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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 729849, 9 pages
Domination Conditions for Families of Quasinearly Subharmonic Functions
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulun Yliopisto, Finland
Received 16 February 2011; Accepted 12 April 2011
Academic Editor: Teodor Bulboacă
Copyright © 2011 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.
Domar has given a condition that ensures the existence of the largest subharmonic minorant of a given function. Later Rippon pointed out that a modification of Domar's argument gives in fact a better result. Using our previous, rather general and flexible, modification of Domar's original argument, we extend their results both to the subharmonic and quasinearly subharmonic settings.
1.1. Results of Domar and Rippon
Suppose that is a domain of, . Let be an upper semicontinuous function. Let be a family of subharmonic functions which satisfy for all . Write and let be the upper semicontinuous regularization of , that is,
Domar gave the following result.
Theorem A. If for some , then is locally bounded above in , and thus is subharmonic in .
See [1, Theorems 1 and 2, pages 430 and 431]. As Domar points out, the original case of subharmonic functions in the result of Theorem 1 is due to Sjöberg  and Brelot  (cf. also ). Observe, however, that Domar also sketches a new proof for Theorem 1 which uses elementary methods and applies to more general functions.
Rippon [5, Theorem 1, page 128] generalized Domar's result in the following form.
Theorem B. Let be an increasing function such that If then is locally bounded above in , and thus is subharmonic in .
As Domar points out, in [1, page 430], the result of his Theorem A holds in fact for more general functions, that is, for functions which by good reasons might be—and indeed already have been—called quasinearly subharmonic functions. See Section 1.2 below for the definition of this function class. In addition, Domar has given a related result for an even more general function class , where the above conditions (1.4) and (1.6) are replaced by a certain integrability condition on the decreasing rearrangement of , see [6, Theorem 1, page 485]. Observe, however, that in the case Domar's class equals the class of nonnegative quasinearly subharmonic functions: if , then is -quasinearly subharmonic. Here (and below) is the Lebesgue measure of the unit ball in . Conversely, if is -quasinearly subharmonic, then .
Below we give a general and at the same time flexible result which includes both Domar's and Rippon's results, Theorems A and B above. See Theorem 2.1, Corollary 2.4, and Remark 2.5 below. For previous preliminary, more or less standard results, see also [7, Theorem 2(d), page 15], [8, Theorem 2, page 71], and [9, Theorem 2.2(vi), page 55] (see Remark 1.2(v)).
Notation 1.1. Our notation is rather standard, see, for example, [7, 9]. For the convenience of the reader we, however, recall the following. is the Lebesgue measure in the Euclidean space , . is always a domain in . Constants will be denoted by and . They are always nonnegative and may vary from line to line.
1.2. Subharmonic Functions and Generalizations
We recall that an upper semicontinuous function is subharmonic if for all closed balls ,
The function is considered subharmonic.
We say that a function is nearly subharmonic, if is Lebesgue measurable, , and for all ,
Observe that in the standard definition of nearly subharmonic functions one uses the slightly stronger assumption that , see, for example, [7, page 14]. However, our above, slightly more general definition seems to be more practical, see, for example [9, Propositions 2.1(iii) and 2.2(vi)-(vii), pages 54 and 55], and also Remark 1.2(i)–(vi) below. The following lemma emphasizes this fact still more.
Lemma 1.1 (see [9, Lemma, page 52]). Let be Lebesgue measurable. Then is nearly subharmonic (in the sense defined above) if and only if there exists a function , subharmonic in such that and almost everywhere in . Here is the upper semicontinuous regularization of :
We say that a Lebesgue measurable function is -quasinearly subharmonic, if and if there is a constant such that for all ,
for all , where . A function is quasinearly subharmonic, if is -quasinearly subharmonic for some .
A Lebesgue measurable function is -quasinearly subharmonic n.s. (in the narrow sense), if and if there is a constant such that for all ,
A function is quasinearly subharmonic n.s., if is -quasinearly subharmonic n.s. for some .
As already pointed out, Domar [1, 6] considered nonnegative quasinearly subharmonic functions. Later on, quasinearly subharmonic functions (perhaps with a different terminology, and sometimes in certain special cases, or just the corresponding generalized mean value inequality (1.10) or (1.11)) have been considered in many papers, see, for example, [8–13] and the references therein.
We recall here only that this function class includes, among others, subharmonic functions, and, more generally, quasisubharmonic and nearly subharmonic functions (see, e.g., [7, pages 14 and 26]), also functions satisfying certain natural growth conditions, especially certain eigenfunctions, and polyharmonic functions. Also, the class of Harnack functions is included, thus, among others, nonnegative harmonic functions as well as nonnegative solutions of some elliptic equations. In particular, the partial differential equations associated with quasiregular mappings belong to this family of elliptic equations.
Remark 1.2. For the sake of convenience of the reader we recall the following, see [9, Propositions 2.1 and 2.2, pages 54 and 55].(i) A -quasinearly subharmonic function n.s. is -quasinearly subharmonic, but not necessarily conversely. (ii) A nonnegative Lebesgue measurable function is -quasinearly subharmonic if and only if it is -quasinearly subharmonic n.s. (iii) A Lebesgue measurable function is 1-quasinearly subharmonic if and only if it is 1-quasinearly subharmonic n.s. and if and only if it is nearly subharmonic (in the sense defined above). (iv) If is -quasinearly subharmonic and is -quasinearly subharmonic, then is -quasinearly subharmonic in . Especially, is -quasinearly subharmonic in . (v)Let be a family of -quasinearly subharmonic (resp., -quasinearly subharmonic n.s.) functions in and let . If is Lebesgue measurable and , then is -quasinearly subharmonic (resp.,-quasinearly subharmonic n.s.) in . (vi)If is quasinearly subharmonic n.s., then either or is finite almost everywhere in , and .
2. The Result
Theorem 2.1. Let . Let and let be increasing functions for which there are , , such that (i)the inverse functions and are defined on , (ii) for all , (iii)the function
is bounded, (iv)the following integral is convergent:
Let be a family of -quasinearly subharmonic functions such that for all , where is a Lebesgue measurable function. If for each compact set , then the family is locally (uniformly) bounded in . Moreover, the function is a -quasinearly subharmonic function. Here where
The proof of the theorem will be based on the following lemma, which has its origin in [1, Lemma 1, pages 431 and 432], see also [14, Proposition 2, pages 257–259]. Observe that we have applied our rather general and flexible lemma already before (unlike previously, now we allow also the value for our “test functions” and ; this does not, however, cause any changes in the proof of our lemma, see [15, pages 5–8]) when considering quasinearly subharmonicity of separately quasinearly subharmonic functions. As a matter of fact, this lemma enabled us to slightly improve Armitage's and Gardiner's almost sharp condition, see [14, Theorem 1, page 256], which ensures a separately subharmonic function to be subharmonic. See [15, Corollary 4.5, page 13], and [12, 13].
Lemma 2.2 (see [15, Lemma 3.2, page 5 and Remark 3.3, page 8]). Let , , and be as in Theorem 2.1. Let be a -quasinearly subharmonic function. Let , , be arbitrary, where Then for each and such that either or where and ,
Proof of Theorem 2.1. Let be an arbitrary compact subset of . Write . Clearly . Write
Then is compact, and . Take arbitrarily, where
Let , say. Take arbitrarily and suppose that , where , say. Using our lemma and the assumption, we get
Since and , the set of values is bounded. Thus also the set of values is bounded.
To show that is -quasinearly subharmonic in , proceed as follows. Take and such that . For each we have then
Since we have
Then just take the upper semicontinuous regularizations on both sides of (2.18) and use Fatou's lemma on the right-hand side (this is of course possible, since is locally bounded in ), say
Since for all , we get the desired inequality
Remark 2.3. If is Lebesgue measurable, it follows that already is -quasinearly subharmonic.
Corollary 2.4. Let be a strictly increasing function such that for some , ,
Let be a family of -quasinearly subharmonic functions such that for all , where is a Lebesgue measurable function.
Let be arbitrary. If for each compact set , then the family is locally (uniformly) bounded in . Moreover, the function is a -quasinearly subharmonic function. Here where
The case and gives Domar's and Rippon's results, Theorems A and B above. For the proof, take arbitrarily, choose , and just check that the conditions (i)–(iv) indeed hold.
Remark 2.5. As already pointed out, Theorem 2.1 is indeed flexible. To get another simple, but still slightly more general corollary, just choose, say, , where is any strictly increasing function which satisfies the following two conditions: (a) satisfies the -condition, (b) for all .
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