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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2010Β (2010), Article IDΒ 382179, 8 pages
http://dx.doi.org/10.1155/2010/382179
Research Article

Bi-Lipschitz Mappings and Quasinearly Subharmonic Functions

1Institute of Applied Mathematics and Mechanics, NASU, R. Luxemburg Street 74, Donetsk 83114, Ukraine
2Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, 80101 Joensuu, Finland

Received 30 November 2009; Accepted 25 December 2009

Academic Editor: StanisΕ‚awa R.Β Kanas

Copyright Β© 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.

Abstract

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. KojiΔ‡ 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, 𝑛β‰₯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, 𝑛β‰₯2, is denoted by π‘šπ‘›. We write 𝐡𝑛(π‘₯,π‘Ÿ) for the ball in ℝ𝑛, with center π‘₯ and radius π‘Ÿ. Recall that π‘šπ‘›(𝐡𝑛(π‘₯,π‘Ÿ))=πœˆπ‘›π‘Ÿπ‘›, where πœˆπ‘›βˆΆ=π‘šπ‘›(𝐡𝑛(0,1)). If 𝐷 is an open set in ℝ𝑛, and π‘₯∈𝐷, then we write 𝛿𝐷(π‘₯) for the distance between the point π‘₯ and the boundary πœ•π· of 𝐷. Our constants 𝐢 are nonnegative, mostly β‰₯1, and may vary from line to line.

1.1. Subharmonic Functions and Generalizations

Let Ξ© be an open set in ℝ𝑛, 𝑛β‰₯2. Let π‘’βˆΆΞ©β†’[βˆ’βˆž,+∞) be a Lebesgue measurable function. We adopt the following definitions.

(i)𝑒 is subharmonic if 𝑒 is upper semicontinuous and if 1𝑒(π‘₯)β‰€πœˆπ‘›π‘Ÿπ‘›ξ€œπ΅π‘›(π‘₯,π‘Ÿ)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)(1.1) for all balls 𝐡𝑛(π‘₯,π‘Ÿ)βŠ‚Ξ©. A subharmonic function may be β‰‘βˆ’βˆž on any component of Ξ©; see [3, page 9] and [4, page 60]. (ii)𝑒 is nearly subharmonic if 𝑒+βˆˆβ„’1loc(Ξ©) and 1𝑒(π‘₯)β‰€πœˆπ‘›π‘Ÿπ‘›ξ€œπ΅π‘›(π‘₯,π‘Ÿ)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)(1.2) for all balls 𝐡𝑛(π‘₯,π‘Ÿ)βŠ‚Ξ©. Observe that this definition, see [5, page 51], is slightly more general than the standard one [3, page 14]. (iii)Let 𝐾β‰₯1. Then 𝑒 is 𝐾-quasinearly subharmonic if 𝑒+βˆˆβ„’1loc(Ξ©) and 𝑒𝐿(𝐾π‘₯)β‰€πœˆπ‘›π‘Ÿπ‘›ξ€œπ΅π‘›(π‘₯,π‘Ÿ)𝑒𝐿(𝑦)π‘‘π‘šπ‘›(𝑦)(1.3) for all 𝐿β‰₯0 and for all balls 𝐡𝑛(π‘₯,π‘Ÿ)βŠ‚Ξ©. Here π‘’πΏβˆΆ=max{𝑒,βˆ’πΏ}+𝐿.

The function 𝑒 is quasinearly subharmonic if 𝑒 is 𝐾-quasinearly subharmonic for some 𝐾β‰₯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 π‘’βˆΆΞ©β†’[0,+∞) is Lebesgue measurable, then 𝑒 is 𝐾-quasinearly subharmonic if and only if π‘’βˆˆβ„’1loc(Ξ©) and 𝐾𝑒(π‘₯)β‰€πœˆπ‘›π‘Ÿπ‘›ξ€œπ΅π‘›(π‘₯,π‘Ÿ)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)(1.4) for all balls 𝐡𝑛(π‘₯,π‘Ÿ)βŠ‚Ξ©.

1.2. Bi-Lipschitz Mappings

Let 𝐷 be an open set in ℝ𝑛, 𝑛β‰₯2. Let 𝑀β‰₯1 be arbitrary. A function π‘“βˆΆπ·β†’β„π‘› is 𝑀-bi-Lipschitz if

||||π‘¦βˆ’π‘₯𝑀≀||||||||𝑓(𝑦)βˆ’π‘“(π‘₯)β‰€π‘€π‘¦βˆ’π‘₯(1.5) for all π‘₯,π‘¦βˆˆπ·. A function is bi-Lipschitz if it is 𝑀-bi-Lipschitz for some 𝑀β‰₯1. It is easy to see that if π‘“βˆΆπ·β†’β„π‘› is 𝑀-bi-Lipschitz, then also π‘“βˆ’1βˆΆπ·ξ…žβ†’β„π‘› is 𝑀-bi-Lipschitz, where π·ξ…žβˆΆ=𝑓(𝐷).

Let Ξ© be an open subset of ℝ𝑛. Let π‘π·βˆˆπ· and π‘₯Ω∈Ω. We write

𝑝𝑀-BiLip𝐷,π‘₯Ξ©ξ€Έξ€½,𝐷,Ω∢=β„ŽβˆΆπ·β†’β„π‘›ξ€·π‘βˆΆβ„Ži𝑠𝑀-bi-Lipschitz,β„Žπ·ξ€Έ=π‘₯Ξ©ξ€Ύ,β„Ž(𝐷)βŠ‚Ξ©.(1.6)

2. On the Generalized Mean Value Inequality

Lemma 2.1. Let 𝐷 be a bounded open set in ℝ𝑛, 𝑛β‰₯2. Fix a point π‘π·βˆˆπ·. Let Ξ© be a domain in ℝ𝑛. Let π‘’βˆΆΞ©β†’[0,+∞) be a 𝐾-quasinearly subharmonic function. Then there is 𝐢=𝐢(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)β‰₯1 such that 𝑒π‘₯Ξ©ξ€Έβ‰€πΆπ‘šπ‘›ξ€œ(β„Ž(𝐷))β„Ž(𝐷)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)(2.1) for every point π‘₯Ω∈Ω and all β„Žβˆˆπ‘€-BiLip(𝑝𝐷,π‘₯Ξ©,𝐷,Ξ©), 𝑀β‰₯1.

Proof. Take π‘₯Ω∈Ω and β„Žβˆˆπ‘€-BiLip(𝑝𝐷,π‘₯Ξ©,𝐷,Ξ©), 𝑀β‰₯1, arbitrarily. (Observe that the set of bi-Lipschitz mappings is (in general) nonempty.) Write π‘…π·βˆΆ=supπ‘¦βˆˆπ·||𝑝𝐷||βˆ’π‘¦,π‘Ÿπ·βˆΆ=𝛿𝐷𝑝𝐷.(2.2) Using the fact that β„Žβˆ£π΅π‘›(𝑝𝐷,π‘Ÿπ·)βˆΆπ΅π‘›(𝑝𝐷,π‘Ÿπ·)β†’β„Ž(𝐡𝑛(𝑝𝐷,π‘Ÿπ·)) is a homeomorphism, one sees easily that 𝐡𝑛(π‘₯Ξ©,π‘Ÿπ·/𝑀)βŠ‚β„Ž(𝐷). Since β„Ž is 𝑀-bi-Lipschitz, it follows from a result of RadΓ³-Reichelderfer, see, for example, [8, Theorem 2.2, page 99], that π‘šπ‘›(β„Ž(𝐷))≀𝑛!π‘€π‘›π‘šπ‘›(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, 𝑒π‘₯Ξ©ξ€Έβ‰€πΎπœˆπ‘›(π‘Ÿπ·/𝑀)π‘›ξ€œπ΅π‘›(π‘₯Ξ©,π‘Ÿπ·/𝑀)𝑒(𝑦)π‘‘π‘šπ‘›(≀𝑦)𝐾𝑀𝑛𝑅𝐷/π‘Ÿπ·ξ€Έπ‘›π‘šπ‘›ξ€·π΅π‘›ξ€·π‘π·,π‘…π·ξ€œξ€Έξ€Έβ„Ž(𝐷)𝑒(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝐾𝑀𝑛𝑅𝐷/π‘Ÿπ·ξ€Έπ‘›π‘šπ‘›ξ€œ(𝐷)β„Ž(𝐷)𝑒(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝐾𝑀𝑛𝑅𝐷/π‘Ÿπ·ξ€Έπ‘›π‘šπ‘›(β„Ž(𝐷))/𝑛!π‘€π‘›ξ€œβ„Ž(𝐷)𝑒(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝑛!𝐾𝑀2𝑛𝑅𝐷/π‘Ÿπ·ξ€Έπ‘›π‘šπ‘›ξ€œ(β„Ž(𝐷))β„Ž(𝐷)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦).(2.3) Thus (2.1) holds with 𝐢=𝐢(𝐾,𝑛,𝑀,𝐷,𝑝𝐷).

Theorem 2.2. Let 𝐷 be an open set in ℝ𝑛, 𝑛β‰₯2, with π‘šπ‘›(𝐷)<+∞. Fix a point π‘π·βˆˆπ·. Let Ξ© be an open set in ℝ𝑛. Let π‘’βˆΆΞ©β†’[0,+∞) be a 𝐾-quasinearly subharmonic function. Then there is a constant 𝐢=𝐢(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)β‰₯1 such that (2.1) holds for every point π‘₯Ω∈Ω and all β„Žβˆˆπ‘€-BiLip(𝑝𝐷,π‘₯Ξ©,𝐷,Ξ©), 𝑀β‰₯1.

Proof. Let 𝑑>1 be arbitrary. It is easy to see that π‘‘π‘šπ‘›(π·βˆ©π΅π‘›(𝑝𝐷,π‘Ÿπ‘‘))β‰₯π‘šπ‘›(𝐷) for some π‘Ÿπ‘‘>0. Write π·π‘‘βˆΆ=π·βˆ©π΅π‘›(𝑝𝐷,π‘Ÿπ‘‘) and 𝑝𝐷𝑑=𝑝𝐷. One sees easily that 𝐷𝑑 satisfies the assumptions of Lemma 2.1; that is, 𝐷𝑑 is a bounded domain, β„Ž(𝐷𝑑)βŠ‚β„Ž(𝐷)βŠ‚Ξ© and β„Ž(𝑝𝐷𝑑)=β„Ž(𝑝𝐷)=π‘₯Ξ©. Hence there is a constant 𝐢1=𝐢1(𝐾,𝑛,𝐷,𝑀,𝑝𝐷)β‰₯1 such that 𝑒π‘₯Ω≀𝐢1π‘šπ‘›ξ€·β„Žξ€·π·π‘‘ξ€œξ€Έξ€Έβ„Ž(𝐷𝑑)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)(2.4) for every point π‘₯Ω∈Ω and all β„Žβˆˆπ‘€-BiLip(𝑝𝐷𝑑,π‘₯Ξ©,𝐷𝑑,Ξ©). Since β„Ž and β„Žβˆ’1 are 𝑀-bi-Lipschitz, it follows that π‘šπ‘›(β„Ž(𝐷))≀𝑛!π‘€π‘›π‘šπ‘›(𝐷) and π‘šπ‘›(𝐷𝑑)≀𝑛!π‘€π‘›π‘šπ‘›(β„Ž(𝐷𝑑)); see again [8, Theorem 2.2, page 99]. Thus for 𝐢2=𝐢2(𝑛,𝑀)=(𝑛!)2𝑀2𝑛, π‘šπ‘›ξ€·π·π‘‘ξ€Έπ‘šπ‘›(𝐷)≀𝐢2β‹…π‘šπ‘›ξ€·β„Žξ€·π·π‘‘ξ€Έξ€Έπ‘šπ‘›.(β„Ž(𝐷))(2.5) Proceed then as follows: 1π‘šπ‘›ξ€·β„Žξ€·π·π‘‘ξ€œξ€Έξ€Έβ„Ž(𝐷𝑑)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)≀𝐢2β‹…π‘šπ‘›(𝐷)π‘šπ‘›ξ€·π·π‘‘ξ€Έβ‹…1π‘šπ‘›(ξ€œβ„Ž(𝐷))β„Ž(𝐷𝑑)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)≀𝐢2β‹…π‘‘π‘šπ‘›ξ€œ(β„Ž(𝐷))β„Ž(𝐷𝑑)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦)≀𝐢2β‹…π‘‘π‘šπ‘›ξ€œ(β„Ž(𝐷))β„Ž(𝐷)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦).(2.6) Therefore 𝑒π‘₯Ω≀𝐢1𝐢2π‘‘π‘šπ‘›ξ€œ(β„Ž(𝐷))β„Ž(𝐷)𝑒(𝑦)π‘‘π‘šπ‘›(𝑦),(2.7) concluding the proof.

3. An Invariance of the Class of Quasinearly Subharmonic Functions

Suppose that 𝐺 and π‘ˆ are open sets in the complex plane β„‚. If π‘“βˆΆπ‘ˆβ†’πΊ is analytic and π‘’βˆΆπΊβ†’[βˆ’βˆž,+∞) is subharmonic, then π‘’βˆ˜π‘“ is subharmonic; see, for example, [3, page 37] and [4, Corollary 3.3.4, page 70]. Using Koebe’s one-quarter and distortion theorems, KojiΔ‡ proved the following partial generalization.

Theorem 3.1 (see [6, Theorem 1, page 245]). Let Ξ© and 𝐺 be open sets in β„‚. Let π‘’βˆΆΞ©β†’[0,+∞) be a 𝐾-quasinearly subharmonic function. If πœ‘βˆΆπΊβ†’Ξ© is conformal, then the composition mapping π‘’βˆ˜πœ‘βˆΆπΊβ†’[0,+∞) is 𝐢-quasinearly subharmonic for some 𝐢=𝐢(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, 𝑛β‰₯2, the class of subharmonic functions is invariant under orthogonal transformations; see [10, page 55].

Theorem 3.2. Let Ξ© and π‘ˆ be open sets in ℝ𝑛,𝑛β‰₯2. Let π‘’βˆΆΞ©β†’[0,+∞) be a 𝐾-quasinearly subharmonic function. If π‘“βˆΆπ‘ˆβ†’Ξ© is 𝑀-bi-Lipschitz, then the composition mapping π‘’βˆ˜π‘“βˆΆπ‘ˆβ†’[0,+∞) is 𝐢-quasinearly subharmonic for some 𝐢=𝐢(𝐾,𝑛,𝑀).

Proof. It is sufficient to show that there exists a constant 𝐢=𝐢(𝐾,𝑛,𝑓)>0 such that (ξ€·π‘₯π‘’βˆ˜π‘“)0ξ€Έβ‰€πΆπ‘šπ‘›ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έπ΅π‘›(π‘₯0,π‘Ÿ0)(π‘’βˆ˜π‘“)(π‘₯)π‘‘π‘šπ‘›(π‘₯)(3.1) for all 𝐡𝑛(π‘₯0,π‘Ÿ0)βŠ‚π‘ˆ. To see this, observe first that 𝐡𝑛π‘₯ξ…ž0,π‘Ÿ0π‘€ξ‚ξ€·π΅βŠ‚π‘“π‘›ξ€·π‘₯0,π‘Ÿ0ξ€Έξ€ΈβŠ‚π΅π‘›ξ€·π‘₯ξ…ž0,π‘€π‘Ÿ0ξ€Έ,(3.2) where π‘₯ξ…ž0=𝑓(π‘₯0).
Then (ξ€·π‘₯π‘’βˆ˜π‘“)0ξ€Έξ€·π‘₯=π‘’ξ…ž0ξ€Έβ‰€πΎπ‘šπ‘›ξ€·π΅π‘›ξ€·π‘₯ξ…ž0,π‘Ÿ0ξ€œ/𝑀𝐡𝑛(π‘₯β€²0,π‘Ÿ0/𝑀)𝑒(𝑦)π‘‘π‘šπ‘›(≀𝐾𝑦)πœˆπ‘›(π‘Ÿ0/𝑀)π‘›ξ€œπ΅π‘›(π‘₯β€²0,π‘Ÿ0/𝑀)𝑓(π‘’βˆ˜π‘“)βˆ’1ξ€Έ(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝐾𝑀2π‘›πœˆπ‘›(π‘€π‘Ÿ0)π‘›ξ€œπ΅π‘›(π‘₯β€²0,π‘Ÿ0/𝑀)𝑓(π‘’βˆ˜π‘“)βˆ’1ξ€Έ(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝐾𝑀2π‘›π‘šπ‘›ξ€·π‘“ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έξ€Έπ‘“(𝐡𝑛(π‘₯0,π‘Ÿ0))𝑓(π‘’βˆ˜π‘“)βˆ’1ξ€Έ(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝐾𝑀2π‘›π‘šπ‘›ξ€·π‘“ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έξ€Έπ‘“(𝐡𝑛(π‘₯0,π‘Ÿ0))𝑓(π‘’βˆ˜π‘“)βˆ’1ξ€Έ||𝐽(𝑦)π‘“βˆ’1||β‹…1(𝑦)||π½π‘“βˆ’1||(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝐾𝑀2π‘›π‘šπ‘›ξ€·π‘“ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έξ€Έπ‘“(𝐡𝑛(π‘₯0,π‘Ÿ0))𝑓(π‘’βˆ˜π‘“)βˆ’1ξ€Έ||𝐽(𝑦)π‘“βˆ’1||β‹…||𝐽(𝑦)π‘“ξ€·π‘“βˆ’1(ξ€Έ||𝑦)π‘‘π‘šπ‘›(≀𝑦)𝐾𝑀2π‘›π‘šπ‘›ξ€·π‘“ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έξ€Έπ‘“(𝐡𝑛(π‘₯0,π‘Ÿ0))𝑓(π‘’βˆ˜π‘“)βˆ’1ξ€Έ||𝐽(𝑦)π‘“βˆ’1||(𝑦)⋅𝑛!π‘€π‘›π‘‘π‘šπ‘›β‰€(𝑦)𝑛!𝐾𝑀3π‘›π‘šπ‘›ξ€·π‘“ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έξ€Έπ‘“(𝐡𝑛(π‘₯0,π‘Ÿ0))𝑓(π‘’βˆ˜π‘“)βˆ’1ξ€Έ||𝐽(𝑦)π‘“βˆ’1||(𝑦)π‘‘π‘šπ‘›β‰€(𝑦)𝑛!𝐾𝑀3π‘›π‘šπ‘›ξ€·π‘“ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έξ€Έπ΅π‘›(π‘₯0,π‘Ÿ0)(π‘’βˆ˜π‘“)(π‘₯)π‘‘π‘šπ‘›β‰€(π‘₯)𝑛!𝐾𝑀3π‘›π‘šπ‘›ξ€·π΅π‘›ξ€·π‘₯ξ…ž0,π‘Ÿ0ξ€œ/𝑀𝐡𝑛(π‘₯0,π‘Ÿ0)(π‘’βˆ˜π‘“)(π‘₯)π‘‘π‘šπ‘›β‰€(π‘₯)𝑛!𝐾𝑀4π‘›π‘šπ‘›ξ€·π΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€œξ€Έξ€Έπ΅π‘›(π‘₯0,π‘Ÿ0)(π‘’βˆ˜π‘“)(π‘₯)π‘‘π‘šπ‘›(π‘₯).(3.3) Above we have used the routineous fact that for 𝑀-bi-Lipschitz mappings, ||π½π‘“ξ€·π‘“βˆ’1ξ€Έ||(𝑦)≀𝑛!𝑀𝑛,(3.4) and the already cited change of variable result of RadΓ³-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 Ξ© be an open set in ℝ𝑛, 𝑛β‰₯2. Let π‘“βˆΆΞ©β†’β„π‘š be continuous. Write

𝐿(π‘₯,𝑓)∢=limsup𝑦→π‘₯||||𝑓(𝑦)βˆ’π‘“(π‘₯)||||.π‘¦βˆ’π‘₯(4.1) The function π‘₯↦𝐿(π‘₯,𝑓) is a Borel function in Ξ©. If 𝑓 is differentiable at π‘₯, then 𝐿(π‘₯,𝑓)=|π‘“ξ…ž(π‘₯)|; see [9, page 11], [11, page 19], and [12, page 93].

A function π‘“βˆΆΞ©β†’β„ is regularly oscillating, if there is 𝐾β‰₯1 such that

𝐿(π‘₯,𝑓)β‰€πΎπ‘Ÿβˆ’1supπ‘¦βˆˆπ΅π‘›(π‘₯,π‘Ÿ)||||,𝑓(𝑦)βˆ’π‘“(π‘₯)𝐡𝑛(π‘₯,π‘Ÿ)βŠ‚Ξ©.(4.2) The class of such functions is denoted by OC1𝐾(Ξ©). The class of all regularly oscillating functions is denoted by RO(Ξ©); see [11, page 19], [13, page 17], [14], [6, page 245], and [12, page 96].

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

Theorem 4.1 (see [6, Theorem 2, page 245]). Let Ξ© and 𝐺 be open sets in β„‚. Let π‘’βˆˆOC1𝐾(Ξ©). If π‘“βˆΆπΊβ†’Ξ© is conformal, then π‘’βˆ˜π‘“βˆˆOC1𝐢(𝐺), where 𝐢 depends only on 𝐾.

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

Theorem 4.2. Let Ξ© and π‘ˆ be open sets in ℝ𝑛,𝑛β‰₯2. Let π‘’βˆˆOC1𝐾(Ξ©). If πœ‘βˆΆπ‘ˆβ†’Ξ© is 𝑀-bi-Lipschitz, 𝑀β‰₯1, then π‘’βˆ˜πœ‘βˆˆOC1𝐾𝑀2(π‘ˆ).

Proof. Let πœ‘βˆΆπ‘ˆβ†’Ξ© be 𝑀-bi-Lipschitz. Take π‘₯0βˆˆπ‘ˆ and π‘Ÿ0>0 arbitrarily such that 𝐡𝑛(π‘₯0,π‘Ÿ0)βŠ‚π‘ˆ. Write π‘₯ξ…ž0=πœ‘(π‘₯0) and π‘₯ξ…ž=πœ‘(π‘₯) for π‘₯βˆˆπ‘ˆ. Then 𝐿π‘₯0ξ€Έ,π‘’βˆ˜πœ‘=limsupπ‘₯β†’π‘₯0||π‘’ξ€·πœ‘ξ€·π‘₯(πœ‘(π‘₯))βˆ’π‘’0||ξ€Έξ€Έ||π‘₯βˆ’π‘₯0||=limsupπ‘₯β†’π‘₯0||ξ€·πœ‘ξ€·π‘₯𝑒(πœ‘(π‘₯))βˆ’π‘’0||ξ€Έξ€Έ||πœ‘ξ€·π‘₯(π‘₯)βˆ’πœ‘0ξ€Έ||β‹…||ξ€·π‘₯πœ‘(π‘₯)βˆ’πœ‘0ξ€Έ||||π‘₯βˆ’π‘₯0||≀limsupπ‘₯β€²β†’π‘₯β€²0||𝑒π‘₯ξ…žξ€Έξ€·π‘₯βˆ’π‘’ξ…ž0ξ€Έ||||π‘₯ξ…žβˆ’π‘₯ξ…ž0||β‹…limsupπ‘₯β†’π‘₯0||ξ€·π‘₯πœ‘(π‘₯)βˆ’πœ‘0ξ€Έ||||π‘₯βˆ’π‘₯0||ξ€·π‘₯=πΏξ…ž0ξ€Έ,𝑒⋅limsupπ‘₯β†’π‘₯0||ξ€·π‘₯πœ‘(π‘₯)βˆ’πœ‘0ξ€Έ||||π‘₯βˆ’π‘₯0||.(4.3) Using (3.2) (for 𝑓=πœ‘), we get 𝐿π‘₯ξ…ž0≀𝐾,π‘’π‘Ÿ0/𝑀supπ‘₯β€²βˆˆπ΅π‘›ξ€·π‘₯β€²0,π‘Ÿ0ξ€Έ/𝑀||𝑒π‘₯ξ…žξ€Έξ€·π‘₯βˆ’π‘’ξ…ž0ξ€Έ||β‰€πΎπ‘€π‘Ÿ0supπ‘₯β€²βˆˆπ΅π‘›ξ€·π‘₯β€²0,π‘Ÿ0ξ€Έ/𝑀||𝑒π‘₯ξ…žξ€Έξ€·π‘₯βˆ’π‘’ξ…ž0ξ€Έ||β‰€πΎπ‘€π‘Ÿ0supπ‘₯β€²ξ€·π΅βˆˆπœ‘π‘›ξ€·π‘₯0,π‘Ÿ0ξ€Έξ€Έ||𝑒π‘₯ξ…žξ€Έξ€·π‘₯βˆ’π‘’ξ…ž0ξ€Έ||β‰€πΎπ‘€π‘Ÿ0supπ‘₯βˆˆπ΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€Έ||ξ€·πœ‘ξ€·π‘₯𝑒(πœ‘(π‘₯))βˆ’π‘’0||β‰€ξ€Έξ€ΈπΎπ‘€π‘Ÿ0supπ‘₯βˆˆπ΅π‘›ξ€·π‘₯0,π‘Ÿ0ξ€Έ||ξ€·π‘₯(π‘’βˆ˜πœ‘)(π‘₯)βˆ’(π‘’βˆ˜πœ‘)0ξ€Έ||.(4.4)
On the other hand, since πœ‘ is 𝑀-bi-Lipschitz, limsupπ‘₯β†’π‘₯0||πœ‘ξ€·π‘₯(π‘₯)βˆ’πœ‘0ξ€Έ||||π‘₯βˆ’π‘₯0||≀limsupπ‘₯β†’π‘₯0𝑀||π‘₯βˆ’π‘₯0||||π‘₯βˆ’π‘₯0||=𝑀<+∞.(4.5) Therefore, 𝐿π‘₯0≀,π‘’βˆ˜πœ‘πΎπ‘€π‘Ÿ0supπ‘₯βˆˆπ΅π‘›(π‘₯0,π‘Ÿ0)||ξ€·π‘₯(π‘’βˆ˜πœ‘)(π‘₯)βˆ’(π‘’βˆ˜πœ‘)0ξ€Έ||≀⋅𝑀𝐾𝑀2π‘Ÿ0supπ‘₯βˆˆπ΅π‘›(π‘₯0,π‘Ÿ0)||ξ€·π‘₯(π‘’βˆ˜πœ‘)(π‘₯)βˆ’(π‘’βˆ˜πœ‘)0ξ€Έ||.(4.6) Thus π‘’βˆ˜πœ‘βˆˆOC1𝐾𝑀2(π‘ˆ).

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 Ξ© be an open set in ℝ𝑛, 𝑛β‰₯2. Let 𝐾β‰₯1. A function π‘“βˆΆΞ©β†’β„ is in HC1𝐾(Ξ©) if

𝐿(π‘₯,𝑓)β‰€πΎπ‘Ÿβˆ’1supπ‘¦βˆˆπ΅π‘›(π‘₯,π‘Ÿ)||||,𝑓(𝑦)𝐡𝑛(π‘₯,π‘Ÿ)βŠ‚Ξ©.(4.7) The class HC1(Ξ©) is the union of all HC1𝐾(Ξ©), 𝐾β‰₯1. Clearly, HC12𝐾(Ξ©)βŠ‚OC1𝐾(Ξ©).

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

Theorem 4.3. Let Ξ© and π‘ˆ be open sets in ℝ𝑛,𝑛β‰₯2. Let π‘’βˆˆHC1𝐾(Ξ©). If πœ‘βˆΆπ‘ˆβ†’β„π‘› is 𝑀-bi-Lipschitz, 𝑀β‰₯1, then π‘’βˆ˜πœ‘βˆˆHC1𝐾𝑀2(π‘ˆ).

Acknowledgment

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

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