Domination Conditions for Families of Quasinearly Subharmonic Functions

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 ofℝ^𝑛, 𝑛≥2. Let 𝐹∶𝐷→[0,+∞] be an upper semicontinuous function. Let ℱ be a family of subharmonic functions 𝑢∶𝐷→[0,+∞) which satisfy 𝑢(𝑥)≤𝐹(𝑥), (1.1) for all 𝑥∈𝐷. Write 𝑤(𝑥)=sup_𝑢∈ℱ𝑢(𝑥),𝑥∈𝐷, (1.2) and let 𝑤^∗∶𝐷→[0,+∞] be the upper semicontinuous regularization of 𝑤, that is, 𝑤^∗(𝑥)∶=limsup_𝑦→𝑥𝑤𝑦. (1.3)

Domar gave the following result.

Theorem A. If for some 𝜀>0, _𝐷log⁺𝐹(𝑥)^{𝑛−1+𝜀}𝑑𝑚_𝑛(𝑥)<+∞, (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^+∞𝑑𝑡𝜑(𝑡)^1/(𝑛−1)<+∞. (1.5) If _𝐷𝜑log⁺𝐹(𝑥)𝑑𝑚_𝑛(𝑥)<+∞, (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 log𝐹, 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 𝜈_𝑛𝐴^𝑛+1-quasinearly subharmonic. Here (and below) 𝜈_𝑛 is the Lebesgue measure of the unit ball 𝐵^𝑛(0,1) in ℝ^𝑛. 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. 𝑚_𝑛 is the Lebesgue measure in the Euclidean space ℝ^𝑛, 𝑛≥2. 𝐷 is always a domain in ℝ^𝑛. 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 𝐵^𝑛(𝑥,𝑟)⊂𝐷,𝑢(𝑥)≤1𝜈_𝑛𝑟^𝑛_𝐵^𝑛_(𝑥,𝑟)𝑢𝑦𝑑𝑚_𝑛𝑦. (1.7)

The function 𝑢≡−∞ is considered subharmonic.

We say that a function 𝑢∶𝐷→[−∞,+∞) is nearly subharmonic, if 𝑢 is Lebesgue measurable, 𝑢⁺∈ℒ¹_loc(𝐷), and for all 𝐵^𝑛(𝑥,𝑟)⊂𝐷, 𝑢(𝑥)≤1𝜈_𝑛𝑟^𝑛_𝐵^𝑛_(𝑥,𝑟)𝑢𝑦𝑑𝑚_𝑛𝑦. (1.8)

Observe that in the standard definition of nearly subharmonic functions one uses the slightly stronger assumption that 𝑢∈ℒ¹_loc(𝐷), 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: 𝑢^∗(𝑥)=limsup_{𝑥′→𝑥}𝑢𝑥′. (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 𝑢⁺∈ℒ¹_loc(𝐷) and if there is a constant 𝐾=𝐾(𝑛,𝑢,𝐷)≥1 such that for all 𝐵^𝑛(𝑥,𝑟)⊂𝐷,𝑢_𝑀(𝑥)≤𝐾𝜈_𝑛𝑟^𝑛_𝐵^𝑛_()𝑥,𝑟𝑢_𝑀𝑦𝑑𝑚_𝑛𝑦, (1.10)

for all 𝑀≥0, where 𝑢_𝑀∶=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 𝑢⁺∈ℒ¹_loc(𝐷) and if there is a constant 𝐾=𝐾(𝑛,𝑢,𝐷)≥1 such that for all 𝐵^𝑛(𝑥,𝑟)⊂𝐷,𝑢(𝑥)≤𝐾𝜈_𝑛𝑟^𝑛_𝐵^𝑛_(𝑥,𝑟)𝑢𝑦𝑑𝑚_𝑛𝑦. (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∈ℱ𝑢. If 𝑤 is Lebesgue measurable and 𝑤⁺∈ℒ¹_loc(𝐷), 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 𝑢∈ℒ¹_loc(𝐷).

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 𝑠₁𝜓+1,+∞∋𝑠⟼⁻¹∘𝜑(𝑠+1)𝜓⁻¹∘𝜑(𝑠)∈ℝ (2.1)is bounded, (iv)the following integral is convergent: _𝑠^+∞₁𝑑𝑠𝜑𝑠−𝑠₀^1/(𝑛−1)<+∞. (2.2)
Let ℱ_𝐾 be a family of 𝐾-quasinearly subharmonic functions 𝑢∶𝐷→[−∞,+∞) such that 𝑢(𝑥)≤𝐹_𝐾(𝑥), (2.3) for all 𝑥∈𝐷, where F_K∶𝐷→[0,+∞] is a Lebesgue measurable function. If for each compact set 𝐸⊂𝐷, _𝐸𝜓𝐹_𝐾(𝑥)𝑑𝑚_𝑛(𝑥)<+∞, (2.4) then the family ℱ_𝐾 is locally (uniformly) bounded in 𝐷. Moreover, the function 𝑤^∗∶𝐷→[0,+∞) is a 𝐾-quasinearly subharmonic function. Here 𝑤^∗(𝑥)∶=limsup_𝑦→𝑥𝑤𝑦, (2.5) where 𝑤(𝑥)∶=sup_{𝑢∈ℱ𝐾}𝑢⁺(𝑥). (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 𝑠₁̃∈ℕ, 𝑠₁̃≥𝑠₃, be arbitrary, where 𝑠₃∶=max{𝑠₁+3,(𝜓⁻¹∘𝜑)(𝑠₁+3)}. Then for each 𝑥∈𝐷 and 𝑟>0 such that 𝐵^𝑛(𝑥,𝑟)⊂𝐷 either 𝑢(𝑥)≤𝜓⁻¹∘𝜑𝑠₁̃+1 (2.7) or Φ(𝑢(𝑥≤𝐶))𝑟^𝑛_𝐵^𝑛_()𝑥,𝑟𝜓𝑢𝑦𝑑𝑚_𝑛𝑦, (2.8) where 𝐶=𝐶(𝑛,𝐾,𝑠₀) and Φ∶[0,+∞)→[0,+∞), Φ(𝑡)∶=^{+∞(𝜑−1(𝑡)∘𝜓)−2}𝑑𝑠𝜑𝑠−𝑠₀^{1/(𝑛−1)1−𝑛},when𝑡≥𝑠_3,𝑡𝑠₃Φ𝑠₃,when0≤𝑡<𝑠₃. (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 𝐸₁∶=_𝑥∈𝐸𝐵^𝑛𝜌𝑥,⁰2. (2.10)
Then 𝐸₁ is compact, and 𝐸⊂𝐸₁⊂𝐷. Take 𝑢∈ℱ⁺_𝐾 arbitrarily, where ℱ⁺_𝐾∶=𝑢⁺∶𝑢∈ℱ_𝐾. (2.11)
Let 𝑠₁̃=𝑠₁+2, say. Take 𝑥∈𝐸 arbitrarily and suppose that 𝑢(𝑥)>𝑠₃̃, where 𝑠₃̃∶=max{𝑠₁̃+3,(𝜓⁻¹∘𝜑)(𝑠₁̃+3)}, say. Using our lemma and the assumption, we get ^+∞_{(𝜑−1∘𝜓)(𝑢(𝑥))−2}𝑑𝑠𝜑(𝑠−𝑠₀)^{1/(𝑛−1)1−𝑛}≤𝐶𝜌₀/2^𝑛_𝐵^𝑛_{(𝑥,𝜌0/2)}𝜓𝑢𝑦𝑑𝑚_𝑛𝑦≤𝐶𝜌₀/2^𝑛_𝐸1𝜓𝐹_𝐾𝑦𝑑𝑚_𝑛𝑦<+∞. (2.12)
Since _𝑠^+∞₁𝑑𝑠𝜑𝑠−𝑠₀^1/(𝑛−1)<+∞, (2.13) and 1−𝑛<0, the set of values 𝜑⁻¹∘𝜓(𝑢(𝑥))−2,𝑥∈𝐸,𝑢∈ℱ⁺_𝐾, (2.14) is bounded. Thus also the set of values 𝑢(𝑥),𝑥∈𝐸,𝑢∈ℱ⁺_𝐾, (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 𝐵^𝑛(𝑥,𝑟)⊂𝐷. For each 𝑢∈ℱ⁺_𝐾 we have then 𝑢(𝑥)≤𝐾𝜈_𝑛𝑟^𝑛_𝐵^𝑛_()𝑥,𝑟𝑢𝑦𝑑𝑚_𝑛𝑦. (2.16)
Since 𝑢(𝑥)≤sup_{𝑢∈ℱ+𝐾}𝑢(𝑥)=𝑤(𝑥)≤𝑤^∗(𝑥), (2.17)we have 𝑤(𝑥)≤𝐾𝜈_𝑛𝑟^𝑛_𝐵^𝑛_()𝑥,𝑟𝑤^∗𝑦𝑑𝑚_𝑛𝑦. (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 limsup_𝑦→𝑥𝑤𝑦≤limsup_𝑦→𝑥𝐾𝜈_𝑛𝑟^𝑛_𝐵^𝑛_(𝑦,𝑟)𝑤^∗(𝑧)𝑑𝑚_𝑛(𝑧)≤limsup_𝑦→𝑥𝐾𝜈_𝑛𝑟^𝑛𝑤^∗(𝑧)𝜒_{𝐵𝑛(𝑦,𝑟)}(𝑧)𝑑𝑚_𝑛(𝑧)≤𝐾𝜈_𝑛𝑟^𝑛𝑤^∗(𝑧)limsup_𝑦→𝑥𝜒_{𝐵𝑛(𝑦,𝑟)}(𝑧)𝑑𝑚_𝑛(𝑧). (2.19)
Since for all 𝑧∈𝐷, limsup_𝑦→𝑥𝜒_{𝐵𝑛(𝑦,𝑟)}(𝑧)≤𝜒_{𝐵𝑛(𝑥,𝑟)}(𝑧), (2.20) we get the desired inequality 𝑤^∗(𝑥)≤𝐾𝜈_𝑛𝑟^𝑛_𝐵^𝑛_(𝑥,𝑟)𝑤^∗𝑦𝑑𝑚_𝑛𝑦. (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, _𝑠^+∞₁𝑑𝑠𝜑𝑠−𝑠₀^1/(𝑛−1)<+∞. (2.22)
Let ℱ_𝐾 be a family of 𝐾-quasinearly subharmonic functions 𝑢∶𝐷→[−∞,+∞) such that 𝑢(𝑥)≤𝐹_𝐾(𝑥), (2.23) for all 𝑥∈𝐷, where 𝐹_𝐾∶𝐷→[0,+∞] is a Lebesgue measurable function.
Let 𝑝>0 be arbitrary. If for each compact set E⊂D, _𝐸𝜑log⁺[𝐹(𝑥)]^𝑝𝑑𝑚_𝑛(𝑥)<+∞, (2.24) then the family ℱ_𝐾 is locally (uniformly) bounded in 𝐷. Moreover, the function 𝑤^∗∶𝐷→[0,+∞) is a 𝐾-quasinearly subharmonic function. Here 𝑤^∗(𝑥)∶=limsup_𝑦→𝑥𝑤𝑦, (2.25) where 𝑤(𝑥)∶=sup_{𝑢∈ℱ𝐾}𝑢⁺(𝑥). (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⁺)(𝑡^𝑝), 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)2𝐾𝜙⁻¹(𝑒^𝑠−𝑠0)≤𝜙⁻¹(𝑒^𝑠) for all 𝑠≥𝑠₁.