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 Classiﬁcation: 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 diﬀerential conditions (see for example [13]§18.3, [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 convexity. 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, first-order differential operator T on ∂D, which allow to rewrite the trace condition in the form described by Theorem 1 and Theorem 2. 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 the condition Z u (L ] ω)dσ = 0 (]) ∂D 1 for every form ω ∈ Cn−2,n(D) such that T (ω) = 0 on ∂D. P j+k−1 Here L ] ω denotes the function j

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 ∈ Cm on Cn and dρ 6= 0 on ∂D. For every α, 0 ≤ α ≤ m, we denote by P hα(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 α P h∂D(D) the space of restrictions to ∂D of pluriharmonic functions in P hα(D) and by ReAα(D) the space of real parts of element of Aα(D).

2.2

Let ν denotes the outerµ unit normal¶ to ∂D and τ = iν. For every F ∈ 1 ∂F ∂F C1(D), we set ∂ F = + i (see [7]§§3.3 and 14.2). n 2 ∂ν ∂τ Then in a neighbourhood of ∂D we have the decomposition of ∂F in the tangential and the normal parts ∂ρ ∂F = ∂bF + ∂nF . |∂ρ| Trace condition for pluriharmonic functions 3

The normal part of ∂F on ∂D can also be expressed in the form ∂nF = P ∂F ∂ρ 1 k or by means of the Hodge ∗-operator and the Lebesgue ∂ζ¯k ∂ζk |∂ρ|

surface measure dσ: ∂nF dσ = ∗∂F |∂D .

2.3 1 1 We shall denote by Harm0(D) the real subspace of Harm (D)

1 1 Harm0(D) = {H ∈ Harm (D): ∂nH is real on ∂D}.

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]§§14 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 ∈ Harm0(D). It was shown in [11] 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). The trace condition is suﬃcient in the case when 1+λ U ∈ C , λ > 0. If the boundary value u = U|∂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. §23) shows 1 that in condition (?) it is suﬃcient to consider functions H ∈ Harm0(D)∩ C∞(D) to obtain the pluriharmonicity of the harmonic extension U of u ∈ Lp(∂D). ∞ 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]§4 it was shown that F ∈ 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] §12). These spaces are the eigenspaces of the Bochner-Martinelli operator. Trace condition for pluriharmonic functions 4

As it was shown in [11]§5, in this case the trace condition for pluriharmonic 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 definition of tangential Cauchy-Riemann operators (see for example [13] §18). A linear first-order differential operator L is tangential to ∂D if (Lρ)(ζ) = 0 for each point ζ ∈ ∂D. A tangential operator of the form n X ∂ L = aj ∂z¯j j=1 is called a tangential Cauchy-Riemann operator. The operators µ ¶ 1 ∂ρ ∂ ∂ρ ∂ Ljk = − , 1 ≤ j < k ≤ n, |∂ρ| ∂z¯k ∂z¯j ∂z¯j ∂z¯k

are tangential and the corresponding vectors at ζ ∈ ∂D span (not inde- pendently when n > 2) the (conjugate) complex tangent space to ∂D at 1 ζ. Then a function f ∈ 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] ∧ ∂z¯k 1 dz¯.P Let ω be a (n − 2, n)-form with C (D)- coeﬃcients. We set ω = cn j

Here we extend by antisymmetry: Kih = −Khi when i > h, Kii = 0. 1 1 Proposition 1. Let H ∈ Harm (D) and the form ω ∈ Cn−2,n(D) be as above. Let L be the tangential Cauchy-Riemann operator deﬁned by L = P j+k j

Then ∂ω|∂D = (L ] ω)dσ on ∂D. Moreover, if ∂ω = ∗∂H on D, then, for every ² < 0 suﬃciently small, the equality

∂nH = L ] ω

holds on ∂D² = {z ∈ D : ρ(z) = ²}.

Proof. On ∂D we have µ ¶ X 1 ∂ρ ∂K ∂ρ ∂K (L ] ω)dσ = (−1)j+k jk − jk dσ ∂zk ∂zj ∂zj ∂zk j

∂ρ (−1)k since dσ equals the restriction of the form −cndz[k] ∧ dz¯ on ∂D ∂zk |∂ρ|

(see for example Lemma 3.5 in [7]). Then (L ] ω)dσ = ∂ω|∂D. If ∂ω = ∗∂H on D, the last assertion of the lemma follows from the ﬁrst part applied on ∂D², since ∂nHdσ = ∗∂H . |∂D²

Remark 2. (i) In the formula defining L ] ω we used the generators Ljk, ∂ρ ∂ρ even if they are not independent for n > 2 (since Ljk + Lki + ∂z¯i ∂z¯j ∂ρ Lij = 0 on ∂D). This choice allows to obtain expressions that are ∂z¯k more symmetric with respect to the coordinates. (ii) If ∗∂H = ∂ω on D, the (n − 2, n)-form ω satisfies the condition ∂∗∂ω = 0 on D, since ∂(∗∂ω) = −∂∂H = 0. When n = 2, the preceding condition is equivalent to ∆ω = 0. If n > 2, ω is harmonic exactly when also the form ∂∂∗ω vanishes. (iii) The equality ∗∂H = ∂ω is equivalent to the following system of differential equations on D:

n ∂H X ∂K = (−1)k (−1)j jk for every k = 1, . . . , n. (1) ∂z¯k ∂zj j=1

3.2 Let T be the real, tangential, first-order differential operator defined on 1 forms ω ∈ Cn−2,n(D) by X −1 j+k T (ω) = Im(L ] ω) = (2i) (−1) (Ljk(Kjk) − Ljk(Kjk)) on ∂D. j

Note that T (ω) depends only on the restriction of the coeﬃcients of ω on ∂D. We can now show that the trace condition for pluriharmonic functions can be reformulated in tangential terms. Trace condition for pluriharmonic functions 6

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 the condition Z u (L ] ω)dσ = 0 (]) ∂D

1 for every form ω ∈ Cn−2,n(D) such that T (ω) = 0 on ∂D. 1 2 0 (b) Let H (D, R) = 0 and ∂D of class C . If u ∈ P h∂D(D), then u satisﬁes (]). 1 0 The same result holds if ω ∈ Cn−2,n(D) and ∂ω ∈ Cn−1,n(D).

Proof. (a) Let U ∈ P hα(D) be the extension of u and F = U + iV ∈ A0(D). Then from Proposition 1 we get Z Z F (L ] ω)dσ = F ∂ω, ∂D ∂D which vanishes since F is a CR function on ∂D. The real part of the ﬁrst integral is the left hand side of (]). (b) Let F = U + iV be as above. We can proceed exactly as in the proof of the second part of Theorem 1 in [11], using Stout’s estimate (see [14]) to get that F belongs to the Hardy space H2(D).

Remark 3. The hypothesis of regularity of the domain D can be weakened using the techniques introduced in [1].

3.3 Here we show that condition (]) characterizes the traces of pluriharmonic functions if the domain D satisfies a global convexity condition. Let the boundary ∂D be at least of class C1+λ, λ > 0. Assume that for every function H ∈ Harm∞(D), the ∂-closed (n − 1, n)-form ∗∂H is ∂-exact with regularity at the boundary: there exists a (n − 2, n)-form ω, with at least C1 coefficients on D, such that ∗∂H = ∂ω. This condition is satisfied for example in the following cases: (i) D is a smooth bounded pseudoconvex domain. (ii) D is a smooth bounded domain that satisfies the Hörmandercon- dition Z(n − 1), i.e., at every point ζ ∈ ∂D the Levi form of ∂D has at least 1 positive eigenvalue (see [9]). Theorem 2. Let D be a domain that satisfies the condition above. If u ∈ 0 0 C (∂D) is a real function that satisfies condition (]), then u ∈ P h∂D(D).

∞ 1 Proof. Let H ∈ Harm0 (D) and let ω be a C (D)-form such that ∂ω = ∗∂H. From Proposition 1, we get that the equality ∂nH = L ] ω is valid on ∂D² for any small ² < 0 and then, by continuity, also on ∂D. ∞ Moreover, H belongs to the space Harm0 (D) exactly when T (ω) = 0. Then condition (]) is equivalent to Z

u ∂nHdσ = 0 ∂D

∞ for every H ∈ Harm0 (D). Trace condition for pluriharmonic functions 7

If u ∈ C1+λ(∂D), then the assertion of the theorem follows from The- orem 2 in [11]. If u is only continuous on ∂D, then it follows from the results obtained in [1].

Remark 4. In view of what has been said in §3.1, the condition (]) needs to be satisﬁed only for every (n − 2, n)-form ω such that ∂∗∂ω = 0 on D, T (ω) = 0 on ∂D.

4 The bidimensional case

When n = 2, the system (1) of diﬀerential equations of §3.1 arising from the equality ∂ω = ∗∂H reduces to the equations  ∂H ∂K  = ∂z¯ ∂z 1 2 (2)  ∂H ∂K  = − ∂z¯2 ∂z1

where K = K12 is the unique coeﬃcient of the (0, 2)-form ω = c2Kdz¯. The preceding equations imply that when H ∈ Harm1(D), also K is ∂K harmonic on D and the derivatives are continuous on D for j = 1, 2. ∂zj From this property it follows that the function µ ¶ 1 ∂ρ ∂K ∂ρ ∂K L ] ω = −L12(K) = − |∂ρ| ∂z1 ∂z2 ∂z2 ∂z1

is continuous on the intersection of D with a neighborhood of ∂D. The tangent vector obtained from L12 at ζ ∈ ∂D is a basis of the complex tangent space to ∂D at ζ. We now show that, in the bidimensional case, we can obtain the result stated in Theorem 2 under a weaker convexity assumption on D. Theorem 3. Let n = 2. Let ∂D be of class C1+λ. Assume that the 1,0 0 Dolbeault cohomology group H∂ (D) vanishes. If u ∈ C (∂D) is a real function that satisﬁes the condition Z

u L12(K)dσ = 0 ∂D ∂K for every complex harmonic function K on D, such that ∈ C0(D) ∂zj 0 (j = 1, 2) and with L12(K) real on ∂D, then u ∈ P h∂D(D). If H1(D, R) = 0 and ∂D is of class C2, the condition stated above is also 0 necessary in order to have u ∈ P h∂D(D).

∞ 1,0 Proof. Let H ∈ Harm0 (D). Since H∂ (D) = 0, we can ﬁnd ω = ∞ c2Kdz¯ ∈ C0,2(D) such that ∂ω = ∗∂H (the form dz¯ can always be fac- tored out). The discussion above says that L12(K) is continuous on a domain {z ∈ D : ²0 < ρ(z)}, and then the equality ∂nH = −L12(K) is valid on ∂D² also for ² = 0. Then Z Z

u ∂nHdσ = − u L12(K)dσ = 0 ∂D ∂D Trace condition for pluriharmonic functions 8

since L12(K) is real on ∂D. Then condition (?) is satisfied by u, and 0 u ∈ P h∂D(D). Conversely, let U ∈ P h0(D) be the extension of u. Then U = Re F , 2 R R with F = U + iV ∈ H (D). The integral F L12(K)dσ = − F ∂ω ∂D² R ∂D² vanishes for every ² < 0 sufficiently small. Then U Re(L12(K))dσ + R ∂D² R V Im(L12(K))dσ = 0. The first integral tends to uL12(K)dσ as ∂D² R ∂D ² tends to 0, while the second has limit ∂D V Im(L12(K))dσ = 0. Remark 5. The functions H and K related by the system (2), can be con- sidered as the two complex components of a (left-)regular function of one quaternionic variable. We refer to [15] for the basic facts of quaternionic analysis. We identify the algebra H of quaternions with the space C2 by means of the map that associates to the pair (z1, z2) = (x1 + iy1, x2 + iy2) the quaternion q = z1 + jz2 = x1 + iy1 + jx2 − ky2, where i, j, k denote the basic quaternions. The equations (2) then become the Cauchy- Riemann-Fueter system for the quaternionic-valued function H + jK of one quaternionic variable q.

5 The case of the unit ball of Cn 5.1 If the domain is the unit ball B of Cn, the trace condition (]) has a more explicit form in terms of spherical harmonics. Let N0 be the real linear 2 projection in L (S) introduced in [11]. It is deﬁned for Ps,t ∈ H(s, t) by   s t Ps,t + P s,t, for t > 0 N (P ) = s + t s + t 0 s,t  Ps,t, for t = 0.

1 1 The space Harm0(B) coincides with Fix(N0) = {H ∈ Harm (B): N0(H) = H}.

5.2 We first consider the bidimensional case. Let H,K ∈ Harm1(B) be a pair of functions that solve the system (2). Then the equality ∂nH = −L12(K) holds on the unit sphere S. The system is satisfied also by the pair −K, H. Then also the equality ∂nK = L12(H) is valid on S. This fact allows to 1 obtain the projection N1 on the space of the functions K ∈ Harm (B) such that L12(K) is real on S. From the equation ∂nK = L12(N0(Ps,t)) for Ps,t ∈ H(s, t), we get that N1 can be defined as follows: s t N (P ) = L (P ) + L (P ), 1 s,t (s + 1)(s + t) 12 s,t (t + 1)(s + t) 12 s,t

∂F ∂F where L12(F ) = z2 − z1 on S. Note that N1(Ps,t) = 0 when s = 0 ∂z¯1 ∂z¯2 or t = 0. Trace condition for pluriharmonic functions 9

Remark 6. Let Us+t be the space of polynomials in the quaternionic variable q that are regular and homogeneous of order s + t over R. In view of what has been said in §4, for every Ps,t ∈ H(s, t), the function R(Ps,t) = N0(Ps,t) + jN1(Ps,t) belongs to Us+t. As a consequence of Theorems 1,2 and 3, we get the following result. Corollary 1. Let B be the unit ball of C2 and let S be the unit sphere. A function u ∈ C0(S) has a pluriharmonic extension to B if and only if Z

u L12(N1(Ps,t))dσ = 0 S

for every s, t > 0 and for every Ps,t ∈ H(s, t) .

5.3 Now we come to the general case n ≥ 2, using the bidimensional case as a guide. An easy computation shows that the (n − 2, n)-form c X ω = − n (−1)j+kL (P )dz[j, k] ∧ dz¯ n + s − 1 jk s,t j

satisﬁes the equation ∂ω = ∗∂Ps,t. Then we can deﬁne, for any s, t ≥ 0, µ ¶ X sL (P ) tL (P ) ω = −c (−1)j+k jk s,t + jk s,t dz[j, k]∧dz.¯ s,t n (n + s − 1)(s + t) (n + t − 1)(s + t) j

The form ωs,t vanishes when s = 0 or t = 0, and the equality ∂ωs,t = n ∗∂N0(Ps,t) holds on C . From Proposition 1, we get that L ] ωs,t = ∂nN0(Ps,t) and T (ωs,t) = 0 on S. As a consequence of Theorems 1 and 2, we get the following result, that reduces to Corollary 1 in the case n = 2. Corollary 2. Let B be the unit ball of Cn and let S be the unit sphere. A function u ∈ C0(S) has a pluriharmonic extension to B if and only if Z

u (L ] ωs,t)dσ = 0 S

for every ωs,t (s, t > 0). Trace condition for pluriharmonic functions 10

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