Higher order boundary integral formula and integro-differential equation on Stein Mainfolds

Transcript

1 Higher order boundary integral formula and integro-differential equation on Stein Mainfolds Lüping Chen, Tongde Zhong and Tao Qian Abstract. By using Stokes formula and integration by parts, we give a new definition of Hadamard principal value of higher order singular integral with Bochner-Martinelli kernel on Stein manifolds. Then we introduce the localization method and the biholomorphic mapping to obtain the Plemelj formula and the composite formula of the higher order singular integral on Stein manifolds. As an application, we discuss the solutions of the corresponding higher order singular integral equations on Stein manifolds. Mathematics Subject Classification (2000). 32A25, 32A40, 45Exx. Keywords. Stein manifolds, higher order singular integral, Bochner-Martinelli integral, Plemelj formula, composite formula, integro-differential equation. 1. Introduction The theory of boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. The existence and L. P. Chen () School of Mathematical Sciences, Xiamen University, Xiamen, Fujian. P.R. China, lpchen@xmu.edu.cn T. D. Zhong School of Mathematical Sciences, Xiamen University, Xiamen, Fujian. P.R. China, zhongtd@xmu.edu.cn T. Qian Faculty of Science and Technology, the University of Macau, 3001, Macau, P. R. China fsttq@umac.mo.

2 2 L. P. Chen, T. D. Zhong and T. Qian multiplicity of solutions for such problems have received a great deal of attention. There are a series of corresponding results on the existence of the solutions to a classes of singular boundary problems with different integral boundary conditions under different integral domains. A number of solving methods, such as the regularization methods, the global and local analysis had been used as special cases in the study of the boundary-value problem. For high dimensional space, such as in the case of manifold, since the geometry structure is complicated, to solve an integro-differential equation directly is difficulty. It requires special techniques and methods. In the present paper we deal with the boundary value properties and the higher order singular integro-differential equations with Bochner-Martinelli kernel in several complex variables. In our discussion, the boundary integral formulas that include the Plemelj formula, the composite formula and others play important roles. In the following we would like to mention a historical sketch about the singular integral and singular integral equations with Bochner-Martinelli kernel. Early in 1957, Qikeng Lu and Tongde Zhong initiated the study of the boundary properties of the singular integral with Bochner-Martinelli kernel on a bounded domain with C (1) smooth boundary in C n, obtained the Plemelj formula of the singular integral [1][2]. Later, Tongde Zhong obtained the Plemelj formula [3] of the singular integral with Bochner-Martinelli kernel on a relatively compact domain with C (1) smooth boundary on Stein Manifold. In 2001, Tao Qian and Tongde Zhong gave a new Hadamard principal value definition of the singular integral with Bochner- Martinelli kernel on a closed smooth orientable manifold in C n, obtained the Plemelj formula and the composite formula [4] of the higher order singular integral. In 2008 and 2010, we gave some other results concern to the higher order singular integral and singular integral equations [12],[13]. In this paper, based on the idea of separating the finite part from the divergent integral by J.Hadamard [5], we use integration by parts and Stokes formula to reduce the order of the higher order singular integral, express the Hadamard principal value by the usual Cauchy principal value. Simultaneously, we introduce the localization method and the biholomorphic mapping, obtain the Plemelj formula and the composite formula of higher order singular integral with Bochner- Martinelli kernel on Stein manifolds. As an application, we use the composite formula to discuss the solutions of the higher order singular integral equations with Bochner-Martinelli kernel on Stein manifolds. This article is organized as follows: Section 2 is devoted to preliminary knowledge, notation and terminology, the necessary definitions that will be used in the latter part of the paper. In section 3 we devote ourselves to the Hadamard principal value and the composite formula for the higher order singular integral on stein Manifolds. In section 4, we describe the regularization method, draw conclusions concerning solutions of higher order singular integral-differential equations on Stein Manifolds.

3 Higher order boundary integral formula 3 2. Definition and Preliminary Knowledge In this section, we introduce some necessary definitions for proving the main theorems of this paper in section 3 and section 4. It is known that G.M.Henkin and J.Leiterer [6] extended the famous Bochner-Martinelli formula from C n to Stein Manifolds. In this paper, what we shall be working with is a generic Stein Manifold, denoted by M. Suppose that T (M) is a complex tangent vector bundle and T (M) is a complex cotangent vector bundle defined on M, T (M M) and T (M M) are pullbacks of T (M) and T (M). Under the pullbacks, M M M and (z, ξ) z respectively. Holomorphic section S(z, ξ) : M M T (M M) generates the analytic sub-sheaf of sheaf M M ϑ, that is (i) For every z M, S(z, z) = 0. The mapping S(z, ) : M T z (M) is biholomorphic in some of the neighborhoods of z. (ii) For every z ξ, S(z, ξ) 0. S(z, ξ) = σs(z, ξ), mapping σ : T (M M) ( T ) (M M) is equivalent to the mapping in C n. Then for a relatively compact domain D whose boundary D is in C (1) or piecewise in C (1), we adopt the following definition [6] : [6] Definition 2.1. Suppose that ν 2κn and ( S, κ) is a Leray section about (D, S, ϕ), the expression (n 1)! (2πi) n ϕν (z, ξ) ω ξ ( S(z, ξ) ω ξ (S(z, ξ)) S(z, ξ) 2n ˆ=Ω(ϕ ν, S, S) (2.1) σ is called a Bochner-Martinelli kernel on the Stein manifold. where ω ξ( S(z, ξ)) = ( 1) k 1 Sk d S 1 [d S k ] d S n, k=1 ω ξ (S(z, ξ)) = d ξ S 1 d S n, norm S(z, ξ) σ = S(z, ξ),s(z, ξ) 1 2 is an Euclidean measure, ϕ(z, ξ) is a holomorphic function on M M. From the proof of the Theorem in [6] and [11], obviously, we have the following results (1) In D M, there are isolated singular points whose orders are 2n 1. (2) For every z, ξ M and ξ z, function ϕ k S 2 σ is C (2). (3) When integer ν 2nκ, for every fixed point z M and ξ M\{z}, formula (2.1) is C (1).

4 4 L. P. Chen, T. D. Zhong and T. Qian (4) For every z M, ϕ(z, z) = 1. Definition 2.2. Suppose that D is a relative compact domain on Stein manifold M, f(ξ) is a function defined on the boundary D, then F () = f(ξ)ω(ϕ ν, S(, ξ), S(, ξ)), D (2.2) D ξ is called a singular integral on D. If f(ξ) satisfies the Hölder continuity condition, the principal value of the integral exists. The Cauchy principal value can be defined as follows [3] where V.P.F () = lim δ 0 δ (,ξ) f(ξ)ω(ϕ ν, S(, ξ), S(, ξ)), D, (2.3) Σ δ (, ξ) = D ξ σ δ (, ξ), σ δ (, ξ) = D ξ B δ (), B δ () = {z : z < δ}. 3. The Hadamard Principal Value and The Composite Formula for Higher Order Singular Integrals on Stein Manifolds In the following, we should first study the higher order Bochner-Martinelli type integral on Stein Manifolds of the form f(ξ) S k (z, ξ) S(z, ξ) 2 Ω(ϕν (z, ξ) S(z, ξ), S(z, ξ)), (3.1) D where f is a differentiable function defined in a neighborhood of the boundary D. On the boundary D, the first-order derivatives of f satisfy the Hölder continuity condition, here the exponent α > 0. For simplicity we denote this function class by H 1 (α). Lemma 3.1. Suppose that f H 1 (α), B(, δ) is a neighborhood of D, for a sufficient small radius δ, we have f(ξ) S k (, ξ) D\B(,δ) S(, ξ) 2 Ω(ϕν (, ξ) S(, ξ), S(, ξ)) = 1 n Cn [f(ξ)ϕ ν (, ξ)] ( D\B(,δ)) ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n + 1 n Cn D\B(,δ) d[f(ξ)ϕ ν (, ξ)] S(, ξ) 2n ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n S(, ξ) 2n, (3.2)

5 Higher order boundary integral formula 5 where C n = (n 1)! (2πi). n Proof. To show the identity we first carry out some differential computation, as follows. { ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n } d S(, ξ) 2n { ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n } = d [ S(, ξ)s(, ξ)] n = ( n)[( S(, ξ)s(, ξ)] (n+1) Sk (, ξ) ( 1) j 1 Sj(, ξ) ( 1) k 1+k 1 ds 1 ds n d S 1 [d S j] d S n +( n)[( S(, ξ)s(, ξ)] (n+1) S j(, ξ) ( 1) j 1 Sj(, ξ) ( 1) k 1 ( 1) n 1+j 1 ds 1 [ds k ] ds n d S 1 d S n 1 + [ S(, ( 1) j 1 ( 1) k 1 ξ)s(, ξ)] n ( 1) n 1+j 1 ds 1 [ds k ] ds n d S 1 d S j d S n S k (, ξ) n ( 1) j 1 Sj(, ξ)ds 1 ds n d S 1 [d S j] d S n = ( n) S(, ξ) 2n+2 ( 1) k+n S j(, ξ) S j(, ξ)ds 1 [ds k ] ds n d S 1 d S n +n +( n) S(, ξ) 2n+2 ( 1) k+n ds 1 [ds k ] ds n d S 1 d S(, ξ) 2n S k (, ξ) n ( 1) j 1 Sj(, ξ)ds 1 ds n d S 1 [d S j] d S n = ( n) S(, ξ) 2n+2 S j(, ξ) 2 ( 1) k+n ds 1 [ds k ] ds n d S 1 d S n +n +( n) S(, ξ) 2n+2 ( 1) k+n ds 1 [ds k ] ds n d S 1 d = ( n) S(, ξ) 2n S k (, ξ) n ( 1) j 1 Sj(, ξ)ds 1 ds n d S 1 [d S j] d S n S n S n S(, ξ) 2n+2 (3.1)

6 6 L. P. Chen, T. D. Zhong and T. Qian where So, S k (, ξ) = ( n) S(, ξ) 2 ϕ ν (, ξ) ϕ ν (, ξ) n ( 1) j 1 Sj(, ξ)ds 1 ds n d S 1 [d S j] d S n S(, ξ) 2n = ( n) (2πi)n S k (, ξ) (n 1)! S(, ξ) 2 ϕ ν (, ξ) Ω(ϕν, S, S), (3.3) Ω(ϕ ν, S, (n 1)! S) = (2πi) n ϕν (, ξ) w ( S(, ξ)) w (S(, ξ)) S(, ξ) 2n. f(ξ) S k (, ξ) D\B(,δ) S(, ξ) 2 Ω(ϕν (, ξ) S(, ξ), S(, ξ)) = 1 n Cn [f(ξ)ϕ ν (, ξ)] D\B(,δ) [ ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n ] d S(, ξ) 2n = 1 [ n Cn d f(ξ)ϕ ν (, ξ) D\B(,δ) ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n ] + 1 n Cn D\B(,δ) d[f(ξ)ϕ ν (, ξ)] S(, ξ) 2n ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n S(, ξ) 2n. (3.4) Applying Stokes formula for the first term of the end of the above equality chain, we immediately obtain the desired formula in the Lemma. The proof is complete. Now we consider the integrals on the right hand side of formula (3.2). Suppose that f is a differentiable function in a neighborhood of the boundary D. On the boundary D, the first-order derivatives of f satisfy the Hölder continuity condition, the exponent α > 0. For the first term, the dimension of the integral region is 2n 2, but the order of a singular point of the integrand is 2n 1, this integral is divergent. The second integral exists in the sense of the Cauchy principal value. In relation to the idea of the Hadamard principal value, we have

7 Higher order boundary integral formula 7 Definition 3.2. Suppose that D is a relatively compact domain on a Stein manifold M, function f belong to H 1 (α) on D. We define the Hadamard principal value FP f(ξ) S k (, ξ) D ξ S(, ξ) 2 Ω(ϕν (, ξ) S(, ξ), S(, ξ)) = PV 1 n Cn D ξ d[f(ξ)ϕ ν (, ξ)] ( 1) j 1 Sj(, ξ)( 1) k 1 ds 1 [ds k ] ds n d S 1 [d S j] d S n S(, ξ) 2n. (3.5) Since the above definition throws away the divergent part of the higher order singular integral and only keeps its finite part of the integral, we can simplify our computation, and use the result of the Cauchy principal value directly. Remark: When the Stein manifold is just the space C n, now S(z, ξ) = ξ z. Our definition above reduces to the Hadamard principal value in C n (refer to Definition 1 of [4]). Lemma 3.3. (Composite formula for Cauchy type singular integrals on Stein manifolds) Suppose that φ(η) C (1) ( D), it can be holomorphically extended to D, then Ω(ϕ ν, S(, ξ), S(, ξ)) φ(η)ω(ϕ µ, S(ξ, η), S(ξ, η)) = 1 φ(). (3.6) D ξ D η 4 For a proof we refer to [7-9]. Theorem 3.4. (Composite formula for higher order singular integral) Suppose that D is a relatively compact domain on Stein manifold M, denote by U( D) a neighborhood of a point on the boundary D. φ(η) is holomorphic in U( D), it can be holomorphically extended into D, then there holds the composite formula Ω(ϕ ν (, ξ), S(, ξ), S(, ξ)) φ(η) S k (ξ, η) D ξ D η S(ξ, η) 2 Ω(ϕµ, S, S) = I(φ()) + 1 φ(), (3.7) 4n k where I(φ()) is constituted by integrals in the ordinary sense and in the Cauchy principal value sense (Also see formulas (3.38) and (3.39) below). If we denote Sφ = 2 Ω(ϕ ν, S(, ξ), S(, ξ)), (3.8) D ξ

8 8 L. P. Chen, T. D. Zhong and T. Qian and where S 1 φ = 2 φ(η)ω 1 (ϕ µ, S(ξ, η), S(ξ, η)) = ψ(ξ), D η (3.9) Ω 1 (ϕ µ, S, S) = S k (ξ, η) S(ξ, η) 2 Ω(ϕµ, S, S), then the composite formula becomes SS 1 φ = 4 Ω(ϕ ν (, ξ) S(, ξ), S(, ξ)) D ξ φ(η)ω 1 (ϕ µ (ξ, η), S(ξ, η), S(ξ, η)) D η = 1 φ() + I(φ()) = S(ψ). (3.10) n k This is an integral-differential equation for which we can obtain a unique solution φ under appropriate boundary value conditions. Proof. We will prove the Theorem by localization method. We need to compute the above integrals and the term Ω(ϕ ν, S(z, η), S(z, η)) in a local coordinate. For a fixed point D, we take δ > 0, V,δ U j is a small coordinate neighborhood of point, V,δ B,δ = {η C n : η < δ} is a biholomorphic mapping. Assume that {U j } is a local finite cover of M which consist of coordinate neighborhoods, W = S(, ). Denoted by H = W (V,δ D) a hypersurface in B,δ which passes through the center. When a point z V,δ D tends to sufficiently, there exists a point z W (V,ε D) such that z = W 1 (z ). Ω(ϕ ν, S(z, η), S(z, η)) = Ω(ϕ ν, ˆ S, Ŝ)(η ) where (n 1)! ˆ S(z = (2πi) n ˆϕν (z, η )ω, η ) ( ˆ S(z, η ) ) 2 (Ŝ(z, η )), (3.11) ˆ S(z, η ) = 1 S(W (z ), W 1 (η )), Ŝ(z, η ) = S(W 1 (z ), W 1 (η )), (3.12) ˆϕ(z, ξ ) = ϕ(w 1 (z ), W 1 (η )). Now we study the relation between the Bochner-Martinelli kernel Ω(ϕ ν, ˆ S, Ŝ)(η ) on the Stein manifold and the kernel Ω( η z, η z ) in C n. Since Ŝ(, ) : B,δ B,δ T (M M), we have ˆ S(, ) : B,δ B,δ T (M M). (3.13)

9 Higher order boundary integral formula 9 Let {u j }, {ū j } be local coordinates of Ŝ, ˆ S. uj (z, η ) are holomorphic functions defined in the convex region B,δ B,δ, u(z, z ) = 0. By division theorem we can obtain the expression u j (z, η ) = γ jk (z, η )(ηk zk), (3.14) k=1 where γ jk (z, η ) is holomorphic for z and η. In addition, u(0, η ) = Ŝ(0, η ) = S(, W 1 (η )) = W (W 1 (η )) = η (here S(, ) = W ), (3.15) when (z, η ) is in a sufficiently small neighborhood of (, ). We might as well suppose this neighborhood is B,δ B,δ, det(γ jk (z, η )) 0. (3.16) By division theorem again, γ jk can be expressed as γ jk (z, η ) = γ jkl (z, η )zl + δ jk, (z, η ) B,δ B,δ. (3.17) l=1 where δ jk = γ jk (0, η ). By formula (3.13)-(3.16) we know kernel Ω( ˆϕ ν, ˆ S, Ŝ)(ξ ) can be expressed as Ω( ˆϕ ν, ˆ S, Ŝ)(η ) = ˆϕ ν (z, η )B(z, η ) + ˆϕ ν (z, η )A(z, η ), (3.18) where the exterior differential formula B does not include dγ jk, but A includes dγ jk. From reference [2],[5],[6] and [10], we have B(z, η ) = A(z, η )) = O( η z 2 2n ), (3.19) (n 1)! (2πi) n det( Γ t Γ)ω [ ( η z ) ω(η z ) l z l )(η j, (3.20) z j )]n where Γ = γ jk (z, η ), Γ t is a transposition of Γ. Moreover, we know the difference between B(z, η ) and the Bochner-Martinelli kernel Ω( η z, η, z ) in C n is B(z, η ) Ω( η z, η z ) = O( η z 2 2n ). (3.21) In addition, we have estimation ˆϕ ν (z, η ) 1 = ˆϕ ν (z, η ) ˆϕ ν (z, z ) = O( η z ). (3.22) When z tends to ξ, we have the following estimations similar to formula (3.21) and (3.22) and B(ξ, η ) Ω( η ξ, η ξ ) = O( η ξ 2 2n ) (3.23) ˆϕ ν (ξ, η ) 1 = O( η ξ ). (3.24)

10 10 L. P. Chen, T. D. Zhong and T. Qian For the kernel S k (z, η) S(z, η) 2 Ω(ϕµ, S(z, η), S(z, η)), based on the previous knowledge, we have the following relations and estimations S k (z, η) S(z, η) 2 Ω(ϕµ, S(z, η), S(z, η)) = ˆ S k Ŝ 2 Ω( ˆϕµ, ˆ S, S)(η ) = ˆϕ µ (z, η ) γ jk ( η j z j ) l z l )(η j z j ) [B(z, η ) + A(z, η )], (3.25) j.k,l γ jk ( η j z j ) l z l )(η j z j ) A(z, η ) = O( η z 1 2n ), (3.26) j.k,l γ jk ( η j z j ) l z l )(η j z j ) B(z, η ) η k z k η z 2 Ω( η z, η z ) = O( η z 1 2n ), j.k,l (3.27) ˆϕ µ (z, η ) 1 = ˆϕ µ (z, η ) ˆϕ µ (z, z ) = O( η z ). (3.28) When z tends to ξ, under the biholomorphic mapping V,δ B,δ, we also have the following estimations in the local coordinate neighborhood of B,δ B,δ, γ jk ( η j ξ j ) l ξ l )(η j ξ j ) B(ξ, η ) η k ξ k η ξ 2 Ω( η ξ, η ξ ) = O( η ξ 1 2n ), (3.29) ˆϕ µ (ξ, η ) 1 = O( η ξ ). (3.30) Let F (z) = Ω(ϕ ν, S(, ξ), S(, ξ)) φ(η) S k (ξ, η) D ξ D η S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)). (3.31)

11 Higher order boundary integral formula 11 Then { F (z) = Ω(ϕ ν, S(, ξ), S(, ξ)) ξ D\V,δ D + [ ˆϕ ν (z, ξ )A(z, ξ ) + ˆϕ ν (z, ξ )(B(z, ξ ) Ω( ξ z, ξ z )) ξ H }{ +( ˆϕ ν (z, ξ ) 1)Ω( ξ z, ξ z ) + Ω( ξ z, ξ z )] φ(η) S k (ξ, η) η D\V ξ,ε D S(ξ, η) 2 [ γ jk ( η j z j ) Ω(ϕ µ, S(ξ, η), S(ξ, η)) + ˆφ(η ) ˆϕ µ (z, η ) η H ξ l z l )(η j z j ) A(z, η ) ( + ˆϕ µ (z, η ) γ jk ( η j z j ) l z l )(η j z j ) B(z, η ) ) η k z k η z 2 Ω( η z, η z ) ]} +( ˆϕ µ (z, η ) 1) η k z k η z 2 Ω( η z, η z ) + η k z k η z 2 Ω( η z, η z ), where (3.32) ˆφ(η ) = φ(w 1 (η )). (3.33) When we use the previously recalled estimations, it is easy to find that the first four terms of both the two brackets in the right hand side of the above expression are proper integrals. The fifth term in the first bracket is a Bochner- Martinelli integral in C n, and the fifth term in the second bracket is a higher order Bochner-Martinelli integral. For simplicity, we separate the second integral in the first bracket of formula (3.32) into two parts, one is Φ(z, ξ ) = [ ˆϕ ν (z, ξ )A(z, ξ ) + ˆϕ ν (z, ξ )(B(z, ξ ) ξ H ξ H Ω( ξ z, ξ z )) + ( ˆϕ ν (z, ξ ) 1)Ω( ξ z, ξ z )], (3.34) the other is ξ H Ω( ξ z, ξ z ),

12 12 L. P. Chen, T. D. Zhong and T. Qian we also separate the second integral in the second bracket of formula (3.32) into two parts, one is [ γ jk ( η j z j ) ˆφ(η )Ψ(z, η ) = ˆφ(η ) ˆϕ µ (z, η ) η H ξ η H ξ l z l )(η j z j ) A(z, η ) ( + ˆϕ µ (z, η ) γ jk ( η j z j ) l z l )(η j z j ) B(z, η ) ) η k z k η z 2 Ω( η z, η z ) ] +( ˆϕ µ (z, η ) 1) η k z k η z 2 Ω( η z, η z ), (3.35) the other is η H ξ then formula (3.32) becomes { F (z) = + ξ D\V,δ D η k z k η z 2 Ω( η z, η z ), Ω(ϕ ν, S(, ξ), S(, ξ)) + ξ D\V,δ D }{ η D\V ξ,ε D ξ H Φ(z, ξ ) Ω( ξ z, ξ z ) φ(η) S k (ξ, η) ξ H η D\V ξ,ε D S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)) } + ˆφ(η )Ψ(z, η ) + ˆφ(η η H ξ η Hξ ) η k z k η z 2 Ω( η z, η z ) = Ω(ϕ ν, S(, ξ), S(, ξ)) φ(η) S k (ξ, η) S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)) + ξ D\V,δ D Ω(ϕ ν, S(, ξ), S(, ξ)) ˆφ(η )Ψ(z, η ) η H ξ Ω(ϕ ν, S(, ξ), S(, ξ)) ˆφ(η η Hξ ) η k z k + ξ D\V,δ D η z 2 Ω( η z, η z ) + Φ(z, ξ ) φ(η) S k (ξ, η) ξ H η D\V ξ,ε D S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)) + Φ(z, ξ ) ˆφ(η )Ψ(z, η ) ξ H η H ξ + Φ(z, ξ ) ˆφ(η ξ H η Hξ ) η k z k η z 2 Ω( η z, η z ) + Ω( ξ z, ξ z ) ˆφ(η ) S k (ξ, η) ξ H η D\V ξ,ε D S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)) + Ω( ξ z, ξ z ) ˆφ(η )Ψ(z, η ) ξ H η H ξ + Ω( ξ z, ξ z ) ˆφ(η ξ H η Hξ ) η k z k η z 2 Ω( η z, η z ). (3.36)

13 Higher order boundary integral formula 13 Suppose that the hypersurface H = W (V,ε D) pass through the origin O, now we set = 0. When z 0, recalling the definition of the Hadamard principal value of higher order singular integrals in C n (Definition 1 in [4]) and using some of the previous estimations, we have Φ() = Ω(ϕ ν, S(, ξ), S(, ξ)) φ(η) ξ D\V,δ D η D\V ξ,ε D S k (ξ, η) S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)) + Ω(ϕ ν, S(, ξ), S(, ξ)) ˆφ(η )Ψ(0, η ) ξ D\V,δ D η H ξ + Ω(ϕ ν, S(, 1 ξ), S(, ξ))pv ˆφ(η ) Ω( η 0, η 0) ξ D\V,δ D η H ξ n ηk + Φ(0, ξ ) ξ H η D\V ξ,ε D + Φ(0, ξ ) ˆφ(η )Ψ(0, η ) ξ H η H ξ + Φ(0, ξ )PV ˆφ(η ) Ω( η 0, η 0) ξ H η H ξ ηk + Ω( ξ 0, ξ 0) ξ H η D\V ξ,ε D + Ω( ξ 0, ξ 0) ˆφ(η )Ψ(0, η ) ξ H η H ξ + Ω( ξ 0, ξ 1 0)PV ξ H η H ξ n φ(η) S k (ξ, η) S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)) ˆφ(η ) S k (ξ, η) S(ξ, η) 2 Ω(ϕµ, S(ξ, η), S(ξ, η)) ˆφ(η ) Ω( η 0, η 0). (3.37) ηk By the previous estimations, we see that the first eight integrals in the right hand side of the above formula are proper integrals exist in the sense of the Cauchy principal value. For simply, denoted by I(φ()) the sum of the first eight integrals, then formula (3.37) becomes Φ() = I(φ()) + Ω( ξ 0, ξ 0)PV ξ H η H ξ 1 n ˆφ(η ) Ω( η 0, η 0). (3.38) ηk For the second term in the expression (3.38), applying the composite formula for singular integral with Bochner-Martinelli kernel in C n (refer to 4, 6 and 8), we have ξ H Ω( ξ 0, ξ 0)PV η H ξ 1 n ˆφ(η ) Ω( η 0, η 0) = 1 φ(). (3.39) ηk 4n k The proof is complete. Remark: In particular, when the Stein manifold M is just the space C n, now

14 14 L. P. Chen, T. D. Zhong and T. Qian the section S(z, ξ) = ξ z, the composite formula (3.7) on the Stein manifold M becomes the composite formula in the space C n which is K(, ξ) φ(η) ξ k η k 1 φ() K(ξ, η) =. (3.40) D ξ D η ξ η 2 4n k Comparing the composite formula (3.7) on the Stein manifold with the composite formula (3.40) in C n, we find that there is an extra term I(φ()) for the Stein manifold case that is caused by the difference of the section S(z, ξ) of the Stein manifold and the section ξ z of C n. In local coordinate, the relationship between the coordinate expression u j (z, ξ ) of S(z, ξ) and ξk z k which is u j (z, ξ ) = n k=1 γ jk (z, ξ )(ξ k z k ), where γ jk(z, ξ ) is holomorphic for z and ξ, this is not a linear relationship. Theorem 3.5. (Plemelj formulas for higher order singular integrals) Suppose that D is a relatively compact domain on the Stein manifold M. f is differentiable in a neighborhood of the boundary D and belong to H 1 (α) on D. For the higher order Bochner-Martinelli type integral, F (z) = f(η) S k (z, η) D η S(z, η) 2 Ω(ϕµ, S(z, η), S(z, η)) z M\ D, (3.41) when z tends to D from the inner part or the outer part of D, we have the Plemelj formulas respectively as follows F i() = FP f(η) S k (, η) η D S(, η) 2 Ω(ϕµ, S(, η), S(, η)) or f()[ ˆϕµ (0, η η 0 ) 1] η [ f() 2 2n k ] + ( 1) n f() 1 F e() = FP f(η) S k (, η) η D S(, η) 2 Ω(ϕµ, S(, η), S(, η)) 1 2 f()[ ˆϕµ (0, η η 0 ) 1] η 0 1 [ f() 2 2n k ] + ( 1) n f() 1 (3.42) (3.43) Proof. Select a small neighbourhood of D which is V,δ, take a biholomorphic mapping W = S(, ) : V,δ B,δ = {η C n : η < δ}, is the center of B,δ. Denote by H = W (V,δ D) a hypersurface in B,δ which passes through the center of B,δ, when z V,δ D is sufficiently close to, there is a point z W (V,δ D) such that z = W 1 (z ). By the proof of Theorem 3.4 we have

15 Higher order boundary integral formula 15 F (z) = f(η) S k (z, η) D η S(z, η) 2 Ω(ϕµ, S(z, η), S(z, η)) = η D\V,δ D [ + ˆf(η ) ˆϕ µ (z, η ) η H ( + ˆϕ µ (z, η ) f(η) S k (, η) S(, η) 2 Ω(ϕµ, S(, η), S(, η)) γ jk ( η j zj ) l z l )(η j z j ) A(z, η ) γ jk ( η j z j ) l z l )(η j z j ) B(z, η ) ) η k z k η z 2 Ω( η z, η z ) +( ˆϕ µ (z, η ) 1) η k z k η z 2 Ω( η z, η z ) ] + η k z k η z 2 Ω( η z, η z ). (3.44) From the estimation formulas (3.26),(3.27),(3.28),(3.29)and(3.30), we find the first and the second terms of the second integral in the above expression are proper integrals. Following we consider the third and the fourth terms of the second integral. The third term is a singular integral with Bochner-Martinelli kernel in C n. For simplicity we suppose the center of B,δ is at the origin O, so = 0. when z tends to 0, using the Plemelj formula in C n, we have lim ˆf(η z 0 η H )( ˆϕ µ (z, η ) 1) η k z k + η z 2 Ω( η z, η z ) = PV ˆf(η η H )( ˆϕ µ (0, η ) 1) η k 0 η 0 2 Ω( η 0, η 0)+ 1 2 ˆf(0)( ˆϕ µ (0, η ) 1) η 0 η 0 2 or (3.45) lim ˆf(η z 0 η H )( ˆϕ µ (z, η ) 1) η k z k η z 2 Ω( η z, η z ) = PV ˆf(η η H )( ˆϕ µ (0, η ) 1) η k 0 η 0 2 Ω( η 0, η 0) 1 2 ˆf(0)( ϕ ˆ µ (0, η ) 1) η 0 η 0, 2 where ˆf(0) = f(). (3.46)

16 16 L. P. Chen, T. D. Zhong and T. Qian The last term in formula (3.44) is a higher order singular integral in C n. According to Theorem 1 of paper [4], we have or lim ˆf(η z 0 η H ) η k z k + η z 2 Ω( η z, η z ) = FP ˆf(η η H ) η k 0 η 0 2 Ω( η 0, η 0) + 1 2n lim ˆf(η z 0 η H ) η k z k η z 2 Ω( η z, η z ) = FP ˆf(η η H ) η k 0 η 0 2 Ω( η 0, η 0) 1 2n Let Hadamard principal value FP f(η) S k (, η) D η S(, η) 2 Ω(ϕµ, S(, η), S(, η)) = f(η) S k (, η) S(, η) 2 Ω(ϕµ, S(, η), S(, η)) η D\V,δ D [ + ˆf(η ) ˆϕ µ (0, η ) η H ( + ˆϕ µ (0, η ) [ f() k [ f() k γ jk ( η j 0) l A(0, η) 0)(ηj 0) γ jk ( η j 0) l 0)(ηj 0) B(0, η ) ] + ( 1) n f(), (3.47) 1 ] + ( 1) n f(). (3.48) 1 ) η k 0 η 0 2 Ω( η 0, η 0) +PV ˆf(η η H )( ˆϕ µ (0, η ) 1) η k 0 η 0 2 Ω( η 0, η 0) ] +FP ˆf(η η H ) η k 0 η 0 2 Ω( η 0, η 0), (3.49) applying the results of formula (3.45),(3.46),(3.47) and (3.48) to formula (3.44), we have the desired formula (3.42) and (3.43). 4. Higher Order Singular Integral-differential Equation on Stein Manifolds Suppose that φ(ξ) is holomorphic in a neighborhood of D, then the composite formula (3.7) holds. We can accordingly solve the higher order singular integral equations with the Bochner-Martinelli kernel as follows. Consider the higher order singular integral equation as + bs 1 φ = ψ, (4.1)

17 Higher order boundary integral formula 17 where a, b are complex value constants, S, S 1 are given by the formulas (3.8) and (3.9). By Definition 3.2, the equation (4.1) is a partial differential singular integral equation. Applying the operator M = ai bs, Iφ = φ (4.2) to the both sides of the characteristic equation, we have That is so a 2 Sφ + abs 1 φ abssφ b 2 SS 1 φ = (ai bs)ψ. a(asφ + bs 1 φ) abssφ b 2 SS 1 φ = (ai bs)ψ. aψ abssφ b 2 SS 1 φ = aψ bsψ, assφ + bss 1 φ = Sψ. (4.3) From Theorem 3.4 (see formula 3.10), we have the following results SSφ = φ, SS 1 φ = I(φ()) + 1 φ(), n k where I(φ()) consists of the proper or the Cauchy principal value integrals. Apply the results above to (4.3), we have aφ + bi(φ()) + b φ() = Sψ. (4.4) n k Under certain boundary value conditions we can solve the integral-differential equation for a unique solution of function φ (Refer to [10]). Acknowledgment Many thanks to the referee for his valuable suggestion. The project was supported by the Fundamental Research Funds for the Central Universities (No ), the research grant of the University of Macau No.RG079/04-05S/QT/FST and Macao Science and Technilogy Development Fund 051/2005/A References [1] Q. K. Lu and T. D. Zhong, An extension of the Privalov Theorem. Acta Math. Sinica, 7, (in Chinese), [2] T. D. Zhong, Integral Representation of Functions of Several Complex Variables and Multidimensional Singular Integral Equations. Xiamen University Press, Xiamen (in Chinese), [3] T. D. Zhong, Holomorphic extension on Stein manifolds, Research Report No. 10, Mittag-Leffler Institute, 1987.

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