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Bi-Lipschitz Mappings and Quasinearly Subharmonic Functions
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 , , these both classes are invariant under bi-Lipschitz mappings.
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 , , is denoted by . We write for the ball in , with center and radius . Recall that , where . If is an open set in , and , then we write for the distance between the point and the boundary of . Our constants are nonnegative, mostly , and may vary from line to line.
1.1. Subharmonic Functions and Generalizations
Let be an open set in , . Let be a Lebesgue measurable function. We adopt the following definitions.(i) is subharmonic if is upper semicontinuous and if for all balls . A subharmonic function may be on any component of ; see [3, page 9] and [4, page 60]. (ii) is nearly subharmonic if and for all balls . Observe that this definition, see [5, page 51], is slightly more general than the standard one [3, page 14]. (iii)Let . Then is -quasinearly subharmonic if and for all and for all balls . Here .
The function is quasinearly subharmonic if is -quasinearly subharmonic for some . 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 , 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 -quasinearly subharmonic. (iii)A nearly subharmonic function is quasinearly subharmonic but not conversely. (iv)If is Lebesgue measurable, then is -quasinearly subharmonic if and only if and for all balls .
1.2. Bi-Lipschitz Mappings
Let be an open set in , . Let be arbitrary. A function is -bi-Lipschitz if
for all . A function is bi-Lipschitz if it is -bi-Lipschitz for some . It is easy to see that if is -bi-Lipschitz, then also is -bi-Lipschitz, where .
Let be an open subset of . Let and . We write
2. On the Generalized Mean Value Inequality
Lemma 2.1. Let be a bounded open set in , . Fix a point . Let be a domain in . Let be a -quasinearly subharmonic function. Then there is such that for every point and all -BiLip, .
Proof. Take and -BiLip, , arbitrarily. (Observe that the set of bi-Lipschitz mappings is (in general) nonempty.) Write Using the fact that is a homeomorphism, one sees easily that . Since is -bi-Lipschitz, it follows from a result of Radó-Reichelderfer, see, for example, [8, Theorem , page 99], that . (Observe that bi-Lipschitz mappings satisfy the property and are differentiable almost everywhere, see, for example, [9, Theorem , page 112, Theorem , page 109].) Therefore, Thus (2.1) holds with .
Theorem 2.2. Let be an open set in , , with . Fix a point . Let be an open set in . Let be a -quasinearly subharmonic function. Then there is a constant such that (2.1) holds for every point and all -BiLip, .
Proof. Let be arbitrary. It is easy to see that for some . Write and . One sees easily that satisfies the assumptions of Lemma 2.1; that is, is a bounded domain, and . Hence there is a constant such that for every point and all -BiLip. Since and are -bi-Lipschitz, it follows that and ; see again [8, Theorem , page 99]. Thus for , Proceed then as follows: Therefore concluding the proof.
3. An Invariance of the Class of Quasinearly Subharmonic Functions
Suppose that and are open sets in the complex plane . If is analytic and is subharmonic, then is subharmonic; see, for example, [3, page 37] and [4, Corollary , page 70]. Using Koebe’s one-quarter and distortion theorems, Kojić proved the following partial generalization.
Theorem 3.1 (see [6, Theorem , page 245]). Let and be open sets in . Let be a -quasinearly subharmonic function. If is conformal, then the composition mapping is -quasinearly subharmonic for some .
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 , , the class of subharmonic functions is invariant under orthogonal transformations; see [10, page 55].
Theorem 3.2. Let and be open sets in . Let be a -quasinearly subharmonic function. If is -bi-Lipschitz, then the composition mapping is -quasinearly subharmonic for some .
Proof. It is sufficient to show that there exists a constant such that
for all . To see this, observe first that
Then Above we have used the routineous fact that for -bi-Lipschitz mappings, and the already cited change of variable result of Radó-Reichelderfer; see [8, Theorem , page 99]. (Recall again that bi-Lipschitz mappings satisfy the property and are differentiable almost everywhere.)
4. An Invariance of Regularly Oscillating Functions
Let be an open set in , . Let be continuous. Write
A function is regularly oscillating, if there is such that
Using again Koebe’s results, Kojić proved also the following result.
Theorem 4.1 (see [6, Theorem , page 245]). Let and be open sets in . Let OC. If is conformal, then OC, where depends only on .
Below we give a partial generalization to Kojić’s above result.
Theorem 4.2. Let and be open sets in . Let OC. If is -bi-Lipschitz, , then OC.
Proof. Let be -bi-Lipschitz. Take and arbitrarily such that . Write and for . Then
Using (3.2) (for ), we get
On the other hand, since is -bi-Lipschitz, Therefore, Thus OC.
In addition of regularly oscillating functions, one sometimes considers so-called HC functions, too; see [11, page 19], [13, page 16], and [12, page 93]. Their definition reads as follows. Let be an open set in , . Let . A function is in HC if
The class HC is the union of all HC, . Clearly, HCOC.
Proceeding as above in the proof of Theorem 4.2 one gets the following result.
Theorem 4.3. Let and be open sets in . Let HC. If is -bi-Lipschitz, , then HC.
The first author was partially supported by the Academy of Finland.
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