DIRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES Alessandro Perotti Abstract. In this paper we consider the Dirichlet problem for real pluriharmonic functions on a smooth, bounded domain in Cn. We start from a result announced by Fichera some years ago, and relate it to the properties of the normal component on @D of the @-operator. We prove, for some classes of domains, that an orthogonality condition, in the L2(@D)-norm, with respect to a suitably chosen space of functions is necessary and sufficient for pluriharmonicity of the harmonic extension of the boundary value. 1. Introduction Let D be a smoothly bounded domain in Cn. In this paper we investigate the Dirichlet problem for real valued pluriharmonic functions on D. Given a real, con- tinuous function u on @D, we examine necessary and sufficient conditions for the pluriharmonicity of the harmonic extension U in D of the boundary value u. There are two different approaches to this problem. The local approach, in which tan- gential differential conditions on u are searched, has been pursued by many authors on domains whose Levi form has at least one positive eigenvalue at every boundary point (see for example [R] 18.3 and [F3] and the references given there). The global approach consists in givingx integral conditions, which impose orthogonality of u, in the L2(@D)-norm, with respect to a suitably chosen space of functions. When D is the unit ball of Cn, a solution to the Dirichlet problem for pluriharmonic functions is contained in the results of Nagel and Rudin [NR] (see also 13 in the book by Rudin [R]). Other results on this line have been given by Dzhuraevx [D]. Our starting point is the following result announced by Fichera in the papers [F1],[F2],[F3]. Let D be a simply connected domain, with boundary of class C1+λ, λ > 0. Let ν be the outer unit normal to @D and τ = iν. Let A, B be a pair of real C1(D) functions, harmonic on D, such that @A @B + = 0 @τ @ν 1991 Mathematics Subject Classification. Primary 32F05; Secondary 35N15, 31C10, 32D15. Key words and phrases. Pluriharmonic functions, Dirichlet problem. The author is a member of the G.N.S.A.G.A. of C.N.R. Typeset by -TEX AMS 1 2 ALESSANDRO PEROTTI on @D. Given u C0(@D), there exists one (and only one) pluriharmonic function 0 2 U C (D) such that U @D = u if and only if 2 j @A @B u dσ = 0 @ν − @τ Z@D for any pair A; B. In this paper, after rewriting the above integral condition in terms of the complex normal component of the @-operator, we show (Theorem 1) that this condition is necessary for pluriharmonicity when @D is of class C1 and u Cα(@D), α > 0, or when @D is of class C2 and u C0(@D). We then prove (Theorem2 2) sufficiency of the condition in the case when2 u C1+λ, λ > 0. If the boundary value u is only continuous, we are able to get the2 result on strongly pseudoconvex domains and on weakly pseudoconvex domains with real analytic boundary (Theorem 3). We give then in 4 a characterization of the pairs A, B appearing in the orthogonality condition in termsx of the Bochner-Martinelli integral operator. In the case of the ball, this result can be further precised and allows to obtain another proof of Nagel-Rudin theorem. 2. Notations and preliminaries 2.1. Let D = z Cn : ρ(z) < 0 be a bounded domain in Cn with boundary of class Cm; m 1.f We2 assume ρ Cgm on Cn and dρ = 0 on @D. ≥ 2 6 We denote by P h(D), A(D) and Harm(D) respectively the spaces of real plurihar- monic, holomorphic and complex harmonic functions on D. For every α; 0 α m, P hα(D) is the space P h(D) Cα(D) and similarly for Aα(D) and Harmα≤(D).≤ We α \ denote by P h@D(D) the space of restrictions to @D of pluriharmonic functions in α α α P h (D) and by ReA@D(D) the restrictions of real parts of functions in A (D). Fi- nally, ReAα(D) is the space of real parts of element of Aα(D). Remark. If D is a simply connected domain or, more precisely, H1(D; R) = 0, then P hα(D) = ReAα(D) for every α 1, since Cauchy-Riemann equations imply that ≥ the harmonic coniugate of a function U P hα(D) belongs to Cα(D). In general, this fact is no longer true when α = 0, but2 it remains valid for 0 < α < 1. In fact, the first derivatives of U P hα(D) satisfy a Hardy-Littlewood estimate (see [L] 2 for example) and then the2 same holds for the harmonic coniugate. x 2.2. For every F C1(D), in a neighbourhood of @D we have the decomposition of 2 @F in the tangential and the normal parts @ρ @F = @bF + @nF @ρ j j 1=2 @ρ 2 @F @ρ 1 where @ρ = and @ F = . j j k n k @ζ @ζk ! @ζk k @ρ P j j P DIRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS 3 If ν denotes the outer unit normal to @D and τ = iν, then we can also write @nF = 1 @F @F + i . The normal part of @F on @D can also be expressed by means of 2 @ν @τ the Hodge -operator and the Lebesgue surface measure dσ: @nF dσ = @F (see ∗ ∗ j@D [K] 3.3 and 14.2). Note that @-Neumann problem for functions xx F = ψ in D, @nF = 0 on @D is equivalent, at least in the smooth case, to the problem @nF = φ on @D (see [K] 14). The compatibility condition for this problem becomes φfdσ = 0 for x @D every f holomorphic in a neighbourhood of D. R 2.3. Now assume that A and B are real harmonic functions on D, of class C1 on D. @A @B Let H = A + iB. Then @ H is real on @D if and only if + = 0 on @D. If this n @τ @ν 1 @A @B is the case, @ H = on @D and Fichera’s condition on u C0(@D) n 2 @ν − @τ 2 becomes the following condition (?) u @nHdσ = 0 Z@D for every complex harmonic function H C1(D) such that @ H is real on @D. 2 n Note that the integral above can be rewritten as u @H, with @H a closed @D ∗ ∗ (n 1; n)-form on D, since @( @H) = @∗@H = H = 2∆H = 0. − ∗1 ∗ − − R 1 We shall denote by Harm0(D) the real subspace of Harm (D) Harm1(D) = H Harm1(D): @ H is real on @D 0 f 2 n g Remark. When n = 1 and H1(D; R) = 0 condition (?) is void: if H = A + iB is as 1 above, let B0 be a harmonic coniugate of A. Then B0 C (D) from Cauchy-Riemann 2 @A @B equations and F = A + iB is holomorphic on D. This implies that + 0 = 0 0 @τ @ν @B @B0 on @D and so = on @D. Then B B0 is a constant and so @ H = @ F = 0 @ν @ν − n n on @D. 3. Condition (?) and pluriharmonic functions 3.1. In this section we examine the validity of condition (?) for boundary values of pluriharmonic functions. 1 1 α Theorem 1. (a) Let H (D; R) = 0 and @D of class C . If u P h@D(D), α > 0, then u satisfies condition (?); 2 1 2 0 (b) Let H (D; R) = 0 and @D of class C . If u P h@D(D), then u satisfies condition (?). 2 4 ALESSANDRO PEROTTI Proof. (a) Let U P hα(D) be the extension of u and F = U + iV A0(D). Then 2 2 F @ Hdσ = F @H = @(F @H) = F@( @H) = 0 n ∗ ∗ ∗ Z@D Z@D ZD ZD The real part of the first integral is @D u@nHdσ. (b) Let U P h0(D) be the extension of u. Then U = Re F , with F = U + iV R holomorphic on2 D. From Stout’s estimate (see [S]) it follows that F belongs to the Hardy space 2(D). Then F has boundary value f L2(@D). Let D = z : ρ(z) < H 2 f for small < 0. We get as before that the integral F @nHdσ vanishes. Then g @D U Re(@nH)dσ+ V Im(@nH)dσ = 0. The first integral tends to u@nHdσ @D @D R @D as tends to 0, while the second has limit V Im(@ H)dσ = 0. R R @D n R Remarks. (i) The proof of (a) shows thatR condition (?) is satisfied by any u 0 2 ReA@D(D) even without the topological restriction on D; 0 0 (ii) If ReA@D(D) is dense in P h@D(D) then condition (?) is satisfied by every u P h0 (D). This happens for example when D is a n-circular domain; 2 @D (iii) If u L2(@D) extends to a pluriharmonic function on D, then the proof of (b) shows that2 (?) still holds. 3.2. Now we show that if the boundary value is sufficiently regular, condition (?) implies pluriharmonicity without any topological assumption on D. Theorem 2. Let @D be of class C1+λ, λ > 0. If u C1+λ(@D) is a real function 2 which satisfies condition (?), then u ReA1+λ(D). In particular, u P h1+λ(D).