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Tangential Form of the Trace Condition for Pluriharmonic Functions in Cn

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Tangential Form of the Trace Condition for Pluriharmonic Functions in Cn Tangential form of the trace condition for pluriharmonic functions in Cn Alessandro Perotti¤ Dipartimento di Matematica, Universit`adegli Studi di Trento, Via Sommarive 14, 38050 Povo-Trento, Italy [email protected] May 13, 2003 Keywords: Pluriharmonic functions, quaternionic regular functions. 2000 Mathematics Subject Classi¯cation: Primary 32U05; Secondary 31C10, 32D15, 30G35. Abstract In this paper we introduce a tangential form of the trace condition for pluriharmonic functions on a smooth bounded domain in Cn that has some convexity. The condition obtained, in the bidimensional case, is related to the complex components of quaternionic regular functions. In the case of the unit ball of Cn, we rewrite the trace condition in terms of spherical harmonics. 1 Introduction Let D be a smooth bounded domain in Cn. Pluriharmonic functions on D with some regularity on D are characterized by a trace condition introduced by Fichera in the papers [4],[5],[6] and investigated in [11], [12] and [1]. This condition has a global character: a real function on @D is the trace of a pluriharmonic function on D if 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 di®erential conditions (see for example [13]x18.3, [2] and [6] and the references given there). This second approach requires some geometrical conditions on the boundary @D, essentially the non-vanishing of the Levi-form at every point of @D. In this paper we search for a characterization in tangential terms of the space of functions orthogonal to pluriharmonic functions. We show that this interpretation is possible on domains having some global convex- ity. In the bidimensional case, the convexity conditions can be weakened. ¤Partially supported by MIUR (Project “Propriet`ageometriche delle variet`areali e com- plesse") and GNSAGA of INdAM. 1 Trace condition for pluriharmonic functions 2 Moreover, in C2 we obtain an interesting relation between the two char- acterizations of the orthogonal space and the complex components of a quaternionic regular function on D. The main results of this paper are the following: there exist tangential Cauchy-Riemann operators Ljk and a real, tangential, ¯rst-order di®er- ential operator T on @D, which allow to rewrite the trace condition in the form described by Theorem 1 and Theorem 2. 1 1 ® Theorem 1. (a) Let H (D; R) = 0 and @D of class C . If u 2 P h@D(D), ® > 0, then u satis¯es the condition Z u (L]!)dσ = 0 (]) @D 1 for every form ! 2 Cn¡2;n(D) such that T (!) = 0 on @D. P j+k¡1 Here L]! denotes the function j<k(¡1) Ljk(Kjk), where Kjk are the coe±cients of the form !. 1 2 0 (b) Let H (D; R) = 0 and @D of class C . If u 2 P h@D(D), then u satis¯es (]). 1 0 The same result holds if ! 2 Cn¡2;n(D) and @! 2 Cn¡1;n(D). Theorem 2. Let D be a smooth bounded pseudoconvex domain or a do- main that satis¯es the HÄormandercondition Z(n ¡ 1), i.e., at every point ³ 2 @D the Levi form of @D has at least 1 positive eigenvalue. If u 2 C0(@D) is a real function that satis¯es condition (]), then 0 u 2 P h@D(D). We refer to xx2 and 3 for notations and for the exact formulation of the results. In x4 we consider in more detail the situation in C2, where the assumption on the domain can be relaxed. In x5 we write the trace condition for the unit sphere of Cn in terms of spherical harmonics. 2 Notations and preliminaries 2.1 Let D = fz 2 Cn : ½(z) < 0g be a bounded domain in Cn with boundary of class Cm; m ¸ 1. We assume ½ 2 Cm on Cn and d½ 6= 0 on @D. For every ®; 0 · ® · m, we denote by P h®(D) the space of real pluri- harmonic functions of class C®(D) and similarly for holomorphic func- tions A®(D) and complex harmonic functions Harm®(D). We denote by ® P h@D(D) the space of restrictions to @D of pluriharmonic functions in P h®(D) and by ReA®(D) the space of real parts of element of A®(D). 2.2 Let º denotes the outerµ unit normal¶ to @D and ¿ = iº. For every F 2 1 @F @F C1(D), we set @ F = + i (see [7]xx3.3 and 14.2). n 2 @º @¿ Then in a neighbourhood of @D we have the decomposition of @F in the tangential and the normal parts @½ @F = @bF + @nF : j@½j Trace condition for pluriharmonic functions 3 The normal part of @F on @D can also be expressed in the form @nF = P @F @½ 1 k or by means of the Hodge ¤-operator and the Lebesgue @³¹k @³k j@½j surface measure dσ: @nF dσ = ¤@F j@D . 2.3 1 1 We shall denote by Harm0(D) the real subspace of Harm (D) 1 1 Harm0(D) = fH 2 Harm (D): @nH is real on @Dg: The space of C1(D)-holomorphic functions on D is the maximal com- 1 1 plex subspace in Harm (D) contained in Harm0(D). This follows from a theorem of Kytmanov and Aizenberg [8] (cf. also [7]xx14 and 15): a 1 C (D)-harmonic function F is holomorphic on D if and only if @nF = 0 on @D. We recall the integral orthogonality condition that characterizes the traces on @D of pluriharmonic functions on D. It was introduced by Fichera in [4] (see also [5],[6]) in a form not involving explicitly the normal component of @. It was investigated in [11] and in [1]: Z U @nHdσ = 0 (?) @D 1 for every H 2 Harm0(D). It was shown in [11] that this condition is necessary for pluriharmonic- ity when @D is of class C1 and U 2 C®(D), ® > 0, or when @D is of class C2 and U 2 C0(D). The trace condition is su±cient in the case when 1+¸ U 2 C , ¸ > 0. If the boundary value u = Uj@D is only continuous, the same result holds on bounded strongly pseudoconvex domains and on bounded weakly pseudoconvex domains with real analytic boundaries. Recently, A. Cialdea improved these results in [1], using some facts from potential theory. He gave a more general theorem allowing data to be in Lp(@D). Note that the proof of this result given in [1] (cf. x23) shows 1 that in condition (?) it is su±cient to consider functions H 2 Harm0(D)\ C1(D) to obtain the pluriharmonicity of the harmonic extension U of u 2 Lp(@D). 1 1 1 We shall denote by Harm0 (D) the space Harm0(D) \ C (D). Remark 1. When n = 1 and H1(D; R) = 0 condition (?) is void, since H 1 belongs to Harm0(D) if and only if H is holomorphic on D (cf. [11]). 2.4 1 The space Harm0(D) can be characterized in terms of Bochner-Martinelli 1 operator M. In [11]x4 it was shown that F 2 Harm0(D) if and only if Im M(F ) = Im F in D. Let B be the unit ball of Cn and 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 s in z and total degree t in z (see [13] x12). These spaces are the eigenspaces of the Bochner-Martinelli operator. Trace condition for pluriharmonic functions 4 As it was shown in [11]x5, in this case the trace condition for pluri- harmonic functions reduces to the orthogonality to the spaces H(s; t), s; t > 0, a theorem proved by Nagel and Rudin in [10]. Other results on this line have been given by Dzhuraev [3]. 2.5 We recall the de¯nition of tangential Cauchy-Riemann operators (see for example [13] x18). A linear ¯rst-order di®erential operator L is tangential to @D if (L½)(³) = 0 for each point ³ 2 @D. A tangential operator of the form n X @ L = aj @z¹j j=1 is called a tangential Cauchy-Riemann operator. The operators µ ¶ 1 @½ @ @½ @ Ljk = ¡ , 1 · j < k · n; j@½j @z¹k @z¹j @z¹j @z¹k are tangential and the corresponding vectors at ³ 2 @D span (not inde- pendently when n > 2) the (conjugate) complex tangent space to @D at 1 ³. Then a function f 2 C (@D) is a CR function if and only if Ljk(f) = 0 on @D for every j; k, or, equivalently, if L(f) = 0 for each tangential Cauchy-Riemann operator L. 3 The trace condition in tangential form 3.1 Let n > 1 and let H be an harmonic function on D. The form ¤@H is ¤ a @-closed (n ¡ 1; n)-form on D, since ¤@(¤@H) = ¡@ @H = ¡¤H = ¤ 2¢H = 0, where @ is the formal L2(D)-adjoint operator to @. Pn k¡1 @H Let cn be the constant such that ¤@H = cn k=1(¡1) dz[k] ^ @z¹k 1 dz¹.P Let ! be a (n ¡ 2; n)-form with C (D)- coe±cients. We set ! = cn j<k Kjkdz[j; k] ^ dz¹, where dz[j; k] denotes the product of the forms dz1; : : : ; dzn with dzj and dzk deleted, and dz¹ = dz¹1 ^ ¢ ¢ ¢ ^ dz¹n. An easy computation shows that Xn Xn i¡1 @Kih @! = cn (¡1) dz[h] ^ dz:¹ @zi h=1 i=1 Here we extend by antisymmetry: Kih = ¡Khi when i > h, Kii = 0.
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