International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article
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New Trends in Geometric Function Theory

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

Oleksiy Dovgoshey, Juhani Riihentaus, "Bi-Lipschitz Mappings and Quasinearly Subharmonic Functions", International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 382179, 8 pages, 2010. 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 𝑛, 𝑛2, these both classes are invariant under bi-Lipschitz mappings.

1. Introduction

Notation. Our notation is rather standard; see, for example, [13] and the references therein. We recall here only the following. The Lebesgue measure in 𝑛, 𝑛2, is denoted by 𝑚𝑛. We write 𝐵𝑛(𝑥,𝑟) for the ball in 𝑛, 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 𝑛, and 𝑥𝐷, 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 𝑛, 𝑛2. Let 𝑢Ω[,+) be a Lebesgue measurable function. We adopt the following definitions.

(i)𝑢 is subharmonic if 𝑢 is upper semicontinuous and if 1𝑢(𝑥)𝜈𝑛𝑟𝑛𝐵𝑛(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+1loc(Ω) and 1𝑢(𝑥)𝜈𝑛𝑟𝑛𝐵𝑛(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 𝐾1. Then 𝑢 is 𝐾-quasinearly subharmonic if 𝑢+1loc(Ω) and 𝑢𝐿(𝐾𝑥)𝜈𝑛𝑟𝑛𝐵𝑛(𝑥,𝑟)𝑢𝐿(𝑦)𝑑𝑚𝑛(𝑦)(1.3) for all 𝐿0 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 𝐾1. For the definition and properties of quasinearly subharmonic functions, see, for example, [1, 47] 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 𝑢Ω[0,+) is Lebesgue measurable, then 𝑢 is 𝐾-quasinearly subharmonic if and only if 𝑢1loc(Ω) and 𝐾𝑢(𝑥)𝜈𝑛𝑟𝑛𝐵𝑛(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 𝑛, 𝑛2. Let 𝑀1 be arbitrary. A function 𝑓𝐷𝑛 is 𝑀-bi-Lipschitz if

||||𝑦𝑥𝑀||||||||𝑓(𝑦)𝑓(𝑥)𝑀𝑦𝑥(1.5) for all 𝑥,𝑦𝐷. A function is bi-Lipschitz if it is 𝑀-bi-Lipschitz for some 𝑀1. It is easy to see that if 𝑓𝐷𝑛 is 𝑀-bi-Lipschitz, then also 𝑓1𝐷𝑛 is 𝑀-bi-Lipschitz, where 𝐷=𝑓(𝐷).

Let Ω be an open subset of 𝑛. Let 𝑝𝐷𝐷 and 𝑥ΩΩ. We write

𝑝𝑀-BiLip𝐷,𝑥Ω,𝐷,Ω=𝐷𝑛𝑝i𝑠𝑀-bi-Lipschitz,𝐷=𝑥Ω,(𝐷)Ω.(1.6)

2. On the Generalized Mean Value Inequality

Lemma 2.1. Let 𝐷 be a bounded open set in 𝑛, 𝑛2. Fix a point 𝑝𝐷𝐷. Let Ω be a domain in 𝑛. Let 𝑢Ω[0,+) be a 𝐾-quasinearly subharmonic function. Then there is 𝐶=𝐶(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)1 such that 𝑢𝑥Ω𝐶𝑚𝑛((𝐷))(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦)(2.1) for every point 𝑥ΩΩ and all 𝑀-BiLip(𝑝𝐷,𝑥Ω,𝐷,Ω), 𝑀1.

Proof. Take 𝑥ΩΩ and 𝑀-BiLip(𝑝𝐷,𝑥Ω,𝐷,Ω), 𝑀1, arbitrarily. (Observe that the set of bi-Lipschitz mappings is (in general) nonempty.) Write 𝑅𝐷=sup𝑦𝐷||𝑝𝐷||𝑦,𝑟𝐷=𝛿𝐷𝑝𝐷.(2.2) Using the fact that 𝐵𝑛(𝑝𝐷,𝑟𝐷)𝐵𝑛(𝑝𝐷,𝑟𝐷)(𝐵𝑛(𝑝𝐷,𝑟𝐷)) is a homeomorphism, one sees easily that 𝐵𝑛(𝑥Ω,𝑟𝐷/𝑀)(𝐷). Since is 𝑀-bi-Lipschitz, it follows from a result of Radó-Reichelderfer, see, for example, [8, Theorem 2.2, page 99], that 𝑚𝑛((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, 𝑢𝑥Ω𝐾𝜈𝑛(𝑟𝐷/𝑀)𝑛𝐵𝑛(𝑥Ω,𝑟𝐷/𝑀)𝑢(𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛𝐵𝑛𝑝𝐷,𝑅𝐷(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛(𝐷)(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛((𝐷))/𝑛!𝑀𝑛(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦)𝑛!𝐾𝑀2𝑛𝑅𝐷/𝑟𝐷𝑛𝑚𝑛((𝐷))(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 𝑛, 𝑛2, with 𝑚𝑛(𝐷)<+. Fix a point 𝑝𝐷𝐷. Let Ω be an open set in 𝑛. Let 𝑢Ω[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 𝑥ΩΩ and all 𝑀-BiLip(𝑝𝐷,𝑥Ω,𝐷,Ω), 𝑀1.

Proof. Let 𝑡>1 be arbitrary. It is easy to see that 𝑡𝑚𝑛(𝐷𝐵𝑛(𝑝𝐷,𝑟𝑡))𝑚𝑛(𝐷) for some 𝑟𝑡>0. Write 𝐷𝑡=𝐷𝐵𝑛(𝑝𝐷,𝑟𝑡) and 𝑝𝐷𝑡=𝑝𝐷. One sees easily that 𝐷𝑡 satisfies the assumptions of Lemma 2.1; that is, 𝐷𝑡 is a bounded domain, (𝐷𝑡)(𝐷)Ω and (𝑝𝐷𝑡)=(𝑝𝐷)=𝑥Ω. Hence there is a constant 𝐶1=𝐶1(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)1 such that 𝑢𝑥Ω𝐶1𝑚𝑛𝐷𝑡(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)(2.4) for every point 𝑥ΩΩ and all 𝑀-BiLip(𝑝𝐷𝑡,𝑥Ω,𝐷𝑡,Ω). Since and 1 are 𝑀-bi-Lipschitz, it follows that 𝑚𝑛((𝐷))𝑛!𝑀𝑛𝑚𝑛(𝐷) and 𝑚𝑛(𝐷𝑡)𝑛!𝑀𝑛𝑚𝑛((𝐷𝑡)); see again [8, Theorem 2.2, page 99]. Thus for 𝐶2=𝐶2(𝑛,𝑀)=(𝑛!)2𝑀2𝑛, 𝑚𝑛𝐷𝑡𝑚𝑛(𝐷)𝐶2𝑚𝑛𝐷𝑡𝑚𝑛.((𝐷))(2.5) Proceed then as follows: 1𝑚𝑛𝐷𝑡(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)𝐶2𝑚𝑛(𝐷)𝑚𝑛𝐷𝑡1𝑚𝑛((𝐷))(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)𝐶2𝑡𝑚𝑛((𝐷))(𝐷𝑡)𝑢(𝑦)𝑑𝑚𝑛(𝑦)𝐶2𝑡𝑚𝑛((𝐷))(𝐷)𝑢(𝑦)𝑑𝑚𝑛(𝑦).(2.6) Therefore 𝑢𝑥Ω𝐶1𝐶2𝑡𝑚𝑛((𝐷))(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 . If 𝑓𝑈𝐺 is analytic and 𝑢𝐺[,+) is subharmonic, then 𝑢𝑓 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 . Let 𝑢Ω[0,+) be a 𝐾-quasinearly subharmonic function. If 𝜑𝐺Ω is conformal, then the composition mapping 𝑢𝜑𝐺[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 𝑛, 𝑛2, 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 𝑛,𝑛2. Let 𝑢Ω[0,+) be a 𝐾-quasinearly subharmonic function. If 𝑓𝑈Ω is 𝑀-bi-Lipschitz, then the composition mapping 𝑢𝑓𝑈[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 (𝑥𝑢𝑓)0𝐶𝑚𝑛𝐵𝑛𝑥0,𝑟0𝐵𝑛(𝑥0,𝑟0)(𝑢𝑓)(𝑥)𝑑𝑚𝑛(𝑥)(3.1) for all 𝐵𝑛(𝑥0,𝑟0)𝑈. To see this, observe first that 𝐵𝑛𝑥0,𝑟0𝑀𝐵𝑓𝑛𝑥0,𝑟0𝐵𝑛𝑥0,𝑀𝑟0,(3.2) where 𝑥0=𝑓(𝑥0).
Then (𝑥𝑢𝑓)0𝑥=𝑢0𝐾𝑚𝑛𝐵𝑛𝑥0,𝑟0/𝑀𝐵𝑛(𝑥0,𝑟0/𝑀)𝑢(𝑦)𝑑𝑚𝑛(𝐾𝑦)𝜈𝑛(𝑟0/𝑀)𝑛𝐵𝑛(𝑥0,𝑟0/𝑀)𝑓(𝑢𝑓)1(𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀2𝑛𝜈𝑛(𝑀𝑟0)𝑛𝐵𝑛(𝑥0,𝑟0/𝑀)𝑓(𝑢𝑓)1(𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢𝑓)1(𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢𝑓)1||𝐽(𝑦)𝑓1||1(𝑦)||𝐽𝑓1||(𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢𝑓)1||𝐽(𝑦)𝑓1||||𝐽(𝑦)𝑓𝑓1(||𝑦)𝑑𝑚𝑛(𝑦)𝐾𝑀2𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢𝑓)1||𝐽(𝑦)𝑓1||(𝑦)𝑛!𝑀𝑛𝑑𝑚𝑛(𝑦)𝑛!𝐾𝑀3𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝑓(𝐵𝑛(𝑥0,𝑟0))𝑓(𝑢𝑓)1||𝐽(𝑦)𝑓1||(𝑦)𝑑𝑚𝑛(𝑦)𝑛!𝐾𝑀3𝑛𝑚𝑛𝑓𝐵𝑛𝑥0,𝑟0𝐵𝑛(𝑥0,𝑟0)(𝑢𝑓)(𝑥)𝑑𝑚𝑛(𝑥)𝑛!𝐾𝑀3𝑛𝑚𝑛𝐵𝑛𝑥0,𝑟0/𝑀𝐵𝑛(𝑥0,𝑟0)(𝑢𝑓)(𝑥)𝑑𝑚𝑛(𝑥)𝑛!𝐾𝑀4𝑛𝑚𝑛𝐵𝑛𝑥0,𝑟0𝐵𝑛(𝑥0,𝑟0)(𝑢𝑓)(𝑥)𝑑𝑚𝑛(𝑥).(3.3) Above we have used the routineous fact that for 𝑀-bi-Lipschitz mappings, ||𝐽𝑓𝑓1||(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 𝑛, 𝑛2. Let 𝑓Ω𝑚 be continuous. Write

𝐿(𝑥,𝑓)=limsup𝑦𝑥||||𝑓(𝑦)𝑓(𝑥)||||.𝑦𝑥(4.1) The function 𝑥𝐿(𝑥,𝑓) is a Borel function in Ω. If 𝑓 is differentiable at 𝑥, then 𝐿(𝑥,𝑓)=|𝑓(𝑥)|; see [9, page 11], [11, page 19], and [12, page 93].

A function 𝑓Ω is regularly oscillating, if there is 𝐾1 such that

𝐿(𝑥,𝑓)𝐾𝑟1sup𝑦𝐵𝑛(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 . Let 𝑢OC1𝐾(Ω). If 𝑓𝐺Ω is conformal, then 𝑢𝑓OC1𝐶(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 𝑛,𝑛2. Let 𝑢OC1𝐾(Ω). If 𝜑𝑈Ω is 𝑀-bi-Lipschitz, 𝑀1, then 𝑢𝜑OC1𝐾𝑀2(𝑈).

Proof. Let 𝜑𝑈Ω be 𝑀-bi-Lipschitz. Take 𝑥0𝑈 and 𝑟0>0 arbitrarily such that 𝐵𝑛(𝑥0,𝑟0)𝑈. Write 𝑥0=𝜑(𝑥0) and 𝑥=𝜑(𝑥) for 𝑥𝑈. Then 𝐿𝑥0,𝑢𝜑=limsup𝑥𝑥0||𝑢𝜑𝑥(𝜑(𝑥))𝑢0||||𝑥𝑥0||=limsup𝑥𝑥0||𝜑𝑥𝑢(𝜑(𝑥))𝑢0||||𝜑𝑥(𝑥)𝜑0||||𝑥𝜑(𝑥)𝜑0||||𝑥𝑥0||limsup𝑥𝑥0||𝑢𝑥𝑥𝑢0||||𝑥𝑥0||limsup𝑥𝑥0||𝑥𝜑(𝑥)𝜑0||||𝑥𝑥0||𝑥=𝐿0,𝑢limsup𝑥𝑥0||𝑥𝜑(𝑥)𝜑0||||𝑥𝑥0||.(4.3) Using (3.2) (for 𝑓=𝜑), we get 𝐿𝑥0𝐾,𝑢𝑟0/𝑀sup𝑥𝐵𝑛𝑥0,𝑟0/𝑀||𝑢𝑥𝑥𝑢0||𝐾𝑀𝑟0sup𝑥𝐵𝑛𝑥0,𝑟0/𝑀||𝑢𝑥𝑥𝑢0||𝐾𝑀𝑟0sup𝑥𝐵𝜑𝑛𝑥0,𝑟0||𝑢𝑥𝑥𝑢0||𝐾𝑀𝑟0sup𝑥𝐵𝑛𝑥0,𝑟0||𝜑𝑥𝑢(𝜑(𝑥))𝑢0||𝐾𝑀𝑟0sup𝑥𝐵𝑛𝑥0,𝑟0||𝑥(𝑢𝜑)(𝑥)(𝑢𝜑)0||.(4.4)
On the other hand, since 𝜑 is 𝑀-bi-Lipschitz, limsup𝑥𝑥0||𝜑𝑥(𝑥)𝜑0||||𝑥𝑥0||limsup𝑥𝑥0𝑀||𝑥𝑥0||||𝑥𝑥0||=𝑀<+.(4.5) Therefore, 𝐿𝑥0,𝑢𝜑𝐾𝑀𝑟0sup𝑥𝐵𝑛(𝑥0,𝑟0)||𝑥(𝑢𝜑)(𝑥)(𝑢𝜑)0||𝑀𝐾𝑀2𝑟0sup𝑥𝐵𝑛(𝑥0,𝑟0)||𝑥(𝑢𝜑)(𝑥)(𝑢𝜑)0||.(4.6) Thus 𝑢𝜑OC1𝐾𝑀2(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 𝑛, 𝑛2. Let 𝐾1. A function 𝑓Ω is in HC1𝐾(Ω) if

𝐿(𝑥,𝑓)𝐾𝑟1sup𝑦𝐵𝑛(𝑥,𝑟)||||,𝑓(𝑦)𝐵𝑛(𝑥,𝑟)Ω.(4.7) The class HC1(Ω) is the union of all HC1𝐾(Ω), 𝐾1. 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 𝑛,𝑛2. Let 𝑢HC1𝐾(Ω). If 𝜑𝑈𝑛 is 𝑀-bi-Lipschitz, 𝑀1, then 𝑢𝜑HC1𝐾𝑀2(𝑈).

Acknowledgment

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

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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.

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