ž0||||ğ‘¥î…žâˆ’ğ‘¥î…ž0||⋅limsup𝑥→𝑥0||𝑥𝜑(𝑥)−𝜑0||||𝑥−𝑥0||𝑥=ğ¿î…ž0,𝑢⋅limsup𝑥→𝑥0||𝑥𝜑(𝑥)−𝜑0||||𝑥−𝑥0||.(4.3) Using (3.2) (for 𝑓=𝜑), we get ğ¿î€·ğ‘¥î…ž0≤𝐾,𝑢𝑟0/𝑀sup𝑥′∈𝐵𝑛𝑥′0,𝑟0/𝑀||ğ‘¢î€·ğ‘¥î…žî€¸î€·ğ‘¥âˆ’ğ‘¢î…ž0||≤𝐾𝑀𝑟0sup𝑥′∈𝐵𝑛𝑥′0,𝑟0/𝑀||ğ‘¢î€·ğ‘¥î…žî€¸î€·ğ‘¥âˆ’ğ‘¢î…ž0||≤𝐾𝑀𝑟0sup𝑥′𝐵∈𝜑𝑛𝑥0,𝑟0||ğ‘¢î€·ğ‘¥î…žî€¸î€·ğ‘¥âˆ’ğ‘¢î…ž0||≤𝐾𝑀𝑟0sup𝑥∈𝐵𝑛𝑥0,𝑟0||𝜑𝑥𝑢(𝜑(𝑥))−𝑢0||≤𝐾𝑀𝑟0sup𝑥∈𝐵𝑛𝑥0,𝑟0||𝑥(𝑢∘𝜑)(𝑥)−(𝑢∘𝜑)0||.(4.4)
On the other hand, since 𝜑 is 𝑀-bi-Lipschitz, limsup𝑥→𝑥0||𝜑𝑥(𝑥)−𝜑0||||𝑥−𝑥0||≤limsup𝑥→𝑥0𝑀||𝑥−𝑥0||||𝑥−𝑥0||=𝑀<+∞.(4.5) Therefore, 𝐿𝑥0≤,𝑢∘𝜑𝐾𝑀𝑟0sup𝑥∈𝐵𝑛(𝑥0,𝑟0)||𝑥(𝑢∘𝜑)(𝑥)−(𝑢∘𝜑)0||≤⋅𝑀𝐾𝑀2𝑟0sup𝑥∈𝐵𝑛(𝑥0,𝑟0)||𝑥(𝑢∘𝜑)(𝑥)−(𝑢∘𝜑)0||.(4.6) Thus 𝑢∘𝜑∈OC1𝐾𝑀2(𝑈).

In addition of regularly oscillating functions, one sometimes considers so-called HC1 functions, too; see [11, page 19], [13, page 16], and [12, page 93]. Their definition reads as follows. Let Ω be an open set in ℝ𝑛, 𝑛≥2. Let 𝐾≥1. A function 𝑓∶Ω→ℝ is in HC1𝐾(Ω) if

𝐿(𝑥,𝑓)≤𝐾𝑟−1sup𝑦∈𝐵𝑛(𝑥,𝑟)||||,𝑓(𝑦)𝐵𝑛(𝑥,𝑟)⊂Ω.(4.7) The class HC1(Ω) is the union of all HC1𝐾(Ω), 𝐾≥1. Clearly, HC12𝐾(Ω)⊂OC1𝐾(Ω).

Proceeding as above in the proof of Theorem 4.2 one gets the following result.

Theorem 4.3. Let Ω and 𝑈 be open sets in ℝ𝑛,𝑛≥2. Let 𝑢∈HC1𝐾(Ω). If 𝜑∶𝑈→ℝ𝑛 is 𝑀-bi-Lipschitz, 𝑀≥1, then 𝑢∘𝜑∈HC1𝐾𝑀2(𝑈).

Acknowledgment

The first author was partially supported by the Academy of Finland.

References

  1. O. Dovgoshey and J. Riihentaus, “A remark concerning generalized mean value inequalities for subharmonic functions,” submitted. View at: Google Scholar
  2. O. Dovgoshey and J. Riihentaus, “Mean value type inequalities for quasi-nearly subharmonic functions,” manuscript. View at: Google Scholar
  3. M. Hervé, Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces, vol. 198 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1971. View at: MathSciNet
  4. D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer, London, UK, 2001.
  5. J. Riihentaus, “Subharmonic functions, generalizations and separately subharmonic functions,” in Proceedings of the 14th Conference on Analytic Functions, Chełm, Poland, July, 2007. View at: Google Scholar
  6. V. Kojić, “Quasi-nearly subharmonic functions and conformal mappings,” Filomat, vol. 21, no. 2, pp. 243–249, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  7. O. Djordjević and M. Pavlović, “ℒp-integrability of the maximal function of a polyharmonic function,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 411–417, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  8. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, vol. 73 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1989. View at: MathSciNet
  9. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, vol. 229 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1971. View at: MathSciNet
  10. W. K. Hayman and P. B. Kennedy, Subharmonic Functions. Vol. I, London Mathematical Society Monographs, Academic Press, London, UK, 1976. View at: MathSciNet
  11. M. Pavlović, “On subharmonic behaviour and oscillation of functions on balls in ℝn,” Institut Mathématique. Publications. Nouvelle Série, vol. 55, no. 69, pp. 18–22, 1994. View at: Google Scholar | MathSciNet
  12. M. Pavlović and J. Riihentaus, “Classes of quasi-nearly subharmonic functions,” Potential Analysis, vol. 29, no. 1, pp. 89–104, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. M. Pavlović, “Subharmonic behaviour of smooth functions,” Matematichki Vesnik, vol. 48, no. 1-2, pp. 15–21, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  14. M. Pavlović, Introduction to Function Spaces on the Disk, vol. 20 of Posebna Izdanja, Matematički Institut SANU, Belgrade, Serbia, 2004. View at: MathSciNet

Copyright © 2010 Oleksiy Dovgoshey and Juhani Riihentaus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

1136 Views | 391 Downloads | 4 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.