A. Sadullaev

Sadullaev, A.

Numérisation de Documents Anciens Mathématiques

On separately subharmonic functions (Lelong’s problem)

Annales de la Faculté des Sciences de Toulouse

ANNALESDE LA FACULTÉDES SCIENCESMathématiquesA. SADULLAEVOn separately subharmonic functions (Lelong’s problem)Tome XX, no S2 (2011), p. 183-187.<http://afst.cedram.org/item?id=AFST_2011_6_20_S2_183_0>© Université Paul Sabatier, Toulouse, 2011, tous droits réservés.L’accès aux articles de la revue « Annales de la faculté des sci-ences de Toulouse Mathématiques » (http://afst.cedram.org/), impliquel’accord avec les conditions générales d’utilisation (http://afst.cedram.org/legal/). Toute reproduction en tout ou partie cet article sous quelqueforme que ce soit pour tout usage autre que l’utilisation à fin strictementpersonnelle du copiste est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Annales de la Faculté des Sciences de Toulouse Vol. XX, n◦ Spécial, 2011pp. 183–187On separately subharmonic functions(Lelong’s problem)A. Sadullaev(1)ABSTRACT. — The main result of the present paper is : every separately-subharmonic function u(x, y), which is harmonic in y, can be representedlocally as a sum two functions, u = u∗ + U , where U is subharmonic andu∗ is harmonic in y , subharmonic in x and harmonic in (x, y) outside ofsome nowhere dense set S.RÉSUMÉ. — Le résultat essentiel de ce papier est le suivant : toute fonc-tion séparément sous-harmonique u(x, y) qui est harmonique en y peutêtre reprétée localement comme la somme de deux fonctions u = u∗ +U ,où U est sous-harmonique et u∗ est harmonique en y, sous-harmoniqueen x et harmonique en (x, y) en dehors d’une ensemble nulle part denseS.1. IntroductionWe will consider functions u(x, y) of two groups of variables x ∈ Rn,y ∈ Rm. If u is separately harmonic, i.e., harmonic in x for fixed y andharmonic in y for fixed x, then u will be harmonic in both variables (Lelong[2],see also [1]). Lelong investigated also separately subharmonic functions,and proved a series of special results in this area. Here originates the questionabout subharmonicity of separately subharmonic functions.(∗) * Partially supported by the fundamental research of Khorezm Mamun Academy,Grant Vuzgorodok, 100174(1) Mathematics department, National University of Uzbekistan, Vu Gorodok, 700174Tashkent, Uzbekistansadullaev@mail.ru– 183 –On separately subharmonic functions (Lelong’s problem)However, Wiegerinck, [3] (see also [4]) has shown that a separatelysubharmonic function need not to be subharmonic in general. He con-structed a separately subharmonic function u (x, y) in the bidisk U2 ={|x| < 1} × {|y| < 1} ⊂ R2x × R2y ≈ C × C, which is not bounded abovenear 0.The problem of subharmonicity of a separately subharmonicfunction u(x, y) that is in addition harmonic in y, is still open.In the present paper we will study the class of these functions. Let usbegin by recalling the following well-known results:1. If a separately subharmonic function is bounded above, then it issubharmonic (Lelong [2], Avanissian [5]);2. If u+ ∈ L1loc, then u is subharmonic (Arsove [6]);3. If u+ ∈ Lploc, p > 0, then u is subharmonic(Riihentaus [7]);4. There are also positive results under weak growth conditions (see [8],[9]).We note that the conditions in the above results are not separated in x andy. The following results demand separate conditions:5. Suppose that u(x, y) is defined on the product domain B = B1 ×B2 ⊂ Rnx × Rmy . If u is subharmonic in x and harmonic in y, thenthere are nowhere dense closed sets S1 ⊂ B1, S2 ⊂ B2 such that uis subharmonic in G = (B1 ×B2) \ (S1 × S2) (Cegrell and Sadullaev[10]);6. If u(x, y) real analytic, subharmonic in x, and harmonic in y, then uis subharmonic (Imomkulov [11]);7. There exists a separately subharmonic function u(x, y), which is realanalytic in x, but which is not subharmonic (Cegrell and Sadullaev[10]);8. If u(x, y) is C2 and subharmonic in x, harmonic in y, then u is sub-harmonic (Ko:lodziej and Thorbiörnson [12]).2. ResultsLet u(x, y) be a separately subharmonic function in the product domainB = B1 × B2, which is harmonic in y. We will assume that u satisfies– 184 –A. Sadullaevthis condition in a slightly larger domain B̃ = B̃1 × B̃2 such that B̃ ⊃B. Then u(x, y) is subharmonic in a domain(B̃1 × B̃2)\ (S1 × S2), whereS1 ⊂ B̃1, S2 ⊂ B̃2 are closed, nowhere dense sets. Moreover, for every fixedy ∈ B̃2 the Laplacian ∆xu (x, y) defines a positive distribution as followsF (ϕ) =∫u (x, y) ∆xϕ (x) dx ϕ ∈ C∞0 ,thus for every test function ϕ (x) ∈ C∞0 (B1), suppϕ ⊂⊂ B1 , ϕ  0 wehave F (ϕ)  0. Hence, ∆xu (x, y) is a Borel measure, depending on theparameter y.Theorem 2.1. — For every test-function ϕ (x) ∈ C∞0 (B1) F (ϕ) is har-monic in y for y ∈ B2\S2. Moreover, if suppϕ⋂S1 = ∅. then F (ϕ) isharmonic in y for all y ∈ B2.We say that the measure ∆xu (x, y) has the harmonic property withrespect to y in the domain G=(B1 ×B2) \ (S1 × S2).Proof. — The result 5) above states that u (x, y) is subharmonic andtherefore u is locally bounded above in G=(B1 ×B2) \ (S1 × S2). Hencethe integralF (ϕ) (y) =∫B1ϕ (x) ∆xu (x, y) =∫B1u (x, y) ∆xϕ (x)is harmonic in B2\S2. If suppϕ⋂S1 = ∅, then this integral is harmonic inall B2. Corollary 2.2 The measure FE (y) =∫E∆xu (x, y) is harmonic inB2 for any E ⊂⊂ B1\S1.Corollary 2.3. — The total measure ‖∆xu (x, y)‖B1 =∫B1∆xu (x, y)is finite (=∞) for every fixed y ∈ B2 and is harmonic function in B2\S2.Theorem 2.4. — The function FB1\S1 (y) =∫B1\S1 ∆xu (x, y) is boundedand positive harmonic in B2.Proof. — Let us take an increasing sequence of compacts Ej ⊂ Ej+1 ⊂⊂B1\S1 such that⋃jEj = B1\S1. Then the functions FEj (y) =∫Ej∆xu (x, y)– 185 –On separately subharmonic functions (Lelong’s problem)are harmonic in B2 and form an increasing sequence in j. By Harnacks’ the-orem either FEj (y)↗ +∞ or (FEj )j converges to a harmonic function. Thefirst possibility is ruled out, because Corollary 2.3 provides a bound on theFEj (y) for every y ∈ B2\S2.Thus limj→∞FEj (y) =∫B1\S1 ∆xu (x, y) is harmonic in B2, which com-pletes the proof. Now we consider the potentialU (x, y) =∫B1\S1K (x− w) ∆wu (w, y) ,where K is the Newtonian kernel,K(w) =12πln |w| , if n = 2− 1(n− 2)σn |w|n−2, if n > 2.The measure ∆xu (x, y) has the harmonic property in (B1\S1) × B2.Moreover, for some constant C the total measure∫B1\S1 ∆xu (x, y)  C, y ∈B2. It follows that the integral∫B1\S1 ϕ (w) ∆wu (w, y) is harmonic in y forevery continuous function ϕ ∈ C(B̄1). Let Kj(w) ∈ C∞(Rn) approximateK from above, Kj (w) ↓ K (w). Then for every fixed x ∈ B1 we have∫B1\S1Kj (x− w) ∆wu (w, y) ↓∫B1\S1K (x− w) ∆wu (w, y)for j →∞, hence U (x, y) is harmonic in y for fixed x ∈ B1. Moreover, U issubharmonic in x and bounded above in B1×B2. It follows by the theoremof Lelong and Avanissian (1), that U is subharmonic in B1 ×B2.Now we take the difference u∗ (x, y) = u (x, y) − U (x, y) . The functionu∗ (x, y) is separately subharmonic and is harmonic in y. Moreover, u∗ (x, y)is harmonic in x outside S1. Thus we haveTheorem 2.5. — Every separately subharmonic function, which is har-monic in y, can locally be represented as a sum of two functions:u (x, y) = u∗ (x, y) + U (x, y) ,where U is a subharmonic function and u∗ is separately subharmonic andharmonic in y, such that the associated measure ∆xu∗ (x, y) is supported onS1 for every fixed y ∈ B2.– 186 –A. SadullaevProblem 2.6. — We finish our discussion by recalling an open prob-lem on the definition of plurisubharmonic functions: in this definition onedemands two conditions.a. The function u (z) is upper semicontinuous;b. For each complex line l the restriction u |l is subharmonic.The above results on separately subharmonic functions seem to indicate,that the condition a. may be implied by b. But this is still open.Bibliography[1] Lelong (P.). — Les fonctions plurisousharmoniques, Ann. Sci. Ecol. Norm. Sup.62, p. 301-328 (1945).[2] Lelong (P.). — Fonctions plurisousharmoniques et fonctions analytiques de vari-ables réelles, Ann. Inst. Fourier 11, p. 515-562 (1961).[3] Wiegerinck (J.). — Separately subharmonic functions need not be subharmonic,Proc. Amer. Math. Soc. 104, p. 770-771 (1988).[4] Wiegerinck (J.) and Zeinstra (R.). — Separately subharmonic functions: whenare they subharmonic, Proc. Sympos. Pure Math. 52:1, p. 245-249 (1991).[5] Avanissian (V.). — Fonctions plurisousharmoniques et fonctions doublementsousharmoniques, Ann. Sci. École Norm. Sup. 78, p. 101-161 (1961).[6] Arsove (M.G.). — On subharmonicity of doubly subharmonic functions, Proc.Amer. Math. Soc. 17, p. 622-626 (1966).[7] Riihentaus (J.). — On a theorem of Avanissian-Arsove, Exposition. Math. 7,p. 69-72 (1989).[8] Riihentaus (J.). — Subharmonic functions, generalizations and separately sub-harmonic functions, arXiv:math / 0610259v5 (2008).[9] Armitage (D.M.) and Gardiner (S.J.). — Conditions for separately subharmonicfunctions to be subharmonic, Potential Anal. 2, p. 225-261 (1993).[10] Cegrell (U.) and Sadullaev (A.). — Uzbek Math. J. 1, p. 78-83 (1993).[11] Imomkulov. — Separately subharmonic functions, S.A. Dokl. UzSSR. 21, p. 8-10(1990).[12] Ko(lodziej (S.) and Thorbiörnson (J.). — Separately harmonic and subharmonicfunctions, Potential Anal. 5, p. 463-466 (1996).– 187 –

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On separately subharmonic functions (Lelong’s problem)

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