Dirichlet Problem for Pluriharmonic Functions of Several Complex Variables

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, continuous 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 tangential 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 giving§ 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 Dzhuraev§ [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.

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1 2 ALESSANDRO PEROTTI

on ∂D. Given u C0(∂D), there exists one (and only one) pluriharmonic function 0 ∈ U C (D) such that U ∂D = u if and only if ∈ | ∂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 (Theorem∈ 2) suﬃciency of the condition in the case when∈ u C1+λ, λ > 0. If the boundary value u is only continuous, we are able to get the∈ 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 terms§ 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.{ We∈ assume ρ C}m on Cn and dρ = 0 on ∂D. ≥ ∈ 6 We denote by P h(D), A(D) and Harm(D) respectively the spaces of real pluriharmonic, 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, but∈ 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 the∈ same holds for the harmonic coniugate. § 2.2. For every F C1(D), in a neighbourhood of ∂D we have the decomposition of ∈ ∂F in the tangential and the normal parts

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 ∗ ∗ |∂D [K] 3.3 and 14.2). Note that ∂-Neumann problem for functions §§

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 § ∂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 ∈ becomes the following condition

(?) u ∂nHdσ = 0 Z∂D for every complex harmonic function H C1(D) such that ∂ H is real on ∂D. ∈ 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 { ∈ n } 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 ∈ ∂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 (?); ∈ 1 2 0 (b) Let H (D, R) = 0 and ∂D of class C . If u P h∂D(D), then u satisﬁes condition (?). ∈ 4 ALESSANDRO PEROTTI

Proof. (a) Let U P hα(D) be the extension of u and F = U + iV A0(D). Then ∈ ∈

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 on∈ 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 ∈ { for small < 0. We get as before that the integral F ∂nHdσ vanishes. Then } ∂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 ∈ 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; ∈ ∂D (iii) If u L2(∂D) extends to a pluriharmonic function on D, then the proof of (b) shows that∈ (?) 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 ∈ which satisfies condition (?), then u ReA1+λ(D). In particular, u P h1+λ(D). ∈ ∂D ∈ ∂D Proof. Let U C1+λ(D) be the harmonic extension of u. Let V C1(D) be a ∈ ∈ ∂V ∂u solution of the (classical) Neumann problem with boundary condition = ∂ν − ∂τ on ∂D. We show that the function F = U + iV is holomorphic on D.

Since ∂nF is real on ∂D, condition (?) implies that ∂D u∂nF dσ = 0. The integral

2 R 2 1 n n ∂F ∂F F ∂F = ∂F ∂F = 2 − i dζ dζ = 2 dv ∗ ∧ ∗ ∂ζ ∧ ∂ζ Z∂D ZD ZD k k ZD k k X X is real. It vanishes because Re ∂D(U iV )∂nF dσ = ∂D u∂nF dσ. Then ∂F = 0 on D. The regularity of F follows from that− of U and Cauchy-Riemann equations. R R Corollary 1. Let ∂D be connected, of class C1+λ, λ > 0. If u C1+λ(∂D), then condition (?) holds if and only if u is the real part of a CR function∈ on ∂D. Remark. Theorem 2 shows that the condition H1(D, R) = 0 can not be omitted in Theorem 1 when the boundary value u is of class C1+λ. 3.3. In the case when D is strongly pseudoconvex, condition (?) implies that the harmonic extension is pluriharmonic even for continuous functions. DIRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS 5

Theorem 3. Let D be a smoothly bounded strongly pseudoconvex domain. If u C0(∂D) is a real function which satisﬁes condition (?), then u P h0 (D). ∈ ∈ ∂D Proof. Let D0 be any simply connected, smoothly bounded domain contained in the 1 interior of D. Let U be the harmonic extension of u on D and let H Harm (D0) ∈ have real normal part ∂n0H of ∂H on ∂D0. Then

U∂n0Hdσ0 = u(η)PD(ζ, η)∂n0H(ζ)dσ(η)dσ0(ζ) = uφdσ Z∂D0 Z∂D0 Z∂D Z∂D where PD(ζ, η) denotes the Poisson kernel for D and φ(η) is the C∞ real valued function on ∂D given by the integral PD(ζ, η)∂n0H(ζ)dσ0(ζ). ∂D0 On D the ∂-Neumann problem ∂RnF = φ can be solved if φ is orthogonal to antiholomorphic functions with respect to integration on ∂D. Let f be holomorphic in a neighbourhood of D. Then

fφdσ = f(η) PD(ζ, η)∂n0H(ζ)dσ0(ζ)dσ(η) Z∂D Z∂D Z∂D0

= ∂n0H(ζ) f(η)PD(ζ, η)dσ(η)dσ0(ζ) Z∂D0 Z∂D = f(ζ)∂n0H(ζ)dσ(ζ) = 0 Z∂D0

since f is pluriharmonic on a neighbourhood of D0. Then there exists F Harm(D) C∞(D) such that ∂nF = φ on ∂D. It follows that ∈ ∩

U∂n0Hdσ0 = u∂nF dσ = 0 Z∂D0 Z∂D

From Theorem 2 we get that U is pluriharmonic on D0, and so on the whole domain D.

Remark. The proof shows that the same result holds on any weakly pseudoconvex domain D for which ∂-Neumann problem can be solved for C∞ forms (see [K] 18). For example, on weakly psudoconvex domains with real analytic boundary. §

1 4. Characterization of Harm0(D) Let M denote the Bochner-Martinelli operator on D Cn, n > 1: ⊂

M(F )(z) = F (ζ)U(ζ, z), for z / ∂D ∈ Z∂D where U(ζ, z) is the Bochner-Martinelli kernel. 6 ALESSANDRO PEROTTI

Proposition 1. Let ∂D be of class C1. Then a function F Harm1(D) is in 1 ∈ Harm0(D) if and only if Im(M(F )) = Im(F ) in D. Proof. We adapt the proof of a theorem of Aronov ([A]) given in [K] (Theorem 14.1). From the Bochner-Martinelli formula for harmonic functions F (z), for z D F (ζ)U(ζ, z) + g0(ζ, z)∂nF (ζ)dσ = ∈ 0, for z Cn D Z∂D Z∂D ∈ \ (n 2)! 2 2n where g (ζ, z) = −n ζ z − . 0 2π | − | If ∂nF is real on ∂D, then Im(M(F )) = Im(F ) in D. Conversely, assume that Im(M(F )) = Im(F ) in D. From the formula we get

Im(∂ F (ζ)) n dσ = 0 ζ z 2n 2 Z∂D | − | − when z D. These integrals vanish also for z / D, because from the jump formula for M(F∈) on ∂D we get that Im(M(F )) = 0 on∈ Cn D. A density argument gives \ Im(∂nF ) = 0 on ∂D. Let N be the real linear operator deﬁned for F Harm1(D) by ∈ N(F ) = Re(F ) + i Im(M(F ))

1 1 Proposition 1 says that Harm0(D) is the set Fix(N) = F Harm (D): N(F ) = F . Note that N(F ) is of class Cα for every α, 0 < α{ <∈1, and that N can be extended} as a real operator from the space L2(∂D) to the Hardy space of harmonic functions h2(D).

5. The case of the ball 5.1. Let B be the unit ball of Cn and let S = ∂B be the unit sphere. We consider the space L2(S) identiﬁed with the space of harmonic extensions of L2 functions from S into B. L2(S) is the direct sum of pairwise orthogonal spaces H(s, t), s 0, t 0, where H(s, t) is the space of harmonic homogeneous polynomials of total≥ degree≥ s in z and total degree t in z (see [R] 12). Romanov [R1] proved that the Bochner- Martinelli operator M is a bounded self-adjoint§ operator in the space L2(S), since n + s 1 M(P ) = − P s,t n + s + t 1 s,t − for any Ps,t H(s, t). This spectral∈ decomposition allows to compute the limit of the iterates of the 2 operator N given in 4. Let N0 be the real linear projection in L (S) deﬁned for P H(s, t) by § s,t ∈ s t Ps,t + Ps,t, for t > 0 N0(Ps,t) = s + t s + t ( Ps,t, for t = 0 DIRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS 7

k 2 Proposition 2. limk N = N0 in the strong operator topology of L (S). →∞ Proof. Assume that M(F ) = λF , M(F ) = µF . Let α = (λ + µ)/2, β = (λ µ)/2. − An easy computation shows that N k(F ) = Re(F ) + iλ(k) Im(F ), where λ(0) = 1, and (k) (k 1) (k) β λ = αλ − + β. If α < 1, λ tends to 1 α as k . Setting F = Ps,t, we β s t | | k − → ∞ get 1 α = s−+t , from which it follows that N (Ps,t) N0(Ps,t) for every s, t. − → 5.2. Taking ρ(ζ) = ζ 2 1 as a deﬁning function for B, we see that ∂ P = | | − n s,t ∂Ps,t st k ζk/ ζ = tPs,t/ ζ . Then ∂nN0(Ps,t) S = (Ps,t + Ps,t) for every s, t. | s+t ∂ζk | | | | P 1 Proposition 3. Let Fix(N0) = F Harm (B): N0(F ) = F . Then Fix(N0) = 1 { ∈ } Harm0(B).

Proof. If N(F ) = F , then N0(F ) = F . Conversely, if F Fix(N0), then ∂nF S = ∈ | ∂nN0(F ) S is real, since ∂nN0(Ps,t) S is real for every s, t. Then F Fix(N) = 1 | | ∈ Harm0(B). This result allows to obtain a proof of a theorem of Nagel and Rudin (see [NR] or [R] 13): § Corollary 2. A real function u C0(S) has a pluriharmonic extension on B if and only if u is orthogonal to the spaces∈ H(s, t) in L2(S) for any s > 0, t > 0. Proof. We show that condition (?) is equivalent to Nagel-Rudin condition. If u satisﬁes condition (?), we can take H = N0(Ps,t) and get that u is orthogonal to Re(Ps,t) for every s > 0, t > 0. Taking iPs,t in place of Ps,t, we get orthogonality with respect to Im(Ps,t) for every s > 0, t > 0, and then with respect to the spaces H(s, t). Conversely, orthogonality with respect to ∂nN0(Ps,t) for any s, t implies that u is orthogonal to ∂nH S for every H = N0(H) Fix(N0). From Proposition 3, this is condition (?). Theorems| 1 and 3 give the result.∈ Remarks. (i) If F Harm1(B) and N (F ) Harm0(B), then the imaginary part of ∈ 0 ∈ N0(F ) is the unique classical solution of the interior Neumann problem with boundary datum ∂ Re F and such that N (F )(0) = F (0). − ∂τ 0 (ii) The real projection N0 is the identity on holomorphic functions, it is the coniugation operator on non-constant antiholomorphic functions and has the property Re N0(F ) = Re F for every F . Assume that N0 is a real linear operator with the 0 same properties on a smooth domain D. Let N0(u) Harm (D), u a real function continuous on ∂D (as usual, u is identiﬁed with its harmonic∈ extension U). Then u 0 ∈ ReA∂D(D) if and only if ∂nN0(u) = 0 on ∂D. In fact, from the above cited theorem of Aronov and from a theorem of Kytmanov and Aizenberg [KA], ∂nN0(u) = 0 on ∂D if and only if N0(u) is holomorphic on D.

Aknowledgments We would like to thank G. Tomassini for many helpful discussions about the subject. We also wish to acknowledge the hospitality of the Mathematics Department of the Trento University. 8 ALESSANDRO PEROTTI

References [A] A.M. Aronov, Functions that can be represented by a Bochner-Martinelli integral, Properties of holomorphic functions of several complex variables, Inst. Fiz. Sibirsk. Otdel. Akad. Nauk SSSR, Krasnoyarsk, 1973, pp. 35–39. (Russian) [D] A. Dzhuraev, On Riemann-Hilbert boundary problem in several complex variables, J. Complex Variables Theory Appl. 29 (1996), 287–303. [F1] G. Fichera, Boundary values of analytic functions of several complex variables, Complex Analysis and Applications (Varna, 1981), Bulgar. Acad. Sci., Sofia, 1984, pp. 167–177. [F2] , Boundary problems for pluriharmonic functions, Proceedings of the Conference held in Honor of the 80th Anniversary of the Birth of Renato Calapso (Messina-Taormina, 1981), Veschi, Rome, 1981, pp. 127–152. (Italian) [F3] , Boundary value problems for pluriharmonic functions, Mathematics today (Luxem- bourg, 1981), Gauthier-Villars, Paris, 1982, pp. 139–151. (Italian) [K] A.M. Kytmanov, The Bochner-Martinelli integral and its applications, BirkhäuserVerlag, Basel, Boston, Berlin, 1995. [KA] A.M. Kytmanov and L.A. Aizenberg, The holomorphy of continuous functions that are rep- resentable by the Bochner-Martinelli integral, Izv. Akad. Nauk Armyan. SSR 13 (1978), 158– 169. (Russian) [L] E. Ligocka, The Hölderduality for harmonic functions, Studia Math. 84 (1986), 269–277. [NR] A. Nagel and W. Rudin, Moebius-invariant function spaces on balls and spheres, Duke Math. J. 43 (1976), 841–865. [R1] A.V. Romanov, Spectral analysis of the Martinelli-Bochner operator for the ball in Cn and its applications, Funkt. Anal. i Prilozhen 12 (1978), 86–87; English transl. in Functional Anal. Appl. 12 (1978), 232–234. [R] W. Rudin, Function theory in the unit ball of Cn, Springer-Verlag, New York, Heidelberg, Berlin, 1980. [S] E.L. Stout, Hp-functions on strictly pseudoconvex domains, Amer. J. Math. 98 (1976), 821– 852.

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