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.

u is subharmonic if u is upper semicontinuous and if
u(x)≤1νnrn∫Bn(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+∈ℒloc1(Ω) and
u(x)≤1νnrn∫Bn(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 K≥1. Then u is K-quasinearly subharmonic if u+∈ℒloc1(Ω) and
uL(x)≤Kνnrn∫Bn(x,r)uL(y)dmn(y)
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.

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 u∈ℒloc1(Ω) and
u(x)≤Kνnrn∫Bn(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 ℝn, n≥2. Let M≥1 be arbitrary. A function f:D→ℝn is M-bi-Lipschitz if

|y-x|M≤|f(y)-f(x)|≤M|y-x|
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-1:D′→ℝn is M-bi-Lipschitz, where D′:=f(D).

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

2. On the Generalized Mean Value InequalityLemma 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)
for every point xΩ∈Ω and all h∈M-BiLip(pD,xΩ,D,Ω), M≥1.

Proof.

Take xΩ∈Ω and h∈M-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).
Using the fact that h∣Bn(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,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!Mn∫h(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 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 h∈M-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)≤C1mn(h(Dt))∫h(Dt)u(y)dmn(y)
for every point xΩ∈Ω and all h∈M-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 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.

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
(u∘f)(x0)≤Cmn(Bn(x0,r0))∫Bn(x0,r0)(u∘f)(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
(u∘f)(x0)=u(x0′)≤Kmn(Bn(x0′,r0/M))∫Bn(x0′,r0/M)u(y)dmn(y)≤Kνn(r0/M)n∫Bn(x0′,r0/M)(u∘f)(f-1(y))dmn(y)≤KM2nνn(Mr0)n∫Bn(x0′,r0/M)(u∘f)(f-1(y))dmn(y)≤KM2nmn(f(Bn(x0,r0)))∫f(Bn(x0,r0))(u∘f)(f-1(y))dmn(y)≤KM2nmn(f(Bn(x0,r0)))∫f(Bn(x0,r0))(u∘f)(f-1(y))|Jf-1(y)|·1|Jf-1(y)|dmn(y)≤KM2nmn(f(Bn(x0,r0)))∫f(Bn(x0,r0))(u∘f)(f-1(y))|Jf-1(y)|·|Jf(f-1(y))|dmn(y)≤KM2nmn(f(Bn(x0,r0)))∫f(Bn(x0,r0))(u∘f)(f-1(y))|Jf-1(y)|·n!Mndmn(y)≤n!KM3nmn(f(Bn(x0,r0)))∫f(Bn(x0,r0))(u∘f)(f-1(y))|Jf-1(y)|dmn(y)≤n!KM3nmn(f(Bn(x0,r0)))∫Bn(x0,r0)(u∘f)(x)dmn(x)≤n!KM3nmn(Bn(x0′,r0/M))∫Bn(x0,r0)(u∘f)(x)dmn(x)≤n!KM4nmn(Bn(x0,r0))∫Bn(x0,r0)(u∘f)(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 ℝn, n≥2. Let f:Ω→ℝm be continuous. Write

L(x,f):=lim supy→x|f(y)-f(x)||y-x|.
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-1supy∈Bn(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], [14], [6, page 245], and [12, page 96].

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

Let Ω and G be open sets in ℂ. Let u∈OCK1(Ω). If f:G→Ω is conformal, then u∘f∈OCC1(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∈OCK1(Ω). If φ:U→Ω is M-bi-Lipschitz, M≥1, then u∘φ∈OCKM21(U).

Proof.

Let φ:U→Ω be M-bi-Lipschitz. Take x0∈U and r0>0 arbitrarily such that Bn(x0,r0)¯⊂U. Write x0′=φ(x0) and x′=φ(x) for x∈U. Then
L(x0,u∘φ)=lim supx→x0|u(φ(x))-u(φ(x0))||x-x0|=lim supx→x0|u(φ(x))-u(φ(x0))||φ(x)-φ(x0)|·|φ(x)-φ(x0)||x-x0|≤lim supx′→x0′|u(x′)-u(x0′)||x′-x0′|·lim supx→x0|φ(x)-φ(x0)||x-x0|=L(x0′,u)·lim supx→x0|φ(x)-φ(x0)||x-x0|.
Using (3.2) (for f=φ), we get
L(x0′,u)≤Kr0/Msupx′∈Bn(x0′,r0/M)|u(x′)-u(x0′)|≤KMr0supx′∈Bn(x0′,r0/M)|u(x′)-u(x0′)|≤KMr0supx′∈φ(Bn(x0,r0))|u(x′)-u(x0′)|≤KMr0supx∈Bn(x0,r0)|u(φ(x))-u(φ(x0))|≤KMr0supx∈Bn(x0,r0)|(u∘φ)(x)-(u∘φ)(x0)|.

On the other hand, since φ is M-bi-Lipschitz,
lim supx→x0|φ(x)-φ(x0)||x-x0|≤lim supx→x0M|x-x0||x-x0|=M<+∞.
Therefore,
L(x0,u∘φ)≤KMr0supx∈Bn(x0,r0)|(u∘φ)(x)-(u∘φ)(x0)|·M≤KM2r0supx∈Bn(x0,r0)|(u∘φ)(x)-(u∘φ)(x0)|.
Thus u∘φ∈OCKM21(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 HCK1(Ω) if

L(x,f)≤Kr-1supy∈Bn(x,r)|f(y)|,Bn(x,r)¯⊂Ω.
The class HC1(Ω) is the union of all HCK1(Ω), K≥1. 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 ℝn,n≥2. Let u∈HCK1(Ω). If φ:U→ℝn is M-bi-Lipschitz, M≥1, then u∘φ∈HCKM21(U).

Acknowledgment

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

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