DIRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES. Alessandro Perotti

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1 IRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES Alessandro Perotti Abstract. In this paper we consider the irichlet problem for real pluriharmonic functions on a smooth, bounded domain in C n. We start from a result announced by Fichera some years ago, and relate it to the properties of the normal component on of the -operator. We prove, for some classes of domains, that an orthogonality condition, in the L 2 ()-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 be a smoothly bounded domain in C n. In this paper we investigate the irichlet problem for real valued pluriharmonic functions on. Given a real, continuous function u on, we examine necessary and sufficient conditions for the pluriharmonicity of the harmonic extension U in 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 L 2 ()-norm, with respect to a suitably chosen space of functions. When is the unit ball of C n, a solution to the irichlet problem for pluriharmonic functions is contained in the results of Nagel and Rudin [NR] (see also 13 in the boo by Rudin [R]). Other results on this line have been given by zhuraev []. Our starting point is the following result announced by Fichera in the papers [F1],[F2],[F3]. Let be a simply connected domain, with boundary of class C 1+λ, λ > 0. Let ν be the outer unit normal to and τ = iν. Let A, B be a pair of real C 1 () functions, harmonic on, such that A τ + B ν = Mathematics Subject Classification. Primary 32F05; Secondary 35N15, 31C10, Key words and phrases. Pluriharmonic functions, irichlet problem. The author is a member of the G.N.S.A.G.A. of C.N.R. Typeset by AMS-TEX 1

2 2 ALESSANRO PEROTTI on. Given u C 0 (), there exists one (and only one) pluriharmonic function U C 0 () such that U = u if and only if ( A u ν B ) dσ = 0 τ 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 is of class C 1 and u C α (), α > 0, or when is of class C 2 and u C 0 (). We then prove (Theorem 2) sufficiency of the condition in the case when u C 1+λ, λ > 0. If the boundary value u is only continuous, we are able to get the result on strongly pseudoconvex domains and on wealy 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 = {z C n : ρ(z) < 0} be a bounded domain in C n with boundary of class C m, m 1. We assume ρ C m on C n and dρ = 0 on. We denote by P h(), A() and Harm() respectively the spaces of real pluriharmonic, holomorphic and complex harmonic functions on. For every α, 0 α m, P h α () is the space P h() C α () and similarly for A α () and Harm α (). We denote by P h α () the space of restrictions to of pluriharmonic functions in P h α () and by ReA α () the restrictions of real parts of functions in Aα (). Finally, ReA α () is the space of real parts of element of A α (). Remar. If is a simply connected domain or, more precisely, H 1 (, R) = 0, then P h α () = ReA α () for every α 1, since Cauchy-Riemann equations imply that the harmonic coniugate of a function U P h α () belongs to C α (). 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 h α () satisfy a Hardy-Littlewood estimate (see [L] 2 for example) and then the same holds for the harmonic coniugate For every F C 1 (), in a neighbourhood of we have the decomposition of F in the tangential and the normal parts where ρ = ( ) ρ 2 1/2 ζ F = b F + n F ρ ρ and n F = F ζ ρ ζ 1 ρ.

3 IRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS 3 ( If ν denotes ) the outer unit normal to and τ = iν, then we can also write n F = 1 F 2 ν + i F. The normal part of F on can also be expressed by means of τ the Hodge -operator and the Lebesgue surface measure dσ: n F dσ = F (see [K] 3.3 and 14.2). Note that -Neumann problem for functions F = ψ in, n F = 0 on is equivalent, at least in the smooth case, to the problem n F = φ on (see [K] 14). The compatibility condition for this problem becomes φfdσ = 0 for every f holomorphic in a neighbourhood of Now assume that A and B are real harmonic functions on, of class C 1 on. Let H = A + ib. Then n H is real on if and only if A τ + B = 0 on. If this ν is the case, n H = 1 ( A 2 ν B ) on and Fichera s condition on u C 0 () τ becomes the following condition ( ) u n Hdσ = 0 for every complex harmonic function H C 1 () such that n H is real on. Note that the integral above can be rewritten as u H, with H a closed (n 1, n)-form on, since ( H) = H = H = 2 H = 0. We shall denote by Harm 1 0() the real subspace of Harm 1 () Harm 1 0() = {H Harm 1 () : n H is real on } Remar. When n = 1 and H 1 (, R) = 0 condition ( ) is void: if H = A + ib is as above, let B be a harmonic coniugate of A. Then B C 1 () from Cauchy-Riemann equations and F = A + ib is holomorphic on. This implies that A τ + B ν = 0 on and so B ν = B ν on. Then B B is a constant and so n H = n F = 0 on. 3. Condition ( ) and pluriharmonic functions 3.1. In this section we examine the validity of condition ( ) for boundary values of pluriharmonic functions. Theorem 1. (a) Let H 1 (, R) = 0 and of class C 1. If u P h α (), α > 0, then u satisfies condition ( ); (b) Let H 1 (, R) = 0 and of class C 2. If u P h 0 (), then u satisfies condition ( ).

4 4 ALESSANRO PEROTTI Proof. (a) Let U P h α () be the extension of u and F = U + iv A 0 (). Then F n Hdσ = F H = (F H) = F ( H) = 0 The real part of the first integral is u nhdσ. (b) Let U P h 0 () be the extension of u. Then U = Re F, with F = U + iv holomorphic on. From Stout s estimate (see [S]) it follows that F belongs to the Hardy space H 2 (). Then F has boundary value f L 2 (). Let ɛ = {z : ρ(z) < ɛ} for small ɛ < 0. We get as before that the integral ɛ F n Hdσ vanishes. Then ɛ U Re( n H)dσ+ ɛ V Im( n H)dσ = 0. The first integral tends to u nhdσ as ɛ tends to 0, while the second has limit V Im( nh)dσ = 0. Remars. (i) The proof of (a) shows that condition ( ) is satisfied by any u ReA 0 () even without the topological restriction on ; (ii) If ReA 0 () is dense in P h0 () then condition ( ) is satisfied by every u P h 0 (). This happens for example when is a n-circular domain; (iii) If u L 2 () extends to a pluriharmonic function on, then the proof of (b) shows that ( ) still holds Now we show that if the boundary value is sufficiently regular, condition ( ) implies pluriharmonicity without any topological assumption on. Theorem 2. Let be of class C 1+λ, λ > 0. If u C 1+λ () is a real function which satisfies condition ( ), then u ReA 1+λ (). In particular, u P h1+λ (). Proof. Let U C 1+λ () be the harmonic extension of u. Let V C 1 () be a solution of the (classical) Neumann problem with boundary condition V ν = u τ on. We show that the function F = U + iv is holomorphic on. Since n F is real on, condition ( ) implies that u nf dσ = 0. The integral F F = F F = 2 1 n i n F 2 ζ dζ dζ = 2 F ζ is real. It vanishes because Re (U iv ) nf dσ = u nf dσ. Then F = 0 on. The regularity of F follows from that of U and Cauchy-Riemann equations. Corollary 1. Let be connected, of class C 1+λ, λ > 0. If u C 1+λ (), then condition ( ) holds if and only if u is the real part of a CR function on. Remar. Theorem 2 shows that the condition H 1 (, R) = 0 can not be omitted in Theorem 1 when the boundary value u is of class C 1+λ In the case when is strongly pseudoconvex, condition ( ) implies that the harmonic extension is pluriharmonic even for continuous functions. 2 dv

5 IRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS 5 Theorem 3. Let be a smoothly bounded strongly pseudoconvex domain. If u C 0 () is a real function which satisfies condition ( ), then u P h 0 (). Proof. Let be any simply connected, smoothly bounded domain contained in the interior of. Let U be the harmonic extension of u on and let H Harm 1 ( ) have real normal part n H of H on. Then U n Hdσ = u(η)p (ζ, η) n H(ζ)dσ(η)dσ (ζ) = uφdσ where P (ζ, η) denotes the Poisson ernel for and φ(η) is the C real valued function on given by the integral P (ζ, η) n H(ζ)dσ (ζ). On the -Neumann problem n F = φ can be solved if φ is orthogonal to antiholomorphic functions with respect to integration on. Let f be holomorphic in a neighbourhood of. Then fφdσ = f(η) P (ζ, η) n H(ζ)dσ (ζ)dσ(η) = n H(ζ) f(η)p (ζ, η)dσ(η)dσ (ζ) = f(ζ) n H(ζ)dσ(ζ) = 0 since f is pluriharmonic on a neighbourhood of. Then there exists F Harm() C () such that n F = φ on. It follows that U n Hdσ = u n F dσ = 0 From Theorem 2 we get that U is pluriharmonic on, and so on the whole domain. Remar. The proof shows that the same result holds on any wealy pseudoconvex domain for which -Neumann problem can be solved for C forms (see [K] 18). For example, on wealy psudoconvex domains with real analytic boundary. 4. Characterization of Harm 1 0() Let M denote the Bochner-Martinelli operator on C n, n > 1: M(F )(z) = F (ζ)u(ζ, z), for z / where U(ζ, z) is the Bochner-Martinelli ernel.

6 6 ALESSANRO PEROTTI Proposition 1. Let be of class C 1. Then a function F Harm 1 () is in Harm0() 1 if and only if Im(M(F )) = Im(F ) in. 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 F (ζ)u(ζ, z) + g 0 (ζ, z) n F (ζ)dσ = 0, for z C n \ where g 0 (ζ, z) = (n 2)! 2π ζ z 2 2n. n If n F is real on, then Im(M(F )) = Im(F ) in. Conversely, assume that Im(M(F )) = Im(F ) in. From the formula we get Im( n F (ζ)) dσ = 0 ζ z 2n 2 when z. These integrals vanish also for z /, because from the jump formula for M(F ) on we get that Im(M(F )) = 0 on C n \. A density argument gives Im( n F ) = 0 on. Let N be the real linear operator defined for F Harm 1 () by N(F ) = Re(F ) + i Im(M(F )) Proposition 1 says that Harm0() 1 is the set Fix(N) = {F Harm 1 () : N(F ) = F }. Note that N(F ) is of class C α for every α, 0 < α < 1, and that N can be extended as a real operator from the space L 2 () to the Hardy space of harmonic functions h 2 (). 5. The case of the ball 5.1. Let B be the unit ball of C n and let S = B be the unit sphere. We consider the space L 2 (S) identified with the space of harmonic extensions of L 2 functions from S into B. L 2 (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 L 2 (S), since M(P s,t ) = n + s 1 n + s + t 1 P s,t for any P s,t H(s, t). This spectral decomposition allows to compute the limit of the iterates of the operator N given in 4. Let N 0 be the real linear projection in L 2 (S) defined for P s,t H(s, t) by { s N 0 (P s,t ) = s + t P s,t + t s + t P s,t, for t > 0 P s,t, for t = 0

7 IRICHLET PROBLEM FOR PLURIHARMONIC FUNCTIONS 7 Proposition 2. lim N = N 0 in the strong operator topology of L 2 (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 (F ) = Re(F ) + iλ () Im(F ), where λ (0) = 1, and λ () = αλ ( 1) + β. If α < 1, λ () β tends to 1 α as. Setting F = P s,t, we β get 1 α = s t s+t, from which it follows that N (P s,t ) N 0 (P s,t ) for every s, t Taing ρ(ζ) = ζ 2 1 as a defining function for B, we see that n P s,t = P s,t ζ / ζ = tp s,t / ζ. Then n N 0 (P s,t ) S = st s+t ζ (P s,t + P s,t ) for every s, t. Proposition 3. Let Fix(N 0 ) = {F Harm 1 (B) : N 0 (F ) = F }. Then Fix(N 0 ) = Harm 1 0(B). Proof. If N(F ) = F, then N 0 (F ) = F. Conversely, if F Fix(N 0 ), then n F S = n N 0 (F ) S is real, since n N 0 (P s,t ) S is real for every s, t. Then F Fix(N) = Harm 1 0(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 C 0 (S) has a pluriharmonic extension on B if and only if u is orthogonal to the spaces H(s, t) in L 2 (S) for any s > 0, t > 0. Proof. We show that condition ( ) is equivalent to Nagel-Rudin condition. If u satisfies condition ( ), we can tae H = N 0 (P s,t ) and get that u is orthogonal to Re(P s,t ) for every s > 0, t > 0. Taing ip s,t in place of P s,t, we get orthogonality with respect to Im(P s,t ) for every s > 0, t > 0, and then with respect to the spaces H(s, t). Conversely, orthogonality with respect to n N 0 (P s,t ) for any s, t implies that u is orthogonal to n H S for every H = N 0 (H) Fix(N 0 ). From Proposition 3, this is condition ( ). Theorems 1 and 3 give the result. Remars. (i) If F Harm 1 (B) and N 0 (F ) Harm 0 (B), then the imaginary part of N 0 (F ) is the unique classical solution of the interior Neumann problem with boundary datum Re F τ and such that N 0 (F )(0) = F (0). (ii) The real projection N 0 is the identity on holomorphic functions, it is the coniugation operator on non-constant antiholomorphic functions and has the property Re N 0 (F ) = Re F for every F. Assume that N 0 is a real linear operator with the same properties on a smooth domain. Let N 0 (u) Harm 0 (), u a real function continuous on (as usual, u is identified with its harmonic extension U). Then u ReA 0 () if and only if nn 0 (u) = 0 on. In fact, from the above cited theorem of Aronov and from a theorem of Kytmanov and Aizenberg [KA], n N 0 (u) = 0 on if and only if N 0 (u) is holomorphic on. Anowledgments We would lie to than G. Tomassini for many helpful discussions about the subject. We also wish to acnowledge the hospitality of the Mathematics epartment of the Trento University.

8 8 ALESSANRO 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. Sibirs. Otdel. Aad. Nau SSSR, Krasnoyars, 1973, pp (Russian) [] A. zhuraev, On Riemann-Hilbert boundary problem in several complex variables, J. Complex Variables Theory Appl. 29 (1996), [F1] G. Fichera, Boundary values of analytic functions of several complex variables, Complex Analysis and Applications (Varna, 1981), Bulgar. Acad. Sci., Sofia, 1984, pp [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 (Italian) [F3], Boundary value problems for pluriharmonic functions, Mathematics today (Luxembourg, 1981), Gauthier-Villars, Paris, 1982, pp (Italian) [K] A.M. Kytmanov, The Bochner-Martinelli integral and its applications, Birhäuser Verlag, Basel, Boston, Berlin, [KA] A.M. Kytmanov and L.A. Aizenberg, The holomorphy of continuous functions that are representable by the Bochner-Martinelli integral, Izv. Aad. Nau Armyan. SSR 13 (1978), (Russian) [L] E. Ligoca, The Hölder duality for harmonic functions, Studia Math. 84 (1986), [NR] A. Nagel and W. Rudin, Moebius-invariant function spaces on balls and spheres, ue Math. J. 43 (1976), [R1] A.V. Romanov, Spectral analysis of the Martinelli-Bochner operator for the ball in C n and its applications, Funt. Anal. i Prilozhen 12 (1978), 86 87; English transl. in Functional Anal. Appl. 12 (1978), [R] W. Rudin, Function theory in the unit ball of C n, Springer-Verlag, New Yor, Heidelberg, Berlin, [S] E.L. Stout, H p -functions on strictly pseudoconvex domains, Amer. J. Math. 98 (1976), ipartimento di Matematica Universita di Milano Via Saldini 50 I Milano ITALY address: perotti@mat.unimi.it