IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation38217910.1155/2010/382179382179Research ArticleBi-Lipschitz Mappings and Quasinearly Subharmonic FunctionsDovgosheyOleksiy1RiihentausJuhani2KanasStanisława R.1Institute of Applied Mathematics and Mechanics, NASUR. Luxemburg Street 74Donetsk 83114Ukraineiamm.ac.donetsk.ua2Department of Physics and MathematicsUniversity of JoensuuP.O. Box 11180101 JoensuuFinlandjoensuu.fi201005012010201030112009251220092010Copyright © 2010This 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.

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,  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 u:Ω[-,+) be a Lebesgue measurable function. We adopt the following definitions.

u is subharmonic if u is upper semicontinuous and if u(x)1νnrnBn(x,r)u(y)dmn(y) 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].

u is nearly subharmonic if u+loc1(Ω) and u(x)1νnrnBn(x,r)u(y)dmn(y) 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].

Let K1. Then u is K-quasinearly subharmonic if u+loc1(Ω) and uL(x)KνnrnBn(x,r)uL(y)dmn(y) 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. [<xref ref-type="bibr" rid="B5">5</xref>, Proposition <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M47"><mml:mrow><mml:mn>2.1</mml:mn></mml:mrow></mml:math></inline-formula>, pages 54-55]).

The following holds.

A subharmonic function is nearly subharmonic but not conversely.

A function is nearly subharmonic if and only if it is 1-quasinearly subharmonic.

A nearly subharmonic function is quasinearly subharmonic but not conversely.

If u:Ω[0,+) is Lebesgue measurable, then u is K-quasinearly subharmonic if and only if uloc1(Ω) and u(x)KνnrnBn(x,r)u(y)dmn(y) 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 f:Dn is M-bi-Lipschitz if

|y-x|M|f(y)-f(x)|M|y-x| 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 f:Dn is M-bi-Lipschitz, then also f-1:Dn is M-bi-Lipschitz, where D:=f(D).

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

M-BiLip(pD,xΩ,D,Ω):={h:Dn:h  is  M-bi-Lipschitz,h(pD)=xΩ,h(D)Ω}.

2. On the Generalized Mean Value InequalityLemma 2.1.

Let D be a bounded open set in n, n2. Fix a point pDD. Let Ω be a domain in n. 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) for every point xΩΩ and all hM-BiLip  (pD,xΩ,D,Ω), M1.

Proof.

Take xΩΩ and hM-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:=δD(pD). Using the fact that hBn(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)nBn(xΩ,rD/M)u(y)dmn(y)KMn(RD/rD)nmn(Bn(pD,RD))h(D)u(y)dmn(y)KMn(RD/rD)nmn(D)h(D)u(y)dmn(y)KMn(RD/rD)nmn(h(D))/n!Mnh(D)u(y)dmn(y)n!KM2n(RD/rD)nmn(h(D))h(D)u(y)dmn(y). 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 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 xΩΩ and all hM-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, 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)C1mn(h(Dt))h(Dt)u(y)dmn(y) for every point xΩΩ and all hM-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)C2·mn(h(Dt))mn(h(D)). Proceed then as follows: 1mn(h(Dt))h(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). Therefore u(xΩ)C1C2tmn(h(D))h(D)u(y)dmn(y), 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 f:UG is analytic and u:G[-,+) 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 [<xref ref-type="bibr" rid="B6">6</xref>, Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M150"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>, page 245]).

Let Ω and G be open sets in . Let u:Ω[0,+) be a K-quasinearly subharmonic function. If φ:GΩ is conformal, then the composition mapping uφ: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 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 u:Ω[0,+) be a K-quasinearly subharmonic function. If f:UΩ is M-bi-Lipschitz, then the composition mapping uf: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 (uf)(x0)Cmn(Bn(x0,r0))Bn(x0,r0)(uf)(x)dmn(x) for all Bn(x0,r0)¯U. To see this, observe first that Bn(x0,r0M)f(Bn(x0,r0))Bn(x0,Mr0), where x0=f(x0).

Then (uf)(x0)=u(x0)Kmn(Bn(x0,r0/M))Bn(x0,r0/M)u(y)dmn(y)Kνn(r0/M)nBn(x0,r0/M)(uf)(f-1(y))dmn(y)KM2nνn(Mr0)nBn(x0,r0/M)(uf)(f-1(y))dmn(y)KM2nmn(f(Bn(x0,r0)))f(Bn(x0,r0))(uf)(f-1(y))dmn(y)KM2nmn(f(Bn(x0,r0)))f(Bn(x0,r0))(uf)(f-1(y))|Jf-1(y)|·1|Jf-1(y)|dmn(y)KM2nmn(f(Bn(x0,r0)))f(Bn(x0,r0))(uf)(f-1(y))|Jf-1(y)|·|Jf(f-1(y))|dmn(y)KM2nmn(f(Bn(x0,r0)))f(Bn(x0,r0))(uf)(f-1(y))|Jf-1(y)|·n!Mndmn(y)n!KM3nmn(f(Bn(x0,r0)))f(Bn(x0,r0))(uf)(f-1(y))|Jf-1(y)|dmn(y)n!KM3nmn(f(Bn(x0,r0)))Bn(x0,r0)(uf)(x)dmn(x)n!KM3nmn(Bn(x0,r0/M))Bn(x0,r0)(uf)(x)dmn(x)n!KM4nmn(Bn(x0,r0))Bn(x0,r0)(uf)(x)dmn(x). Above we have used the routineous fact that for M-bi-Lipschitz mappings, |Jf(f-1(y))|n!Mn, 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 f:Ωm be continuous. Write

L(x,f):=lim supyx|f(y)-f(x)||y-x|. 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 f:Ω is regularly oscillating, if there is K1 such that

L(x,f)Kr-1supyBn(x,r)|f(y)-f(x)|,Bn(x,r)¯Ω. The class of such functions is denoted by OCK1(Ω). The class of all regularly oscillating functions is denoted by RO(Ω); see [11, page 19], [13, page 17], , [6, page 245], and [12, page 96].

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

Theorem 4.1 (see [<xref ref-type="bibr" rid="B6">6</xref>, Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M198"><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math></inline-formula>, page 245]).

Let Ω and G be open sets in . Let uOCK1(Ω). If f:GΩ is conformal, then ufOCC1(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 uOCK1(Ω). If φ:UΩ is M-bi-Lipschitz, M1, then uφOCKM21(U).

Proof.

Let φ:UΩ 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 L(x0,uφ)=lim supxx0|u(φ(x))-u(φ(x0))||x-x0|=lim supxx0|u(φ(x))-u(φ(x0))||φ(x)-φ(x0)|·|φ(x)-φ(x0)||x-x0|lim supxx0|u(x)-u(x0)||x-x0|·lim supxx0|φ(x)-φ(x0)||x-x0|=L(x0,u)·lim supxx0|φ(x)-φ(x0)||x-x0|. Using (3.2) (for f=φ), we get L(x0,u)Kr0/MsupxBn(x0,r0/M)|u(x)-u(x0)|KMr0supxBn(x0,r0/M)|u(x)-u(x0)|KMr0supxφ(Bn(x0,r0))|u(x)-u(x0)|KMr0supxBn(x0,r0)|u(φ(x))-u(φ(x0))|KMr0supxBn(x0,r0)|(uφ)(x)-(uφ)(x0)|.

On the other hand, since φ is M-bi-Lipschitz, lim supxx0|φ(x)-φ(x0)||x-x0|lim supxx0M|x-x0||x-x0|=M<+. Therefore, L(x0,uφ)KMr0supxBn(x0,r0)|(uφ)(x)-(uφ)(x0)|·MKM2r0supxBn(x0,r0)|(uφ)(x)-(uφ)(x0)|. Thus uφOCKM21(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 f:Ω is in HCK1(Ω) if

L(x,f)Kr-1supyBn(x,r)|f(y)|,Bn(x,r)¯Ω. The class HC1(Ω) is the union of all HCK1(Ω), K1. Clearly, HC2K1(Ω)OCK1(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 uHCK1(Ω). If φ:Un is M-bi-Lipschitz, M1, then uφHCKM21(U).

Acknowledgment

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

DovgosheyO.RiihentausJ.A remark concerning generalized mean value inequalities for subharmonic functionssubmittedDovgosheyO.RiihentausJ.Mean value type inequalities for quasi-nearly subharmonic functionsmanuscriptHervéM.Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces1971198Berlin, GermanySpringervi+90Lecture Notes in MathematicsMR0466602ArmitageD. H.GardinerS. J.Classical Potential Theory2001London, UKSpringerRiihentausJ.Subharmonic functions, generalizations and separately subharmonic functionsProceedings of the 14th Conference on Analytic FunctionsJuly, 2007Chełm, PolandKojićV.Quasi-nearly subharmonic functions and conformal mappingsFilomat200721224324910.2298/FIL0702243KMR2360893DjordjevićO.PavlovićM.p-integrability of the maximal function of a polyharmonic functionJournal of Mathematical Analysis and Applications2007336141141710.1016/j.jmaa.2007.03.006MR2348514ReshetnyakYu. G.Space Mappings with Bounded Distortion198973Providence, RI, USAAmerican Mathematical Societyxvi+362Translations of Mathematical MonographsMR994644VäisäläJ.Lectures on n-Dimensional Quasiconformal Mappings1971229Berlin, GermanySpringerxiv+144Lecture Notes in MathematicsMR0454009HaymanW. K.KennedyP. B.Subharmonic Functions. Vol. I1976London, UKAcademic Pressxvii+284London Mathematical Society MonographsMR0460672PavlovićM.On subharmonic behaviour and oscillation of functions on balls in nInstitut Mathématique. Publications. Nouvelle Série199455691822MR1324970PavlovićM.RiihentausJ.Classes of quasi-nearly subharmonic functionsPotential Analysis20082918910410.1007/s11118-008-9089-1MR2421496ZBL1158.31002PavlovićM.Subharmonic behaviour of smooth functionsMatematichki Vesnik1996481-21521MR1410667ZBL0944.31003PavlovićM.Introduction to Function Spaces on the Disk200420Belgrade, SerbiaMatematički Institut SANUvi+184Posebna IzdanjaMR2109650