Bi-Lipschitz Mappings and Quasinearly Subharmonic Functions

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 ℝ^𝑛, 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 ℝ^𝑛, 𝑛≥2, is denoted by 𝑚_𝑛. We write 𝐵^𝑛(𝑥,𝑟) for the ball in ℝ^𝑛, with center 𝑥 and radius 𝑟. Recall that 𝑚_𝑛(𝐵^𝑛(𝑥,𝑟))=𝜈_𝑛𝑟^𝑛, where 𝜈_𝑛∶=𝑚_𝑛(𝐵^𝑛(0,1)). If 𝐷 is an open set in ℝ^𝑛, and 𝑥∈𝐷, then we write 𝛿_𝐷(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 ℝ^𝑛, 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 𝑢(𝑥)≤1𝜈_𝑛𝑟^𝑛_{𝐵𝑛(𝑥,𝑟)}𝑢(𝑦)𝑑𝑚_𝑛(𝑦)(1.1) for all balls 𝐵^𝑛(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⁺∈ℒ^1loc(Ω) and 𝑢(𝑥)≤1𝜈_𝑛𝑟^𝑛_{𝐵𝑛(𝑥,𝑟)}𝑢(𝑦)𝑑𝑚_𝑛(𝑦)(1.2) for all balls 𝐵^𝑛(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⁺∈ℒ^1loc(Ω) and 𝑢_𝐿(𝑥)≤𝐾𝜈_𝑛𝑟^𝑛_{𝐵𝑛(𝑥,𝑟)}𝑢_𝐿(𝑦)𝑑𝑚_𝑛(𝑦)(1.3) for all 𝐿≥0 and for all balls 𝐵^𝑛(𝑥,𝑟)⊂Ω. Here 𝑢_𝐿∶=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∈ℒ^1loc(Ω) and 𝑢(𝑥)≤𝐾𝜈_𝑛𝑟^𝑛_{𝐵𝑛(𝑥,𝑟)}𝑢(𝑦)𝑑𝑚_𝑛(𝑦)(1.4) for all balls 𝐵^𝑛(𝑥,𝑟)⊂Ω.

1.2. Bi-Lipschitz Mappings

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

||𝑦−𝑥||𝑀≤||𝑓(𝑦)−𝑓(𝑥)||≤𝑀||𝑦−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→ℝ^𝑛 is 𝑀-bi-Lipschitz, then also 𝑓⁻¹∶𝐷^→ℝ^𝑛 is 𝑀-bi-Lipschitz, where 𝐷^∶=𝑓(𝐷).

Let Ω be an open subset of ℝ^𝑛. Let 𝑝_𝐷∈𝐷 and 𝑥_Ω∈Ω. We write

𝑀-BiLip𝑝_𝐷,𝑥_Ω,𝐷,Ω∶=ℎ∶𝐷→ℝ^𝑛∶ℎi𝑠𝑀-bi-Lipschitz,ℎ𝑝_𝐷=𝑥_Ω,ℎ(𝐷)⊂Ω.(1.6)

2. On the Generalized Mean Value Inequality

Lemma 2.1. Let 𝐷 be a bounded open set in ℝ^𝑛, 𝑛≥2. Fix a point 𝑝_𝐷∈𝐷. Let Ω be a domain in ℝ^𝑛. Let 𝑢∶Ω→[0,+∞) be a 𝐾-quasinearly subharmonic function. Then there is 𝐶=𝐶(𝐾,𝑛,𝐷,𝑀,𝑝_𝐷)≥1 such that 𝑢𝑥_Ω≤𝐶𝑚_𝑛(ℎ(𝐷))_ℎ(𝐷)𝑢(𝑦)𝑑𝑚_𝑛(𝑦)(2.1) for every point 𝑥_Ω∈Ω and all ℎ∈𝑀-BiLip(𝑝_𝐷,𝑥_Ω,𝐷,Ω), 𝑀≥1.

Proof. Take 𝑥_Ω∈Ω and ℎ∈𝑀-BiLip(𝑝_𝐷,𝑥_Ω,𝐷,Ω), 𝑀≥1, arbitrarily. (Observe that the set of bi-Lipschitz mappings is (in general) nonempty.) Write 𝑅_𝐷∶=sup_𝑦∈𝐷||𝑝_𝐷−𝑦||,𝑟_𝐷∶=𝛿_𝐷𝑝_𝐷.(2.2) Using the fact that ℎ∣𝐵^𝑛(𝑝_𝐷,𝑟_𝐷)∶𝐵^𝑛(𝑝_𝐷,𝑟_𝐷)→ℎ(𝐵^𝑛(𝑝_𝐷,𝑟_𝐷)) is a homeomorphism, one sees easily that 𝐵^𝑛(𝑥_Ω,𝑟_𝐷/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 𝑚_𝑛(ℎ(𝐷))≤𝑛!𝑀^𝑛𝑚_𝑛(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𝑥_Ω≤𝐾𝜈_𝑛(𝑟_𝐷/𝑀)^𝑛_{𝐵𝑛(𝑥Ω,𝑟𝐷/𝑀)}≤𝐾𝑀_𝑛𝑢(𝑦)𝑑𝑚_𝑛(𝑦)𝑅_𝐷/𝑟_𝐷^𝑛𝑚_𝑛𝐵^𝑛𝑝_𝐷,𝑅_𝐷^ℎ(𝐷)≤𝐾𝑀_𝑛𝑢(𝑦)𝑑𝑚_𝑛(𝑦)𝑅_𝐷/𝑟_𝐷^𝑛𝑚_𝑛(𝐷)_ℎ(𝐷)≤𝐾𝑀_𝑛𝑢(𝑦)𝑑𝑚_𝑛(𝑦)𝑅_𝐷/𝑟_𝐷^𝑛𝑚_𝑛(ℎ(𝐷))/𝑛!𝑀^𝑛_ℎ(𝐷)≤𝑛!𝐾𝑀_2𝑛𝑢(𝑦)𝑑𝑚_𝑛(𝑦)𝑅_𝐷/𝑟_𝐷^𝑛𝑚_𝑛(ℎ(𝐷))_ℎ(𝐷)𝑢(𝑦)𝑑𝑚_𝑛(𝑦).(2.3) Thus (2.1) holds with 𝐶=𝐶(𝐾,𝑛,𝑀,𝐷,𝑝_𝐷).

Theorem 2.2. Let 𝐷 be an open set in ℝ^𝑛, 𝑛≥2, with 𝑚_𝑛(𝐷)<+∞. Fix a point 𝑝_𝐷∈𝐷. Let Ω be an open set in ℝ^𝑛. Let 𝑢∶Ω→[0,+∞) be a 𝐾-quasinearly subharmonic function. Then there is a constant 𝐶=𝐶(𝐾,𝑛,𝐷,𝑀,𝑝_𝐷)≥1 such that (2.1) holds for every point 𝑥_Ω∈Ω and all ℎ∈𝑀-BiLip(𝑝_𝐷,𝑥_Ω,𝐷,Ω), 𝑀≥1.

Proof. Let 𝑡>1 be arbitrary. It is easy to see that 𝑡𝑚_𝑛(𝐷∩𝐵^𝑛(𝑝_𝐷,𝑟_𝑡))≥𝑚_𝑛(𝐷) for some 𝑟_𝑡>0. Write 𝐷_𝑡∶=𝐷∩𝐵^𝑛(𝑝_𝐷,𝑟_𝑡) and 𝑝_𝐷𝑡=𝑝_𝐷. One sees easily that 𝐷_𝑡 satisfies the assumptions of Lemma 2.1; that is, 𝐷_𝑡 is a bounded domain, ℎ(𝐷_𝑡)⊂ℎ(𝐷)⊂Ω and ℎ(𝑝_𝐷𝑡)=ℎ(𝑝_𝐷)=𝑥_Ω. Hence there is a constant 𝐶₁=𝐶₁(𝐾,𝑛,𝐷,𝑀,𝑝_𝐷)≥1 such that 𝑢𝑥_Ω≤𝐶¹𝑚_𝑛ℎ𝐷_𝑡^ℎ(𝐷_𝑡⁾𝑢(𝑦)𝑑𝑚_𝑛(𝑦)(2.4) for every point 𝑥_Ω∈Ω and all ℎ∈𝑀-BiLip(𝑝_𝐷𝑡,𝑥_Ω,𝐷_𝑡,Ω). Since ℎ and ℎ⁻¹ are 𝑀-bi-Lipschitz, it follows that 𝑚_𝑛(ℎ(𝐷))≤𝑛!𝑀^𝑛𝑚_𝑛(𝐷) and 𝑚_𝑛(𝐷_𝑡)≤𝑛!𝑀^𝑛𝑚_𝑛(ℎ(𝐷_𝑡)); see again [8, Theorem 2.2, page 99]. Thus for 𝐶₂=𝐶₂(𝑛,𝑀)=(𝑛!)²𝑀^2𝑛, 𝑚_𝑛𝐷_𝑡𝑚_𝑛(𝐷)≤𝐶²⋅𝑚^𝑛ℎ𝐷_𝑡𝑚_𝑛(ℎ(𝐷)).(2.5) Proceed then as follows: 1𝑚_𝑛ℎ𝐷_𝑡^ℎ(𝐷_𝑡⁾𝑢(𝑦)𝑑𝑚_𝑛(𝑦)≤𝐶₂⋅𝑚^𝑛(𝐷)𝑚_𝑛𝐷_𝑡⋅𝑚_𝑛(ℎ1(𝐷))_{ℎ(𝐷𝑡)}≤𝐶₂𝑢(𝑦)𝑑𝑚_𝑛(𝑦)⋅𝑡𝑚_𝑛(ℎ(𝐷))_{ℎ(𝐷𝑡)}≤𝐶₂𝑢(𝑦)𝑑𝑚_𝑛(𝑦)⋅𝑡𝑚_𝑛(ℎ(𝐷))_ℎ(𝐷)𝑢(𝑦)𝑑𝑚_𝑛(𝑦).(2.6) Therefore 𝑢𝑥_Ω≤𝐶¹𝐶₂𝑡𝑚_𝑛(ℎ(𝐷))_ℎ(𝐷)𝑢(𝑦)𝑑𝑚_𝑛(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 ℝ^𝑛, 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 ℝ^𝑛,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 (𝑢∘𝑓)𝑥₀≤𝐶𝑚_𝑛𝐵^𝑛𝑥₀,𝑟₀^{𝐵𝑛(𝑥}₀^,𝑟₀⁾(𝑢∘𝑓)(𝑥)𝑑𝑚_𝑛(𝑥)(3.1) for all 𝐵^𝑛(𝑥₀,𝑟₀)⊂𝑈. To see this, observe first that 𝐵^𝑛𝑥^0,𝑟⁰𝑀⊂𝑓𝐵^𝑛𝑥₀,𝑟₀⊂𝐵^𝑛𝑥^0,𝑀𝑟₀,(3.2) where 𝑥^0=𝑓(𝑥₀).
Then (𝑢∘𝑓)𝑥₀=𝑢𝑥^0≤𝐾𝑚_𝑛𝐵^𝑛𝑥^0,𝑟₀/𝑀_𝐵^𝑛_(𝑥^′0_,𝑟0/𝑀)≤𝐾𝑢(𝑦)𝑑𝑚_𝑛(𝑦)𝜈_𝑛(𝑟₀/𝑀)^𝑛_{𝐵𝑛(𝑥′0,𝑟0/𝑀)}≤𝐾𝑀_2𝑛(𝑢∘𝑓)𝑓⁻¹(𝑦)𝑑𝑚_𝑛(𝑦)𝜈_𝑛(𝑀𝑟₀)^𝑛_{𝐵𝑛(𝑥′0,𝑟0/𝑀)}≤𝐾𝑀_2𝑛(𝑢∘𝑓)𝑓⁻¹(𝑦)𝑑𝑚_𝑛(𝑦)𝑚_𝑛𝑓𝐵^𝑛𝑥₀,𝑟₀^{𝑓(𝐵𝑛(𝑥}₀^,𝑟₀⁾⁾≤𝐾𝑀_2𝑛(𝑢∘𝑓)𝑓⁻¹(𝑦)𝑑𝑚_𝑛(𝑦)𝑚_𝑛𝑓𝐵^𝑛𝑥₀,𝑟₀^{𝑓(𝐵𝑛(𝑥}₀^,𝑟₀⁾⁾(𝑢∘𝑓)𝑓⁻¹(𝑦)||𝐽_𝑓⁻¹(𝑦)||⋅1||𝐽_𝑓−1(𝑦)||𝑑𝑚^𝑛≤𝐾𝑀_2𝑛(𝑦)𝑚_𝑛𝑓𝐵^𝑛𝑥₀,𝑟₀^{𝑓(𝐵𝑛(𝑥}₀^,𝑟₀⁾⁾≤𝐾𝑀_2𝑛(𝑢∘𝑓)𝑓⁻¹(𝑦)||𝐽_𝑓⁻¹(𝑦)||⋅||𝐽_𝑓𝑓⁻¹(𝑦)||𝑑𝑚_𝑛(𝑦)𝑚_𝑛𝑓𝐵^𝑛𝑥₀,𝑟₀^{𝑓(𝐵𝑛(𝑥}₀^,𝑟₀⁾⁾≤𝑛!𝐾𝑀_3𝑛(𝑢∘𝑓)𝑓⁻¹(𝑦)||𝐽_𝑓⁻¹(𝑦)||⋅𝑛!𝑀^𝑛𝑑𝑚_𝑛(𝑦)𝑚_𝑛𝑓𝐵^𝑛𝑥₀,𝑟₀^{𝑓(𝐵𝑛(𝑥}₀^,𝑟₀⁾⁾≤𝑛!𝐾𝑀_3𝑛(𝑢∘𝑓)𝑓⁻¹(𝑦)||𝐽_𝑓⁻¹(𝑦)||𝑑𝑚_𝑛(𝑦)𝑚_𝑛𝑓𝐵^𝑛𝑥₀,𝑟₀^{𝐵𝑛(𝑥}₀^,𝑟₀⁾≤𝑛!𝐾𝑀_3𝑛(𝑢∘𝑓)(𝑥)𝑑𝑚_𝑛(𝑥)𝑚_𝑛𝐵^𝑛𝑥^0,𝑟₀/𝑀_𝐵^𝑛_{(𝑥0,𝑟0)}≤𝑛!𝐾𝑀_4𝑛(𝑢∘𝑓)(𝑥)𝑑𝑚_𝑛(𝑥)𝑚_𝑛𝐵^𝑛𝑥₀,𝑟₀^{𝐵𝑛(𝑥}₀^,𝑟₀⁾(𝑢∘𝑓)(𝑥)𝑑𝑚_𝑛(𝑥).(3.3) Above we have used the routineous fact that for 𝑀-bi-Lipschitz mappings, ||𝐽_𝑓𝑓⁻¹(𝑦)||≤𝑛!𝑀^𝑛,(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 ℝ^𝑛, 𝑛≥2. Let 𝑓∶Ω→ℝ^𝑚 be continuous. Write

𝐿(𝑥,𝑓)∶=limsup_𝑦→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)=|𝑓^(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⁻¹sup_{𝑦∈𝐵𝑛(𝑥,𝑟)}||𝑓(𝑦)−𝑓(𝑥)||,𝐵^𝑛(𝑥,𝑟)⊂Ω.(4.2) The class of such functions is denoted by OC^1𝐾(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∈OC^1𝐾(Ω). If 𝑓∶𝐺→Ω is conformal, then 𝑢∘𝑓∈OC^1𝐶(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 ℝ^𝑛,𝑛≥2. Let 𝑢∈OC^1𝐾(Ω). If 𝜑∶𝑈→Ω is 𝑀-bi-Lipschitz, 𝑀≥1, then 𝑢∘𝜑∈OC^1𝐾𝑀₂(𝑈).

Proof. Let 𝜑∶𝑈→Ω be 𝑀-bi-Lipschitz. Take 𝑥₀∈𝑈 and 𝑟₀>0 arbitrarily such that 𝐵^𝑛(𝑥₀,𝑟₀)⊂𝑈. Write 𝑥^0=𝜑(𝑥₀) and 𝑥^=𝜑(𝑥) for 𝑥∈𝑈. Then 𝐿𝑥₀,𝑢∘𝜑=limsup_𝑥→𝑥0||𝑢(𝜑(𝑥))−𝑢𝜑𝑥₀||=limsup_𝑥→𝑥0||𝑥−𝑥₀||||𝑢(𝜑(𝑥))−𝑢𝜑𝑥₀||||𝜑(𝑥)−𝜑𝑥₀||⋅||𝜑(𝑥)−𝜑𝑥⁰||≤limsup_{𝑥′→𝑥′0}||𝑥−𝑥₀||||𝑢𝑥^−𝑢𝑥^0||||𝑥^−𝑥^0||⋅limsup_𝑥→𝑥0||𝜑(𝑥)−𝜑𝑥₀||=𝐿𝑥_0||𝑥−𝑥₀||,𝑢⋅limsup_𝑥→𝑥0||𝜑(𝑥)−𝜑𝑥₀||||𝑥−𝑥₀||.(4.3) Using (3.2) (for 𝑓=𝜑), we get 𝐿𝑥^0,𝑢≤𝐾𝑟₀/𝑀sup_{𝑥′∈𝐵𝑛𝑥′0,𝑟0/𝑀}≤𝐾𝑀||𝑢𝑥^−𝑢𝑥^0||𝑟₀≤𝐾𝑀sup_{𝑥′∈𝐵𝑛𝑥′0,𝑟0/𝑀}||𝑢𝑥^−𝑢𝑥^0||𝑟₀≤𝐾𝑀sup_{𝑥′𝐵∈𝜑𝑛𝑥0,𝑟0}||𝑢𝑥^−𝑢𝑥^0||𝑟₀≤𝐾𝑀sup_{𝑥∈𝐵𝑛𝑥0,𝑟0}||𝑢(𝜑(𝑥))−𝑢𝜑𝑥₀||𝑟₀sup_{𝑥∈𝐵𝑛𝑥0,𝑟0}||(𝑢∘𝜑)(𝑥)−(𝑢∘𝜑)𝑥₀||.(4.4)
On the other hand, since 𝜑 is 𝑀-bi-Lipschitz, limsup_𝑥→𝑥0||𝜑(𝑥)−𝜑𝑥₀||||𝑥−𝑥₀||≤limsup_𝑥→𝑥0𝑀||𝑥−𝑥₀||||𝑥−𝑥₀||=𝑀<+∞.(4.5) Therefore, 𝐿𝑥₀,𝑢∘𝜑≤𝐾𝑀𝑟₀≤𝐾𝑀₂sup_{𝑥∈𝐵𝑛(𝑥0,𝑟0)}||(𝑢∘𝜑)(𝑥)−(𝑢∘𝜑)𝑥₀||⋅𝑀𝑟₀sup_{𝑥∈𝐵𝑛(𝑥0,𝑟0)}||(𝑢∘𝜑)(𝑥)−(𝑢∘𝜑)𝑥₀||.(4.6) Thus 𝑢∘𝜑∈OC^1𝐾𝑀₂(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 ℝ^𝑛, 𝑛≥2. Let 𝐾≥1. A function 𝑓∶Ω→ℝ is in HC^1𝐾(Ω) if

𝐿(𝑥,𝑓)≤𝐾𝑟⁻¹sup_{𝑦∈𝐵𝑛(𝑥,𝑟)}||𝑓(𝑦)||,𝐵^𝑛(𝑥,𝑟)⊂Ω.(4.7) The class HC¹(Ω) is the union of all HC^1𝐾(Ω), 𝐾≥1. Clearly, HC^12𝐾(Ω)⊂OC^1𝐾(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 ℝ^𝑛,𝑛≥2. Let 𝑢∈HC^1𝐾(Ω). If 𝜑∶𝑈→ℝ^𝑛 is 𝑀-bi-Lipschitz, 𝑀≥1, then 𝑢∘𝜑∈HC^1𝐾𝑀₂(𝑈).

Acknowledgment

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