THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOTH CLOSED ORIENTABLE MANIFOLDS 1

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1 00,B():-8 THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOTH CLOSED ORIENTABLE MANIFOLDS Qian Tao ( ) Faculty of Science and Technology, The University of Macau P.O.Box 300, Macau(Via Hong Kong) fstt@wkgl.umac.mo Zhong Tongde ( ) Institute of Mathematics, Xiamen University, Xiamen 36005, China Abstract Using integration by parts and Stokes formula the authors give a new definition of Hadamard principal value of higher order singular integrals with Bochner-Martinelli kernel on smooth closed orientable manifolds in C n. The Plemelj formula and composite formula of higher order singular integral are obtained. Differential integral equations on smooth closed orientable manifolds are treated by using the composite formula. Key words Bochner-Martinelli kernel, Plemelj formula, Composite formula, Higher order singular integral, Differential integral Equation 99 MR Subject Classification 3A5, 3A40 Introduction Since the limit value formula, viz. the Plemelj formula, of the Cauchy type integral with Bochner-Martinelli kernel was proved in 957 [, it has been successfully used to the study of singular integral equations, solving the b -equation, holomorphic extension, -closed extension and C R manifolds [ 5. Evidently, the research of higher order singular integrals with Bochner-Martinelli kernel itself also has important significance. In 95, J. Hadamard first defined the principal value of higher order singular integrals, that is the so called Hadamard principal value, by using which he solved the Cauchy problem of hyperbolic partial differential equation. The idea of Hadamard is to separate its finite part, i.e. the principal value [6, from a divergent integral on real axis. In 957, C. Fox extended the idea of Hadamard to the case of higher order singular integrals in the complex plane [7. The calculations of all those principal values of higher order singular integrals are very complicated, and they are very difficult to be extended to the several complex variables case. In 990 Wang Xiaoqin studied the higher order singular integrals on complex hypersphere in C n and the higher order singular integrals with Bochner-Martinelli kernel. She obtained some remarkable results [8 with simpler methods, but the calculations are still complicated. In [9 by using integration by parts and Stokes formula we Project supported by the Bilateral Science and Technology Collaboration Program of Australia 998 and the Natural Science Foundation of China (No )

2 ACTA MATHEMATICA SCIENTIA Vol. Ser.B reduce the order of singularity of higher order singular integrals, and we finally use the Cauchy singular integral to express the higher order singular integrals. The Hadamard principal values therefore are expressed by the Cauchy principal value. For simplicity, we consider higher order singular integrals on the boundary of a bounded domain D with smooth boundary in the complex plane πi f(ζ) dζ, z D, () (ζ z) where f(ζ) H (α), that is the set of the functions whose first order derivatives satisfy the Hölder condition of index α. Singular integral () under Cauchy principal value is divergent. Using integration by parts and Stokes formula we have { [( ) ( ) } f(ζ) πi (ζ z) dζ = d f(ζ) df(ζ) πi ζ z ζ z = πi f (ζ) dζ, z D. ζ z f(ζ) Therefore the study of the Cauchy type integral πi (ζ z) dζ is equivalent to the study f (ζ) of the Cauchy type integral πi ζ z dζ. Since f (ζ) H(α), when z = η, the Cauchy principal value of the right hand side exists. Therefore, if we denote the Hadamard principal value by FP, the Cauchy principal value by PV, then we may define FP πi f(ζ) dζ = PV (ζ η) πi f (ζ dζ, η. () ζ η This definition agrees with the original idea of Hadamard to separate the finite part from the divergent integral. Moreover, in our definition the Hadamard principal value is expressed by the Cauchy principal value that gives us the convenience of utilizing the results of the Cauchy principal value. In the following we will use this idea to study the higher order singular integral with Bochner-Martinelli kernel in C n. Hadamard Principal Value and Plemelj Formula Let D be a bounded domain in C n, its boundary be a smooth orientable manifold of dimension n of class C (). If f A c (D), then we have the well-known Bochner-Martinelli integral f(z) = f(ζ)k(ζ, z), z C n \, (3) where K(ζ, z) = Ω(ζ z, ζ z) := (n )! (πi) n w (ζ z) w(ζ z) (ζ z, ζ z) n, (4) w (v) : = j= ( )j+ v j dv d[v j dv n, w(v) : = dv dv n, < v, u >: = j= v ju j. (5)

3 No. Qian & Zhong: DIFFERENTIAL INTEGRAL EQUATIONS ON MANIFOLDS 3 In order to study the higher order singular integral with Bochner-Martinelli kernel, firstly we study the following higher order Bochner-Martinelli type integral f(ζ) ζ k z k ζ z K(ζ, z), z Cn \, (6) where f is differentiable and belongs to H (α) on. We have Lemma Let f be differentiable and belong to H (α) on D, then for any w and any ball B(w, ǫ) of radius ǫ (ǫ is a sufficient small positive number) centred at w, we have f(ζ) ξ k ω k \B(ω,ǫ) ζ ω K(ζ, ω) = n C n f(ξ) (\B(ω,ǫ)) j= ( )j (ξ j ω j )( ) k dζ [dζ k dζ n dζ [dζ j dζ n + n C n \B(ω,ǫ) df(ζ) ζ ω n j= ( )j (ξ j ω j )( ) k dζ [dζ k dζ n dζ [dζ j dζ n ζ ω n, where C n = (n )! (πi). n Proof By integration by parts we have f(ζ) ζ k ω k ζ ω K(ζ, ω) = \B(ω,ǫ) n \B(ω,ǫ) Cn f(ζ) [ j= ( )j (ζ j ω j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n = n Cn d [ f(ζ) + n Cn \B(ω,ǫ) ζ ω n j= ( )j (ζ j ω j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n df(ζ) \B(ω,ǫ) ζ ω n j= ( )j (ζ j ω j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n ζ ω n. Applying Stokes formula to the first term of right hand side, (7) follows immediately. Now we consider the first term on the right hand side of (7). Since the dimension of integration domain is n, but the order of singularity of the integrand is n, this integral is divergent. Moreover, since f H (α) on, the integral of the second term exists in the sense of the Cauchy principal value. By the idea of Hadamard principal value, we have Definition If f is a function defined and differentiable on and f H (α) on, we define FP f(ζ) ζ k ω k K(ζ, ω) ζ ω (7)

4 4 ACTA MATHEMATICA SCIENTIA Vol. Ser.B = PV n C n df(ζ) j= ( )j (ζ j ω j )( ) k dζ [dζ k dζ n dζ [dζ j dζ n ζ ω n. Obviously, the above definition of Hadamard principal value is simpler and clearer than that in [8. This definition has some advantages: ) it has discarded the divergent part of the higher order singular integral directly, and only kept the finite part. In this way we can avoid complicated calculations in applications; ) it is defined at a general point ω ; and 3) the Hadamard principal value is in terms of the Cauchy principal value, so we can utilize the results of the Cauchy principal value directly. As example, by the Plemelj formula of the Cauchy singular integral on (see[ 4), we have Theorem (The Plemelj formula for higher order singular integral) Under the assumptions in Definition, if z approaches ω from the inner part and the outer part of D, then for the higher order Bochner-Martinelli type integral F(z) = it follows the Plemelj formula F i (ω) = FP F e (ω) = FP f(ζ) ζ k z k ζ z K(ζ, z) z Cn \ (9) f(ζ) ζ k ω k K(ζ, z) + ζ ω n f(ζ) ζ k ω k K(ζ, z) ζ ω n [ f (ω) + ( ) n f ω k [ f (ω) + ( ) n f (ω) ω k ω (ω) ω (8), (0). () Proof By integration by parts and Stokes formula we have F(z) = f(ζ) ζ k z k K(ζ, z) ζ z = n Cn f(ζ) [ j= d ( )j (ζ j z j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n ζ z n = n Cn d [ f(ζ) j= ( )j (ζ j z j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n ζ z n + n Cn df(ζ) j= ( )j (ζ j z j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n ζ z n = n Cn df(ζ) j= ( )j (ζ j z j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n. ζ z n

5 No. Qian & Zhong: DIFFERENTIAL INTEGRAL EQUATIONS ON MANIFOLDS 5 Therefore, F i(ω) = lim z ω + n Cn = lim df(ζ) j= ( )j (ζ j z j)( ) k dζ [dζ k dζ n dζ [dζ j dζ n z ω + n n j= ζ z n f (ζ)k(ζ, z) + ( ) n ζ k n Cn lim z ω + ζ z n f z j (ζ j z j)dζ [dζ k dζ n dζ dζ n. Owing to [8, lim C z ω +( )n n ζ z n n j= f ζ j (ζ j ζ j )( ) k dζ [dζ k dζ n dζ dζ n = ( ) n n f C n PV ζ ω n (ζ j ω j )( ) k dζ ζ j j= [dζ k dζ n + A f j (ω), ω j where A j = ( ) n, j = ; A j = 0, j =,,n. Then by Definition we have F i (ω) = lim z ω + n [PV ζ z n f (ζ)k(ζ, ω) + ( ) n C n PV ζ k n f (ζ j ω j )( ) k dζ [dζ k dζ n ζ j j= dζ dζ n + n [ f f (ω) + A j (ω) ω k ω j = FP f(ζ) ζ k ω [ k f K(ζ, ω) + (ω) + ( ) n f (ω), ζ ω n ω k ω which is (0). The equality () is proved similarly. 3 Composite Formula Lemma (Composite formula for the Cauchy singular integral) Suppose that φ(ζ) C () (), and that it can be holomorphically extended into D, then K(ζ, η) φ(ξ)k(η, ξ) = φ(ζ). () η ξ 4 For this Lemma we refer the reader to [3,4,0.

6 6 ACTA MATHEMATICA SCIENTIA Vol. Ser.B Theorem (Composite formula for higher order singular integrals) Suppose that φ(ξ) is holomorphic in a neighbourhood of, that is φ(ξ) = 0 in U(), then the composite ξ formula K(ζ, η) φ(ξ) η k ζ k η ξ η ξ K(η, ξ) = φ(ζ) (3) 4 ζ k holds. In the notation Sφ = φ(ξ)k(η, ξ) (4) ξ S φ = φ(ξ)k (η, ξ), K (η, ξ) = η k ξ k K(η, ξ), (5) xi η ξ the composite formula can be written as SS φ = 4 K(ζ, η) φ(ξ)k (η, ξ) = φ(ζ). (6) η ξ n ζ k Proof The assumption φ(ζ) = 0 in U() implies that φ(ζ) can be holomorphically ζ extended into D. Hence the composite formula () for φ(ζ) holds. Owing to Definition we have K(ζ, η) φ(ξ) η k ξ k K(η, ξ) η η ξ ξ = K(ζ, η) η n Cndφ(ξ) j= ( )j ( ) k (ξ j η j )dξ [dξ k dξ ndξ [dξ k dξ n η ξ n = φ(ξ) j= K(ζ, η) C ( )j (ξ j η j )d ξ dξ n dξ [dξ k dξ n n n η ξ k η ξ n ξ = φ(ξ) K(ζ, η) K(η, ξ) = φ(ζ). n η ξ k 4n ζ k ξ ξ The proof is complete. Remark Assume that D is a complex ball and φ = P s,t is a homogeneous harmonic polynomial of degree s with respect to z, and degree t with respect to z. Then for n >, the composite formula () in the case of the Cauchy singular integral cannot hold, unless when P s,t is a holomorphic polynomial, i.e. when t = 0. So in Lemma we assume that φ(ζ) C () (), and that it can be holomorphically extended into D. Likewise, in Theorem we assume that φ(ξ) is holomorphic in a neighbourhood of. It is a sufficient condition for the composite formula (3) to hold. Recall that S φ = φ(ξ)k (η, ξ) = ψ(η), ξ ζ k K (η, ξ) = η k ξ k K(η, ξ). (7) η ξ

7 No. Qian & Zhong: DIFFERENTIAL INTEGRAL EQUATIONS ON MANIFOLDS 7 Applying the operator to both sides of (7), we have Sφ = φ(ξ)k(η, ξ) ξ SS φ = φ(ζ) = Sψ = n ζ k η ψ(η)k(ζ, η), (8) Under suitable boundary conditions we can uniquely solve the above partial differential equation for φ. 4 Higher Order Singular Integral Equations and Partial Differential Integral Equations In this section we discuss the linear space L which consists of complex value differentiable functions whose partial derivatives satisfy the Hölder condition on. In fact, the higher order Bochner-Martinelli singular integrals induce linear operators on L. In the following we solve the higher order singular integral equations with Bochner-Martinelli kernel. We consider the higher order singular integral equation asφ + bs φ + Tφ = ψ, (9) where a, b are complex constants, ψ L is a given function, Tφ = ξ φ(ξ)l(η, ξ), where the kernel L(η, ξ) is a complex exterior differential form of degree n and L(η, ξ) H, i.e. the first order partial devivatives of L(η, ξ) with respect to η and ξ satisfy the Hölder condition on. Firstly, we consider the characteristic equation of (9) asφ + bs φ = ψ. (0) Indeed, by Definition, it is a differential and singular integral equation. Applying the operator where M = ai bs, () Iφ = φ (φ L), () to both sides of (0), and utilizing the composite formula (6), we have aφ + b φ(ζ) = Sψ. (3) n ζ k Solving this partial differential equation under suitable boundary value conditions we can obtain the unknown function φ uniquely. The general equation (9) can be reduced to a equivalent Fredholm equation by using the operator M. In fact, applying the operator M to equation (9) and using composite formula (6), we have aφ + b φ(ζ) + STφ = Sψ. (4) n ζ k

8 8 ACTA MATHEMATICA SCIENTIA Vol. Ser.B Applying the theorem of exchanging order of integration for singular integral and ordinary integral ([ Th.4.5.), we have STφ = K(ζ, η) φ(ξ)l(η, ξ) η ξ = φ(ξ) L(η, ξ)k(ζ, η). ξ η We can then prove that η L(η, ξ)k(ζ, η) satisfies the Hölder condition with respect to ξ and ζ ([ p.9). Therefore, (4) is a Fredholm equation. Equation (4) and (9) are equivalent. To show this we need only prove that the solution of (4) satisfies equation (9). In fact, applying the operator M = ai + bs (5) to both sides of (4) and use composite formula (6) we obtain (9) immediately. To summarise, we have Theorem 3 Suppose that in equation (9), a, b are complex constants, ψ L, and the kernel L(η, ξ) of operator T belongs to H, then (i) The solution of the characteristic equation (0) of (9) can be obtained uniquely in L by solving the partial differential equation (3) under suitable boundary value conditions. (ii) In L, the higher order singular integral equation (9) is equivalent to the Fredholm equation (4). Acknowledgment The second author wishes to express his sincere gratitude to the University of New England and Professor Tao Qian for the hospitality and good working conditions provided during his stay in Armidale in August 998. References Lu Qi-keng, Zhong Tongde. An Extension of the Privalov theorem (in Chinese). Acta Math Sinica, 957,7: Zhong Tongde. Integral Representation of Functions of Several Complex Variables and Multidimensional Singular Integral Equations (in Chinese). Xiamen: Xiamen University Press Zhong Tongde. Singular integrals and integral representations in several complex variables. Contemporary Mathematics. The American Mathematical Society, 993,4: Kytmanov A M. Bochner-Mortinelli Integral and Its Applications (in Russian). Siberia: Science Press, 99 5 Wang Yuan, et al ed. Encyclopaedia of Mathematical Sciences (in Chinese). Beijing: Science Press, 994,: Hadamard J. Lectures on Cauchy s Problem in Linear Partial Differential Equations. New York, 95 7 Fox C. A Generalization of the Cauchy principal value. Canadian J Math, 9: Wang Xiaogin. Singular integrals and analyticity theorems in several complex variables. Doctoral Dissertation, Sweden: Uppsala University, Tao Qian, Zhong Tongde. Transformation formula of higher order singular integral on complex hypershere. Accepted to appear in Journal of the Australian Mathematical Society 0 Zhong Tongde. Singular integral equations on Stein manifolds. Journal of Xiamen University (Natural Science), 99,30(3): 3-34