Some applications of the trace condition for pluriharmonic functions in Cn

Publicacions Matem`atiques, Vol. 44 (2000), 449–456 SOME APPLICATIONS OF THE TRACE CONDITION FOR PLURIHARMONIC FUNCTIONS IN Cn Alessandro Perotti Abstract In this paper we investigate some applications of the trace condition for pluriharmonic functions on a smooth, bounded domain in Cn. This condition, related to the normal component on ∂D of the ∂-operator, permits us to study the Neumann problem for pluriharmonic functions and the ∂-problem for (0,1)-forms on D with solutions having assigned real part on the boundary. 1. Introduction Let Dbe a smoothly bounded domain in Cn. We are interested in some results that can be obtained from the trace condition for pluriharmonic functions introduced by Fichera in the papers [F1], [F2], [F3] and investigated in [P]. This condition has a global character: a real function on ∂D is the trace of a pluriharmonic function on Dif it is orthogonal, in the L2(∂D)-norm, with respect to a suitably chosen space of functions. This approach is alternative to the local study of tangential differential conditions (see for example [R,§18.3] and [F3] and the references given there). In this paper we first consider the Neumann problem for pluriharmonic functions. Let λ>0. Given a real function φon the boundary, of class Cλ, we show (Theorem 1) that the solutions of the classical Neumann problem ∂U ∂ν =φon ∂D are pluriharmonic on Dif and only if φis orthogonal in L2(∂D)tothe subspace of real harmonic functions that admit a decomposition H1+iH2 1991 Mathematics Subject Classification. Primary 32F05; Secondary 32F20, 35N15, 32D15. Key words. Pluriharmonic functions, Neumann problem, ∂-problem. Partially supported by MURST (Project “Propriet`a geometriche delle variet`a reali e complesse”) and GNSAGA of CNR. 450 A. Perotti with H1,H 2∈Harm1 0(D) (see §§2,3 for the precise definitions). When the domain is the unit ball of Cn, this result can be expressed in terms of spherical harmonics. We thus obtain (Corollary 1) another proof of a theorem given by Dzhuraev in [D2]: Uis pluriharmonic if and only if it satisfies the Gauss compatibility condition Sφdσ = 0 and the trace condition. In §4 we investigate the ∂-problem for (0,1)-forms on Dwith a boundary condition. Given a ∂-closed form f∈C0 0,1(D) and a real Cλfunction gon ∂D, we look for a solution u∈C1(D) of the problem ∂u =fon D, Re u=gon ∂D. We prove (Theorem 2) that a solution exists if and only if ∂D g ∂nHdσ equals the real part of Df∧∗∂H for every H∈Harm1 0(D). For this problem too, we can rewrite the compatibility condition on the unit ball in terms of spherical harmonics (Corollary 2). A similar approach to this problem on the ball of Cnappears in [D1]. 2. Notations and preliminaries 2.1. Let D={z∈Cn:ρ(z)<0}be a bounded domain in Cnwith boundary of class Cm,m≥1. We assume ρ∈Cmon Cnand dρ =0 on ∂D. For every α,0≤α≤m, we denote by Phα(D) the space of real pluriharmonic functions of class Cα(D) and similarly for holomorphic functions Aα(D) and complex harmonic functions Harmα(D). We denote by Phα ∂D(D) the space of restrictions to ∂D of pluriharmonic functions in Phα(D) and by Re Aα(D) the space of real parts of element of Aα(D). 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 ∂F =∂bF+∂nF∂ρ |∂ρ| where ∂nF=k∂F ∂ζk ∂ρ ∂ζk 1 |∂ρ|. If νdenotes the outer unit normal to ∂D and τ=iν, then we can also write ∂nF=1 2∂F ∂ν +i∂F ∂τ . The normal part of ∂F on ∂D can also be expressed by means of the Hodge ∗-operator and the Lebesgue surface measure dσ:∂nFdσ=∗∂F|∂D (see [K,§§3.3 and 14.2]). Some Applications of The Trace Condition 451 2.3. We shall denote by Harm1 0(D) the real subspace of Harm1(D) Harm1 0(D)={H∈Harm1(D):∂nHis real on ∂D}. This space can be characterized in terms of Bochner-Martinelli operator M.In[P,§4] it was shown that F∈Harm1 0(D) if and only if Im M(F)=ImFin D. We recall the integral orthogonality condition that characterizes the traces on ∂D of pluriharmonic functions on D which was introduced (in a different form) by Fichera in [F1] (see also [F2], [F3]) and that was studied in [P]: ∂D U ∂nHdσ=0() for every H∈Harm1 0(D). It was shown in [P] that this condition is necessary for pluriharmonicity when ∂D is of class C1and U∈Cα(D), α>0, or when ∂D is of class C2and U∈C0(D). The trace condition is sufficient in the case when U∈C1+λ,λ>0. If the boundary value uis only continuous, the same result holds on strongly pseudoconvex domains and on weakly pseudoconvex domains with real analytic boundary. Remark. When n= 1 and H1(D,R) = 0 condition () is void, since H belongs to Harm1 0(D) if and only if His holomorphic on D(cf. [P]). In general (n≥1), the space of C1(D)-holomorphic functions on Dis the maximal complex subspace in Harm1(D) that contains Harm1 0(D). This follows from a theorem of Kytmanov and Aizenberg [KA] (cf. also [K,§§14 and 15]): a C1(D)-harmonic function Fis holomorphic on D if and only if ∂nF=0on∂D. 2.4. Let Bbe the unit ball of Cnand let S=∂B. The space 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 sin zand total degree tin z(see [R,§12]). These spaces are the eigenspaces of the Bochner-Martinelli operator. In this case, the trace condition for pluriharmonic functions reduces to the orthogonality to the spaces H(s, t), s, t > 0. This is the content of a theorem of Nagel and Rudin [NR] (see also [P,§5]). 3. The Neumann problem for pluriharmonic functions 3.1. In this section we study the Neumann problem for pluriharmonic functions: 452 A. Perotti Given φon the boundary ∂D, find necessary and sufficient conditions for the existence of a pluriharmonic function Uon Dsuch that ∂U ∂ν =φ on ∂D. We start from the following result announced by Fichera in [F1], [F2], [F3]. Let Dbe a simply connected domain, with boundary of class C1+λ, λ>0. Let A,B,Cbe real functions of class C1(D), harmonic on D, such that ∂A ∂τ +∂B ∂ν =0,∂A ∂ν −∂C ∂τ =0 on ∂D. Given φ∈Cλ(∂D), there exists a pluriharmonic function U∈ C1(D) such that ∂U ∂ν =φon ∂D if and only if ∂D φ(B−C)dσ =0 for any triplet A, B, C. The preceding condition is equivalent to the following ∂D φ(H1+iH2)dσ =0() for every pair H1,H 2∈Harm1 0(D) such that H1+iH2is real. This can be seen setting H1=−C+iA,H2=−A−iB. Then H1,H 2∈Harm1 0(D) and H1+iH2=B−C. Theorem 1. Let H1(D, R)=0and ∂D of class C1+λ.Letφ∈Cλ(∂D) be a real function, with λ>0. Then there exists U∈Ph1(D)such that ∂U ∂ν =φon ∂D if and only if φsatisfies condition (). Proof: We relate the condition () to the trace condition (). Let H1, H2be as before. From the complex version of Green’s formula (see for example [K,§11.3]) we get ∂D U(∂nH1+i∂nH2)dσ =∂D ∂nU(H1+iH2)dσ for every real harmonic function U∈C1(D). Taking real parts, we obtain ∂D U∂nH1dσ =1 2∂D ∂U ∂ν (H1+iH2)dσ. Now assume that φsatisfies condition (). When H1=1,H2=0 () becomes the Gauss compatibility condition for the classical Neumann problem for harmonic functions. Let λbe any positive number smaller than λ. Let U∈Harm1+λ (D) be real, such that ∂U ∂ν =φon ∂D (determined up to an additive constant). We show that Usatisfies the trace condition. If H∈Harm1+λ 0(D), we can set H1=H, Some Applications of The Trace Condition 453 H2=−Im H+iG, where G∈C1(D) is a real harmonic solution of ∂G ∂ν =∂Im H ∂τ on ∂D. Since H1,H 2∈Harm1 0and H1+iH2is real, the integral ∂D φ(H1+iH2)dσ =2∂D U∂nHdσvanishes. From Theorem 2 in [P] we get that Uis the real part of a holomorphic function. Note that in the proof of the cited theorem it is sufficient to consider functions H∈Harm1+λ 0(D). Conversely, if Uis pluriharmonic on D, then Theorem 1 in [P]says that Usatisfies condition () and then ∂D φ(H1+iH2)dσ =0. Remarks. (i) In effect the proof shows that the condition () implies that U∈Re A1+λ(D) even without the topological assumption on D. (ii) If n=1,H1,H 2belong to Harm1 0(D) if and only if they are holomorphic functions on D. Then H1+iH2is a real valued holomorphic function, that is a constant. As is expected, condition () reduces to the Gauss compatibility condition for the Neumann problem. 3.2. If Bis the unit ball of Cnand Sthe unit sphere, the preceeding result can be rewritten in terms of the spaces of harmonic homogeneous polynomials H(s, t). Let N0be the real linear projection in L2(S) introduced in [P]. It is defined for Ps,t ∈H(s, t)by N0(Ps,t)=   s s+tPs,t +t s+tPs,t,for t>0 Ps,t,for t=0 . The space Harm1 0(B) coincides with Fix(N0)={F∈Harm1(B): N0(F)=F}. We show that Theorem 1 in the case of the ball reduces to a result given by Dzhuraev in [D2]. Corollary 1. Let φ∈Cλ(S)be a real function, with λ>0. Then there exists U∈C1(B), pluriharmonic on Band such that ∂U ∂ν =φon Sif and only if Sφdσ =0and φis orthogonal to the spaces H(s, t)in L2(S) for any s, t > 0. Proof: If s, t > 0, we set H1=N0(Ps,t) and H2=N0(−Im H1). Note that for s, t > 0wehaveN0(Re F)=N0(F), N0(Im F)=N0(−iF ). An easy computation gives H1+iH2=4st (s+t)2Re Ps,t. Replacing Ps,t with iPs,t, we get H1+iH2=4st (s+t)2Im Ps,t. Then the condition ()is equivalent to the orthogonality of φto the spaces H(s, t) and Theorem 1 gives the result. 454 A. Perotti 4. ∂-problem with assigned real part on the boundary 4.1. In this section we study the ∂-problem for (0,1)-forms on Dwith a boundary condition. Let f∈C0 0,1(D)bea∂-closed form with continuous coefficients on Dand let gbe a real continuous function on ∂D.Welook for a function u∈C1(D) such that ∂u =fon D, Re u=gon ∂D. The solution, if it exists, is unique up to an imaginary constant. This problem was considered by Dzhuraev in [D1] and [D2] in the case of the unit ball. If H1(D,R) = 0 and there exists a solution wof ∂w =fwhich is continuous on D, the problem can be reduced to the trace condition for pluriharmonic functions, since then it amounts to finding a holomorphic function hon Dsuch that Re(h+w)=gon ∂D.If∂D is of class C2and Dis strongly pseudoconvex, there exists a solution operator Sq:C0,q(D)→C0,q−1(D) such that if ∂f = 0 and fis of class Ck(D), then ∂Sq(f)=fand Sq(f)∈Ck+1/2 0,q−1(D) (see for example [RA] and the references given there). If ∂D is of class C∞and Dis strongly pseudoconvex, another well-known solution operator is given by the Neumann operator Nrelated to .If∂f = 0, then ∂∗N(f) is the unique solution of minimal L2(D)-norm of the equation ∂u =f. Theorem 2. Let Dbe a smoothly bounded strongly pseudoconvex domain. Assume that H1(D,R)=0.Letf∈C0 0,1(D)and let gbe a real Cλfunction on ∂D (λ>0). Then there exists u∈C1(D)such that ∂u =fon D,Re u=gon ∂D if and only if ∂f =0and for every H∈Harm1 0(D) ∂D g ∂nHdσ=ReD f∧∗∂H. Proof: Let α<λbe a positive number with α<1 2. The function g− Re S1(f)∈Cα(∂D) is the trace of a pluriharmonic function if and only if it satisfies the trace condition (). This follows from Theorems 1 and 3in[P]. For H∈Harm1 0(D), we transform the integral on ∂D involving fin an integral on D: ∂D S1(f)∂nHdσ=∂D S1(f)∗∂H =D ∂S1(f)∧∗∂H =D f∧∗∂H. Here we have used the fact that ∗∂H is a closed (n−1,n)-form, since ∗∂(∗∂H)=−∂∗∂H =−H=2∆H=0. Some Applications of The Trace Condition 455 The last integral is the Hodge product (f,∂H)onD. Since ∂nHis real on ∂D, the trace condition becomes 0=∂D (g−Re S1(f)) ∂nHdσ =∂D g ∂nHdσ−Re ∂D S1(f)∂nHdσ =∂D g ∂nHdσ−Re D f∧∗∂H. Note that Phα(D)=ReAα(D), since the first derivatives of U∈Phα(D) satisfy a Hardy-Littlewood estimate (see [L,§2] for example) and then the same holds for the harmonic conjugate. 4.2. If Dis the unit ball, the condition given in the theorem has the following more explicit form. Corollary 2. There exists u∈C1(B)such that ∂u =fon B,Re u=g on Sif and only if ∂f =0and for every s, t > 0and Ps,t ∈H(s, t) Re S gPs,t dσ =ReB n  k=1 fk ∂ ∂zkPs,t t+Ps,t sdv Im S gPs,t dσ =ImB n  k=1 fk ∂ ∂zkPs,t t−Ps,t sdv. Proof: For H=N0(Ps,t) the left integral in Theorem 2 becomes Sg2st s+tRe Ps,t dσ, while the right integrand f∧∗∂N0(Ps,t) is equal to  k fkdzk∧2 s+ti 2n  k (−1)k−1∂ ∂zk (sPs,t +tPs,t)dz[k]∧dz. Since dv =i 2ndz ∧dz, we get the first condition. Replacing Ps,t with iPs,t we get the second one. References [D1] A. Dzhuraev, On Riemann-Hilbert boundary problem in several complex variables, Complex Variables Theory Appl. 29(4) (1996), 287–303. [D2] A. Dzhuraev, On linear boundary value problems in the unit ball of Cn,J. Math. Sci. Univ. Tokyo 3(2) (1996), 271–295. 456 A. Perotti [F1] G. Fichera, Boundary values of analytic functions of several complex variables, in: “Complex analysis and applications ’81 (Varna, 1981)”, Bulgar. Acad. Sci., Sofia, 1984, pp. 167–177. [F2] G. Fichera, Boundary problems for pluriharmonic functions, (Italian), in: “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. [F3] G. Fichera, Boundary value problems for pluriharmonic functions, (Italian), in: “Mathematics today (Luxembourg, 1981)”, Gauthier Villars, Paris, 1982, pp. 139–151. [K] A. M. Kytmanov,“The Bochner-Martinelli integral and its applications”, Birkh¨auser Verlag, Basel, 1995. [KA] A. M. Kytmanov and L. A. Aizenberg, The holomorphy of continuous functions that are representable by the BochnerMartinelli integral, (Russian), Izv. Akad. Nauk Armjan. SSR Ser. Mat. 13(2) (1978), 158–169, 173. [L] E. Ligocka, The H¨older duality for harmonic functions, Studia Math. 84(3) (1986), 269–277. [NR] A. Nagel and W. Rudin, Moebius-invariant function spaces on balls and spheres, Duke Math. J. 43(4) (1976), 841–865. [P] A. Perotti, Dirichlet problem for pluriharmonic functions of several complex variables, Comm. Partial Differential Equations 24(3–4) (1999), 707–717. [RA] R. M. Range,“Holomorphic functions and integral representations in several complex variables”, Graduate Texts in Mathematics 108, Springer-Verlag, New York-Berlin, 1986. [R] W. Rudin,“Function theory in the unit ball of Cn”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science] 241, Springer-Verlag, New York-Berlin, 1980. Dipartimento di Matematica e Applicazioni Universit`a degli Studi di Milano Bicocca Via Bicocca degli Arcimboldi 8 I-20126 Milano Italy E-mail address:[email protected] Primera versi´o rebuda el 15 d’octubre de 1999, darrera versi´o rebuda el 26 de gener de 2000.