International Journal of Mathematics and Mathematical Sciences
Volume 2011, Issue 1 729849
Research Article
Open Access

Domination Conditions for Families of Quasinearly Subharmonic Functions

Juhani Riihentaus, 

Corresponding Author

Juhani Riihentaus

Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulun Yliopisto, Finland oulu.fi

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First published: 15 June 2011
https://doi.org/10.1155/2011/729849
Academic Editor: Teodor Bulboacă

Abstract

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. Introduction

1.1. Results of Domar and Rippon

Suppose that D is a domain of  ℝn, n ≥ 2. Let F : D → [0, +∞] be an upper semicontinuous function. Let ℱ be a family of subharmonic functions u : D → [0, +∞) which satisfy
(1.1)
for all x ∈ D. Write
(1.2)
and let w* : D → [0, +∞] be the upper semicontinuous regularization of w, that is,
(1.3)

Domar gave the following result.

Theorem A. If for some ε > 0,

(1.4)
then w is locally bounded above in D, and thus w* is subharmonic in D.

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 [2] and Brelot [3] (cf. also [4]). 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 φ : [0, +∞] → [0, +∞] be an increasing function such that

(1.5)
If
(1.6)
then w is locally bounded above in D, and thus w* is subharmonic in D.

As pointed out by Domar [1, pages 436–440] and by Rippon [5, page 129], the above results are for many particular cases sharp.

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 K(A, α), where the above conditions (1.4) and (1.6) are replaced by a certain integrability condition on the decreasing rearrangement of log F, see [6, Theorem 1, page 485]. Observe, however, that in the case α = n Domar′s class K(A, n) equals the class of nonnegative quasinearly subharmonic functions: if u ∈ K(A, n), then u is νnAn+1-quasinearly subharmonic. Here (and below) νn is the Lebesgue measure of the unit ball Bn(0,1) in ℝn. Conversely, if u ≥ 0 is C-quasinearly subharmonic, then u ∈ K(C, n).

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. mn is the Lebesgue measure in the Euclidean space ℝn, n ≥ 2. D is always a domain in ℝn. Constants will be denoted by C and K. They are always nonnegative and may vary from line to line.

1.2. Subharmonic Functions and Generalizations

We recall that an upper semicontinuous function u : D → [  − ∞, +∞) is subharmonic if for all closed balls ,
(1.7)

The function u ≡ −∞ is considered subharmonic.

We say that a function u : D → [  − ∞, +∞) is nearly subharmonic, if u is Lebesgue measurable, , and for all ,
(1.8)

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 u : D → [  − ∞, +∞) be Lebesgue measurable. Then u is nearly subharmonic (in the sense defined above) if and only if there exists a function u*, subharmonic in D such that u* ≥ u and u* = u almost everywhere in D. Here u* is the upper semicontinuous regularization of u:

(1.9)

The proof follows at once from [7, proof of Theorem 1, pages 14 and 15], (and referring also to [9, Propositions 2.1(iii) and 2.2(vii), pages 54 and 55]).

We say that a Lebesgue measurable function u : D → [  − ∞, +∞) is K-quasinearly subharmonic, if and if there is a constant K = K(n, u, D) ≥ 1 such that for all ,
(1.10)

for all M ≥ 0, where uM : = max {u, −M} + M. A function u : D → [  − ∞, +∞) is quasinearly subharmonic, if u is K-quasinearly subharmonic for some K ≥ 1.

A Lebesgue measurable function u : D → [  − ∞, +∞) is K-quasinearly subharmonic n.s. (in the narrow sense), if and if there is a constant K = K(n, u, D) ≥ 1 such that for all ,
(1.11)

A function u : D → [  − ∞, +∞) is quasinearly subharmonic n.s., if u is K-quasinearly subharmonic n.s. for some K ≥ 1.

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 K-quasinearly subharmonic function n.s. is K-quasinearly subharmonic, but not necessarily conversely.

  • (ii)

    A nonnegative Lebesgue measurable function is K-quasinearly subharmonic if and only if it is K-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 u : D → [−∞, +∞) is K1-quasinearly subharmonic and v : D → [  − ∞, +∞) is K2-quasinearly subharmonic, then max {u, v} is max {K1, K2}-quasinearly subharmonic in D. Especially, u+ : = max  {u, 0} is K1-quasinearly subharmonic in D.

  • (v)

    Let ℱ be a family of K-quasinearly subharmonic (resp., K-quasinearly subharmonic n.s.) functions in D and let w : = sup u∈ℱ u. If w is Lebesgue measurable and , then w is K-quasinearly subharmonic (resp.,  K-quasinearly subharmonic n.s.) in D.

  • (vi)

    If u : D → [  − ∞, +∞) is quasinearly subharmonic n.s., then either u ≡ −∞ or u is finite almost everywhere in D, and .

2. The Result

Theorem 2.1. Let K ≥ 1. Let φ : [0, +∞] → [0, +∞] and let ψ : [0, +∞] → [0, +∞] be increasing functions for which there are s0,   s1 ∈ ℕ, s0 < s1, such that

  • (i)

    the inverse functions φ−1 and ψ−1 are defined on [min {φ(s1 − s0), ψ(s1 − s0)}, +∞],

  • (ii)

    2K(ψ−1∘φ)(s − s0)≤(ψ−1∘φ)(s) for all s ≥ s1,

  • (iii)

    the function

    (2.1)

  •  

    is bounded,

  • (iv)

    the following integral is convergent:

    (2.2)

Let ℱK be a family of K-quasinearly subharmonic functions u : D → [  − ∞, +∞) such that

(2.3)
for all x ∈ D, where FK : D → [0, +∞] is a Lebesgue measurable function. If for each compact set E ⊂ D,
(2.4)
then the family ℱK is locally (uniformly) bounded in D. Moreover, the function w* : D → [ 0, +∞) is a K-quasinearly subharmonic function. Here
(2.5)
where
(2.6)

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” φ and ψ; 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 K, φ, ψ and s0, s1 ∈ ℕ be as in Theorem 2.1. Let u : D → [ 0, +∞) be a K-quasinearly subharmonic function. Let , , be arbitrary, where Then for each x ∈ D and r > 0 such that either

(2.7)
or
(2.8)
where C = C(n, K, s0) and Φ : [ 0, +∞)→[0, +∞),
(2.9)

Proof of Theorem 2.1. Let E be an arbitrary compact subset of D. Write ρ0∶ = dist  (E, ∂D). Clearly ρ0 > 0. Write

(2.10)

Then E1 is compact, and E ⊂ E1 ⊂ D. Take arbitrarily, where

(2.11)

Let , say. Take x ∈ E arbitrarily and suppose that , where , say. Using our lemma and the assumption, we get

(2.12)

Since

(2.13)
and 1 − n < 0, the set of values
(2.14)
is bounded. Thus also the set of values
(2.15)
is bounded.

To show that w* is K-quasinearly subharmonic in D, proceed as follows. Take x ∈ D and r > 0 such that . For each we have then

(2.16)

Since

(2.17)
we have
(2.18)

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 w* is locally bounded in D), say

(2.19)

Since for all z ∈ D,

(2.20)
we get the desired inequality
(2.21)

Remark 2.3. If w is Lebesgue measurable, it follows that already w is K-quasinearly subharmonic.

Corollary 2.4. Let φ : [0, +∞] → [0, +∞] be a strictly increasing function such that for some s0, s1 ∈ ℕ, s0 < s1,

(2.22)

Let ℱK be a family of K-quasinearly subharmonic functions u : D → [  − ∞, +∞) such that

(2.23)
for all x ∈ D, where FK : D → [0, +∞] is a Lebesgue measurable function.

Let p > 0 be arbitrary. If for each compact set E ⊂ D,

(2.24)
then the family ℱK is locally (uniformly) bounded in D. Moreover, the function w* : D → [ 0, +∞) is a K-quasinearly subharmonic function. Here
(2.25)
where
(2.26)

The case p = 1 and K = 1 gives Domar′s and Rippon′s results, Theorems A and B above. For the proof, take p > 0 arbitrarily, choose ψ(t) = (φ∘log +)(tp), 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, ψ(t) = (φ∘log +)(ϕ(t)), where ϕ : [0, +∞]→[0, +∞] is any strictly increasing function which satisfies the following two conditions:

  • (a)

    ϕ−1 satisfies the Δ2-condition,

  • (b)

    for all s ≥ s1.

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