Separately Subharmonic Functions and Quasi-Nearly Subharmonic Functions

Separately subharmonic functions and quasi-nearly subharmonic functions

Juhani RIIHENTAUS Department of Physics and Mathematics, University of Joensuu P.O. Box 111, FI-80101 Joensuu, Finland

ABSTRACT R2n, n ≥ 1. Constants will be denoted by C and K. They will be nonnegative and may vary from line to line. First, we give the definition for quasi-nearly subharmonic functions. Second, after recalling the existing subharmonic- 1.3 Nearly subharmonic functions ity results of separately subharmonic functions, we give cor- We recall that an upper semicontinuous function u : D → responding counterparts for separately quasi-nearly subhar- [−∞,+∞) is subharmonic if for all BN (x,r) ⊂ D, monic functions, thus generalizing previous results of Ar- 1 mitage and Gardiner, of ours, of Arsove, of Avanissian, and u(x) ≤ N u(y)dmN (y). νN r Z of Lelong. BN (x,r) Key words: Subharmonic, Quasi-nearly subharmonic, Sep- The function u ≡−∞ is considered subharmonic. arately subharmonic, Integrability condition. We say that a function u : D → [−∞,+∞) is nearly + L1 subharmonic, if u is Lebesgue measurable, if u ∈ loc(D), 1. INTRODUCTION and for all BN (x,r) ⊂ D, 1 u(x) ≤ N u(y)dmN (y). 1.1 Previous results νN r Z BN x r It is a well-known problem whether a separately subhar- ( , ) monic function is subharmonic or not. As far as we know, it Observe that in the standard definition of nearly subharmonic was Lelong [9, Théorème 1 bis, p. 315] who gave the first re- functions one uses the slightly stronger assumption that u ∈ L1 sult related to this problem. Much later Wiegerinck [24], see loc(D), see e.g. [7, p. 14]. However, our above, slightly also [25, Theorem 1, p. 246], showed that a separately sub- more general definition seems to be more practical, see [21, harmonic function need not be subharmonic. On the other Proposition 2.1 (iii) and Proposition 2.2 (vi) and (vii)]. hand, Armitage and Gardiner [1, Theorem 1, p. 256], gave an “almost sharp” condition, which ensures that a separately 1.4 Quasi-nearly subharmonic functions subharmonic function u of a domain Ω in Rm+n, m ≥ n ≥ 2, Let K ≥ 1. A Lebesgue measurable function u : D → [−∞,+∞) is subharmonic. Armitage’s and Gardiner’s condition was + L1 is K-quasi-nearly subharmonic, if u ∈ loc(D) and if there the following: is a constant K = K(N,u,D) ≥ 1 such that for all BN (x,r) ⊂ φ(log+ u+) is locally integrable in Ω, where φ : [0,+∞) → D, [0,+∞) is an increasing function such that K uM(x) ≤ N uM(y)dmN (y) νN r Z N +∞ B (x,r) s(n−1)/(m−1)(φ(s))−1/(m−1) ds < +∞. for all M ≥ 0. Here u := max{u,−M} + M. A function Z M 1 u : D → [−∞,+∞) is quasi-nearly subharmonic, if u is K- quasi-nearly subharmonic for some K ≥ 1. A Lebesgue measurable function u : D → [−∞,+∞) The purpose of this paper is the following. First, we is K-quasi-nearly subharmonic n.s. (in the narrow sense), if list the existing results on this problem. Second, we extend + L1 u ∈ loc(D) and if there is a constant K = K(N,u,D) ≥ 1 these results to the more general setup of so called quasi- such that for all BN (x,r) ⊂ D, nearly subharmonic functions. We begin with the notation and necessary definitions. K u(x) ≤ N u(y)dmN (y). νN r Z 1.2 Notation. BN (x,r)

Our notation is rather standard, see e.g. [21] and [7]. mN A function u : D → [−∞,+∞) is quasi-nearly subharmonic is the Lebesgue measure in the Euclidean space RN , N ≥ n.s., if u is K-quasi-nearly subharmonic n.s. for some K ≥ 1. 2. We write νN for the Lebesgue measure of the unit ball Quasi-nearly subharmonic functions (perhaps with N N N B (0,1) in R , thus νN = mN (B (0,1)). D is a domain of a different termonology) have previously been considered at RN . The complex space Cn is identified with the real space least in [13], [12], [16], [18], [19], [14], [21], [8] and [4]. We recall here only that this function class includes, among oth- 2. ON THE SUBHARMONICITY OF ers, subharmonic functions, and, more generally, quasisub- OF SEPARATELY SUBHARMONIC FUNCTIONS harmonic (see e.g. [9, p. 309], [3, p. 136], [7, p. 26]) and also nearly subharmonic functions (see e.g. [7, p. 14]), also func- 2.1 Armitage’s and Gardiner’s result tions satisfying certain natural growth conditions, especially Armitage and Gardiner 1993: Let Ω be a domain in certain eigenfunctions, and polyharmonic functions. Also, Rm+n, m ≥ n ≥ 2. Let u : Ω → [−∞,+∞) be such that the class of Harnack functions is included, thus, among oth- n ers, nonnegative harmonic functions as well as nonnegative (a) for each y ∈ R the function solutions of some elliptic equations; in particular, the par- Ω(y) 3 x 7→ u(x,y) ∈ [−∞,+∞) tial differential equations associated with quasiregular mappings belong to this family of elliptic equations, see [23]. is subharmonic, Observe that already Domar in [5, p. 430] has pointed out (b) for each x ∈ Rm the function the relevance of the class of (nonnegative) quasi-nearly subharmonic functions. For, at least partly, an even more gen- Ω(x) 3 y 7→ u(x,y) ∈ [−∞,+∞) eral function class, see [6]. is subharmonic, For basic properties of quasi-nearly subharmonic func- (c) φ(log+ u+) is locally integrable in Ω, where φ : tions, see the above references, especially [14] and [21]. [0,+∞) → [0,+∞) is an increasing function such Here we recall only the following: that (i) A K-quasi-nearly subharmonic function n.s. is K- +∞ quasi-nearly subharmonic, but not necessarily con- s(n−1)/(m−1)(φ(s))−1/(m−1) ds < +∞. versely. Z (ii) A nonnegative Lebesgue measurable function is K- 1 quasi-nearly subharmonic if and only if it is K- Then u is subharmonic in Ω. quasi-nearly subharmonic n.s. (iii) A Lebesgue measurable function is 1-quasi-nearly 2.2 Previous results subharmonic if and only if it is 1-quasi-nearly sub- Below we list the previous results of Lelong [9, Théorème harmonic n.s. and if and only if it is nearly subhar- 1 bis, p. 315], of Avanissian [3, Théorème 9, p. 140], see monic (in the sense defined above). also [7, Theorem, p. 31], of Arsove [2, Theorem 1, p. 622] (iv) If u : D → [0,+∞) is quasi-nearly subharmonic and and of ours [15, Theorem 1, p. 69]. Observe that though Ar- ψ : [0,+∞) → [0,+∞) is permissible, then ψ ◦ u is mitage’s and Gardiner’s result includes all of them, its proof quasi-nearly subharmonic in D. is, however, based either on the previous result of Avanis- (v) Harnack functions are quasi-nearly subharmonic. sian, or of Arsove, or of Riihentaus. Observe here that all Recall that a function ψ : [0,+∞) → [0,+∞) is per- these three results have different and independent proofs. missible, if there exist an increasing (strictly or not), con- Lelong 1945: Let Ω be a domain in Cn, n ≥ 2, and let vex function ψ1 : [0,+∞) → [0,+∞) and a strictly increas- u : Ω → [−∞,+∞) be separately subharmonic (i.e. sub- ing surjection ψ2 : [0,+∞) → [0,+∞) such that ψ = ψ2 ◦ψ1 harmonic with respect to each complex variable z j, when and such that the following conditions are satisfied: the other variables z1,...,z j−1,z j+1,...,zn are fixed, j = (a) ψ1 satisfies the ∆2-condition, 1,2,...,n). If u is locally bounded above in Ω, then u is −1 subharmonic. (b) ψ2 satisfies the ∆2-condition, See also [10, Thèoréme 1 b, p. 290] and [11, Propo- (c) the function t 7→ ψ2(t) is quasi-decreasing, i.e. there t sition 3, p. 24]. is a constant C = C(ψ ) > 0 such that 2 Avanissian 1961: As the result of Armitage and Gardiner ψ (s) ψ (t) 1993, except that (c) is now replaced with the stronger con- 2 ≥ C 2 s t dition: (c’) u is locally bounded above in Ω. for all 0 ≤ s ≤ t. Arsove 1966: As the result of Armitage and Gardiner 1993, except that (c) is now replaced with the stronger condition: Recall also that a function ϕ : [0,+∞) → [0,+∞) satisfies (c”) u ∈ L1 (Ω). the ∆ -condition, if there is a constant C = C(ϕ) ≥ 1 such loc 2 Riihentaus 1989: As the result of Armitage and Gardiner that ϕ(2t) ≤ C ϕ(t) for all t ∈ [0,+∞). p 1993, except that (c) is now replaced with the stronger con- Examples of permissible functions are: ψ1(t)= t , L p pα pγ β dition: (c”’) u ∈ loc(Ω) for some p > 0. p > 0, and ψ2(t)= ct [log(δ + t )] , c > 0, 0 < α < 1, δ ≥ 1, β,γ ∈ R such that 0 < α + βγ < 1, and p ≥ 1. And also functions of the form ψ3 = φ ◦ ϕ, where φ : [0,+∞) → 3. ON QUASI-NEARLY SUBHARMONICITY [0,+∞) is a concave surjection whose inverse φ−1 satisfies OF SEPARATELY QUASI-NEARLY the ∆2-condition and ϕ : [0,+∞) → [0,+∞) is an increasing, SUBHARMONIC FUNCTIONS convex function satisfying the ∆2-condition. See e.g. [16], [18], [19] and [14, Lemma 1 and Remark 1]. 3.1 A counterpart to Armitage’s and Gardiner’s result Next a counterpart to the cited result of Armitage and Gar- Then u is quasi-nearly subharmonic in Ω. diner [1, Theorem 1, p. 256] for quasi-nearly subharmonic Though the next corollary does not contain Armitage’s functions. We will present our result with a complete proof and Gardiner’s result, it nevertheless improves the cited pre- elsewere. We mention here only that the method of proof vious results of Lelong, of Avanissian, of Arsove and of is more or less a straitforward and technical, though by no ours, and as such it might be of some interest. As a matter means easy, modification of Armitage’s and Gardiner’s ar- of fact, it is already a corollary of our previous result [21, gument [1, proof of Proposition 2, pp. 257-259, proof of Theorem 3.1], but it can be considered as a consequence of Theorem 1, pp. 258-259] and of Domar’s argument [5, Lem- the present Theorem, too. ma 1, pp. 431-432 and 430]. Our result gives a slight gener- Corollary 2. ([21, Corollary 3.3]) Let Ω be a domain in alization, at least seemingly and formally, also for the clas- Rm+n, m,n ≥ 2. Let u : Ω → [−∞,+∞) be such that sical situation of separately subharmonic functions. (a) for each y ∈ Rn the function 3.2 The result Ω(y) 3 x 7→ u(x,y) ∈ [−∞,+∞) Theorem. ([22, Theorem 4.2]) Let Ω be a domain in Rm+n, m ≥ n ≥ 2, and let K ≥ 1. Let u : Ω → [−∞,+∞) be a Lebesgue is nearly subharmonic, and, for almost every y ∈ Rn measurable function. 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