ß¼ 0384 9200852727 UDC Î ± À» An Integral Equation Problem With Shift of Several Complex Variables Û Ò ÖÞ Ô ²» Ý Õ Ø ³ÇÀ ¼ 2 0 º 4 Ñ ³ÇÙÐ 2 0 º Ñ Ä ¼ 2 0 º Ñ ÄÞ Ê Ã Ö 20 5 Â
Å ¾ º ½ É É Ç ¹ ¹Ý É ½ ÚÓÉ ÅÞ ¹Ý µ È É Æ ½ Ï Åµ ÀÊ ÀÊÉ Á Æ Ë
Å ¾ Æ É Çº Ø» ½ Û Å Å»± Ü Ù ÉÅ ÔÑ ½ ÊÎ Ã Å ½ µ ¹ ͱ ½ Í ß Ú Å ½ Ì Í ØÐ Ù Å ½ Ý Â Đ ½ Ð ÛÅ ½ Æ Ð Å 2 ³ Æ Â µ Þ Æ ÅÁ Ë Ë Á Ë Ë
¼ Á.... III Á.... IV ¼.... À Ã Ç ß ¼²....3 À Ã Ó W,S,WS... 5 2. Ò W,S,WS...5 2.2 Ò W,S,WS ÕÆ...5 ÀÎÃ Đ± ¼ É ¼ Ö ¼. 3..... 3.2» È».... 5 3.3» Õ»«È» Ñ. 7 3.4 Û» Ã.... 2 µ «...25 Ë....26 I
Contents Chinese Abstract... III English Abstract...IV Introduction...... Chapter. Preliminaries and basic lemma... 3 Chapter 2. The operators W,S,WS...5 2. The operators W,S,WS... 5 2.2 Estimates of the operators W,S,WS...5 Chapter 3. Singular integral equation with shift, corresponding equation set and adjoint equation.... 3. Basic definitions... 3.2 Relation of the solvability between the singular integral equation with shift and his corresponding equation set... 5 3.3 Relationship for the numbers of solutions of singular integral equation with shift,corresponding equation set and adjoint equation......7 3.4 Existence of solution for a concrete singular integral equation......2 References.......25 Acknowledgements....26 II
Á [] ÑÄ Ü Ö ¹Ë » [2-5] Å ¹Ë » ÆĐ ÈÊĐ Å Ö ¹Ë Â Æ ¹Ë ÂÍ Î ¼ «¾  À ÆÙ ²ÍÙ ± Ñ Å Ö ¹Ë  ÎÙ Æ Ì Ü Ìß Ü ĐÛ Ðµ Û µå Ù ²Í Û È» Ü Ì ÕÜ ¹Ë ½ ½ ¹Ë ½ Ò ÄÆ H µ (Γ) Ö ½ W,S,WS ±» ÌÖ Ñ ÈÊ Ü Ì ß ¹ ¹Ö Ü Ö Ñ ¹Ë ÂÍÎ ¼ «¾ ÂÙ ÀÆ Ñ Å Ö Ü Ö ¹Ë  Π¼ «ÆÙ Ñ ÔÑ ¹ Ë ÂÆÙ Æ º À Ü Ö²É ¹Ë  Π¾ ÂÍ ¼ «٠± ¾Ñ Ù Ã ¾Ñ Ü¹Ö Å Ö ¹Ë  ÎÙ Æ Å ¹Ë  ½ III
ABSTRACT For one complex variable case, singular integral equation and boundary value problem with shift had been introduced in[], while for several complex variables case, been studied in[2-5]. Based on it, we discuss singular integral equation with shift in several complex variables and generalizes the definitions of singular integral equation with shift, corresponding equation set and adjoint equation. Moreover we study the relation of the solvability and number of solutions among them, and prove existence of solution for a concrete singular integral equation. The paper is organized as follows: In chapter, we introduce some definitions and natations,including generalized polydisc,cauchy integral formulas on the generalized polydisc domains. Also some basic lemmas has been included. In chapter 2, we calculate the norm estimates of the operator W,S and WS in H µ (Γ). In chapter 3, we get the main results. The generalized definitions of singular integral equation with shift,corresponding equation set and adjoint equation are introduced in section. For section 2, we study the relation of the solvability between the singular integral equation with shift and the corresponding equation set. And then we obtain a relationship for the numbers of solutions of the singular integral equation with shift, corresponding equation set and adjoint equation. Finally, we prove existence of solution for a concrete singular integral equation. operators. Keywords: several complex variables; singular integral equation;shift; IV
¾ÔÕ ÕÏ Ê À Í ³ º ¾ ¼ » Æ ÝĐ ² Ë ÆÒ Ö µ ¼ Đ ÔÑ Ð Ë Û Þ Å «Ç - ² ÆÏ Â» ²ÍÉ Ù ÇÆ ÈÌ Hilbert Haseman ²Í Carleman»» ± A. Ƶ Ë È µ Ü Å Ì ¼«Û Ë [] ÑÄ Ü Ö ¹Ë » ß «Å ¹Ë  [2]» [3,4] Ô Ë ¹Ë  [5,6] ÕÞ Å Ö ¹Ë Â Æ ¹Ë ÂÍÎ ¼ «¾  À Æ Ù ²ÍÙ ± Ñ Â Å Ö ¹Ë  ÎÙ Æ Ì Ü Ìß Ü ĐÛ Ðµ Û µå µå Đ Cauchy Ë ¹Ë ½ ½ ²Í Û È» Ü ÌÆ H µ (Γ) Ö ½ W,S,WS ±» ÌÖ Ñ ÈÊ Ü Ì ß ¹ ¹Ö Ü Ö Ñ ¹Ë ÂÍÎ ¼ «¾ ÂÙ ÀÆ Ñ Å Ö Ü Ö ¹Ë ÂÆÙ Æ º À Î ¼ «ÆÙ Ü Ö²É ¹Ë  Π¾ ÂÍ ¼ «٠± n, n 2, n ¾Ñ
¾ÔÕ ÕÏ Ê À 2 n = n + n 2 Ü¹Ö Å Ö ¹Ë  ÎÙ Æ a(ζ)ϕ(ζ) + b(ζ)wϕ(ζ) + c(ζ)sϕ(ζ) + e(ζ)wsϕ(ζ) = g(ζ)f(ζ, ϕ(ζ), Wϕ(ζ), Sϕ(ζ), WSϕ(ζ)) ()
¾ÔÕ ÕÏ Ê À 3 Á Ä È ¾ ÕÜ Đе ²Í Û È» Ä. [7] D k ± z k (k =, 2, n) É Û z, z 2,, z n ¾Û Æ D, D 2,, D n ĐÆ ±«(z, z 2,, z n ) Ð º C n Ö µå ² (D, D 2,, D n ) Ï (D, D 2,, D n ) = D D 2 D n Đ µå (D, D 2,, D n ) n É D, D 2,, D n ÈË Ë C 2 Ö ² D = D D 2 D k»û Γ k ¹ Ù (k =, 2) Γ = Γ Γ 2 Ä.2 [2] ± ϕ(ζ) Æ Γ Đ Hölder À Û ξ, ζ Γ ϕ(ζ) ϕ(ξ) C ζ ξ µ, ÎÖ 0 < µ < C ± ζ ξ = ( 2 ζ k ξ k ) 2. Û µ = Æ ϕ(ζ) Æ Γ Đ Lipschitz À H µ (Γ) Æ Γ Đ ± µ Hölder À ±º е H µ (Γ) Ö ± k= ϕ(ζ) ϕ(ξ) ϕ µ = max ϕ(ζ) + sup, ζ Γ ζ ξ µ É H µ (Γ) º Banach ζ,ξ Γ ϕ + ϕ 2 µ ϕ + ϕ 2, ϕ ϕ 2 µ C ϕ ϕ 2 (.) ÎÖ C ± ± еÆÔ ϕ(ζ) ϕ(ξ) ϕ µ ζ ξ µ. Ä.3 [] L ¹ Ù d(z) Ù L Ñ Â à  à d(z) : L L Ô ± d(z) Ö ÃÇÎ
¾ÔÕ ÕÏ Ê À 4 ±Å Ù L Ý ÇÎ Ú Ý» ¹Ë Â Ò Ò Ñ α k Γ k (k =, 2) Ñ ζ, ξ Γ Lipschitz À α k (ζ k ) α k (ξ k ) M k ζ k ξ k, ÎÖ k =, 2; M k ± α(ζ) = (α (ζ ), α 2 (ζ 2 )), É α(ζ) Lipschitz À Đ α(ζ) α(ξ) = (α (ζ ), α 2 (ζ 2 )) (α (ξ ), α 2 (ξ 2 )) = [ α (ζ ) α (ξ ) 2 + α 2 (ζ 2 ) α 2 (ξ 2 ) 2 ] 2 [M 2 ζ ξ 2 + M 2 2 ζ 2 ξ 2 2 ] /2 [(max{m, M 2 }) 2 ( ζ ξ 2 + ζ 2 ξ 2 2 )] 2 = max{m, M 2 } ζ ξ C 2 ζ ξ (.2) [7] [8] ½³. D = (D, D 2 ) µå Î»Û Γ k (k =, 2) ØÅ ÚÒ Æ ¹ Ù«f(z) Æ (D, D 2 ) ÐÏ ÆÎ»Û É z D D 2 µå Cauchy f(z) = f(ζ, ζ 2 ) (2πi) 2 Γ Γ 2 (ζ z )(ζ 2 z 2 ) dζ dζ 2. ½³.2 ϕ(ζ) Æ Γ Đ É Cauchy Ë Φ(z) = ϕ(ζ, ζ 2 ) (2πi) 2 Γ Γ 2 (ζ z )(ζ 2 z 2 ) dζ dζ 2, (z k / Γ k ) z, z 2 Ù ± Φ(z, ) = Φ(, z 2 ) = Φ(, ) = 0. ºÉ [9] 2πi Γ k dζ k (ζ k ξ k ) = 2.(ξ k Γ k, k =, 2) (.3) ½³.3 [0] ¹ «Ï Ñ Ú ÎÑ
¾ÔÕ ÕÏ Ê À 5 Á Ä Ô W,S,WS ÌÑ Ü ß Í ¹Ë ½ ½ ¹Ë ½ Đ½ ±» ÒÈÊ 2. Ó W,S,WS Cauchy Ë ϕ(ξ, ξ 2 ) Φ(z) = (2πi) 2 (ξ z )(ξ 2 z 2 ) dξ dξ 2, z = (z, z 2 ) / Γ. ¹Ë ½ (Sϕ)(ζ) = Γ (πi) 2 Γ ϕ(ξ, ξ 2 ) (ξ ζ )(ξ 2 ζ 2 ) dξ dξ 2, ζ = (ζ, ζ 2 ) Γ. Ë Æ Cauchy ß µò Ù ± ϕ(ξ, ξ 2 ) H µ (Γ Γ 2 ).»Ü ½ (Wϕ)(ζ) = ϕ[α(ζ)] = ϕ(α (ζ ), α 2 (ζ 2 )), (WSϕ)(ζ) = ϕ(ξ, ξ 2 ) (πi) 2 [ξ α (ζ )][ξ 2 α 2 (ζ 2 )] dξ dξ 2. 2.2 Ó W,S,WS Ö Γ Ò Æ H µ (Γ) Ö ½ W,S,WS ±» Ü 2. α k (ζ):γ k Γ k (k =, 2) Lipschitz À ϕ H µ (Γ), É Æ ϕ ¾ ± C 3 Ô Æ ± е Wϕ µ = max ζ Γ Wϕ µ C 3 ϕ µ. (2.) Wϕ(ζ) Wϕ(ξ) Wϕ(ζ) + sup, ζ,ξ Γ ζ ξ µ
º α(ζ) ½ ² Ï ¾ÔÕ ÕÏ Ê À 6 max Wϕ(ζ) = max ϕ(α (ζ ), α 2 (ζ 2 )) = max ϕ(ζ) ζ Γ ζ Γ ζ Γ ϕ[α(ζ)] ϕ[α(ξ)] α(ζ) α(ξ) µ ϕ(ζ) ϕ(ξ) sup ζ,ξ Γ ζ ξ µ ϕ(ζ) ϕ(ξ) ϕ[α(ζ)] ϕ[α(ξ)] sup α(ζ) α(ξ) µ ζ ξ µ ζ,ξ Γ ² (.2) ξ, ζ Γ ² «Wϕ(ζ) Wϕ(ξ) = ϕ[α(ζ)] ϕ[α(ξ)] ϕ(ζ) ϕ(ξ) sup α(ζ) α(ξ) µ ζ ξ µ ζ,ξ Γ ϕ(ζ) ϕ(ξ) sup (C ζ ξ µ 2 ζ ξ ) µ ζ,ξ Γ Wϕ(ζ) Wϕ(ξ) ϕ(ζ) ϕ(ξ) sup sup C µ ζ ξ µ ζ ξ µ 2. ζ,ξ Γ ζ,ξ Γ Wϕ µ max ϕ(ζ) + ζ Γ Cµ 2 sup ϕ(ζ) ϕ(ξ) ζ ξ µ ζ,ξ Γ max{, C µ 2 }[max ϕ(ζ) ϕ(ξ) ϕ(ζ) + sup ] ζ Γ ζ ξ µ = max{, C µ 2 } ϕ µ C 3 ϕ µ ζ,ξ Γ
¾ÔÕ ÕÏ Ê À 7 Ü 2.2 [7] ϕ H µ (Γ), É Æ ϕ ¾ ± C 4 Ô Æ ± е Sϕ µ = max ζ Γ Ò Õ» Sϕ(ζ) Sϕ(η) ζ η µ Sϕ µ C 4 ϕ µ. (2.2) Sϕ(ζ) Sϕ(η) Sϕ(ζ) + sup, ζ,η Γ ζ η µ ζ η. ζ, η Γ, t = ζ η = ( ζ η 2 + ζ 2 η 2 2 ) 2 (t 2 + t 2 2) 2 ² ζ Ö ζ 3t O(ζ, 3t), Γ k O(ζ, 3t) Γ k0 Å Γ k.(k =, 2). º Γ Γ 2 = Γ (Γ 20 +Γ 2 ) = Γ Γ 20 +Γ Γ 2 = Γ Γ 20 +(Γ 0 +Γ ) Γ 2 = Γ Γ 20 + Γ 0 Γ 2 + Γ Γ 2. ² Sϕ(ζ) Sϕ(η) = (πi) 2( Γ Γ 20 + ϕ(ξ)dξ dξ 2 (ξ η )(ξ 2 η) ] Γ 0 Γ 2 + I Γ Γ 20 + I Γ0 Γ 2 + I Γ Γ 2. Γ Γ 2 )[ ϕ(ξ)dξ dξ 2 (ξ ζ )(ξ 2 ζ 2 ) «ϕ(ξ, ξ 2 )dξ dξ 2 I Γ Γ 20 = (πi) 2 Γ Γ 20 (ξ ζ )(ξ 2 ζ 2 ) ϕ(ξ, ξ 2 )dξ dξ 2 (πi) 2 Γ Γ 20 (ξ η )(ξ 2 η 2 ) ϕ(ξ, ξ 2 ) ϕ(ξ, ζ 2 ) = (πi) 2 Γ Γ 20 (ξ ζ )(ξ 2 ζ 2 ) dξ dξ 2 ϕ(ξ, ξ 2 ) ϕ(ξ, η 2 ) (πi) 2 Γ Γ 20 (ξ η )(ξ 2 η 2 ) dξ dξ 2 + ϕ(ξ, ζ 2 ) (πi) 2 Γ Γ 20 (ξ ζ )(ξ 2 ζ 2 ) dξ dξ 2 ϕ(ξ, η 2 ) (πi) 2 Γ Γ 20 (ξ η )(ξ 2 η 2 ) dξ dξ 2
(i) ± ÐµÆ ¾ÔÕ ÕÏ Ê À 8 ϕ(ξ, ξ 2 ) ϕ(ξ, ζ 2 ) (πi) 2 Γ Γ 20 (ξ ζ )(ξ 2 ζ 2 ) dξ dξ 2 ξ 2 ζ 2 ϕ µ (πi) Γ Γ20 µ 2 (ξ ζ )(ξ 2 ζ 2 ) dξ dξ 2 (2.3) º Γ 20 Γ 2 O(ζ, 3t) ² ξ 2 ζ 2 3t 2. «(ii) ± ÐµÆ (2.3) ϕ µ π 3t2 0 ξ 2 ζ 2 µ dξ 2 ϕ(ξ, ξ 2 ) ϕ(ξ, η 2 ) (πi) 2 Γ Γ 20 (ξ η )(ξ 2 η 2 ) dξ dξ 2 ξ 2 η 2 ϕ µ (πi) Γ Γ20 µ dξ 2 dξ 2 (2.4) ξ η º ξ 2 η 2 ξ 2 ζ 2 + ζ 2 η 2 3t 2 + t 2 = 4t 2. ² (.3) ÆÔ (2.4) ϕ µ π (iii) ϕ(ξ, ζ 2 ) ξ 2 ¾ ²  = = = 4t2 0 ξ 2 η 2 µ dξ 2 ϕ(ξ, ζ 2 ) (πi) 2 Γ Γ 20 (ξ ζ )(ξ 2 ζ 2 ) dξ dξ 2 ϕ(ξ, ζ 2 ) dξ 2 dξ πi Γ ξ ζ πi Γ 20 ξ 2 ζ 2 ϕ(ξ, ζ 2 ) dξ πi Γ ξ ζ ϕ(ξ, η 2 ) (πi) 2 Γ Γ 20 (ξ η )(ξ 2 η 2 ) dξ dξ 2 ϕ(ξ, η 2 ) dξ πi Γ ξ η
¾ÔÕ ÕÏ Ê À 9 (i)(ii)(iii) Æ 3t2 4t2 I Γ Γ 20 ϕ µ ξ 2 ζ 2 µ dξ 2 + ϕ µ ξ 2 η 2 µ dξ 2 π 0 π 0 + ϕ(ξ, η 2 ) dξ ϕ(ξ, η 2 ) dξ πi Γ ξ η πi Γ ξ η N ϕ µ t µ Â I Γ0 Γ 2 N 2 ϕ µ t µ I Γ Γ 2 = [ (πi) 2 Γ Γ 2 (ξ ζ )(ξ 2 ζ 2 ) (ξ η )(ξ 2 η 2 ) ]ϕ(ξ, ξ 2 )dξ dξ 2 ϕ(ξ, ξ 2 ) = (πi) 2 Γ Γ 2 (ξ ζ )(ξ 2 ζ 2 )(ξ η )(ξ 2 η 2 ) [(ξ η )(ξ 2 η 2 ) (ξ ζ )(ξ 2 ζ 2 )]dξ dξ 2 ϕ(ξ, ξ 2 ) = (πi) 2 Γ Γ 2 (ξ ζ )(ξ 2 ζ 2 )(ξ η )(ξ 2 η 2 ) {[(ξ ζ ) + (ζ η )] [(ξ 2 ζ 2 ) + (ζ 2 η 2 )] (ξ ζ )(ξ 2 ζ 2 )}dξ dξ 2 ϕ(ξ, ξ 2 ) = (πi) 2 Γ Γ 2 (ξ ζ )(ξ 2 ζ 2 )(ξ η )(ξ 2 η 2 ) [(ζ η )(ξ 2 ζ 2 ) +(ξ ζ )(ζ 2 η 2 ) + (ζ η )(ζ 2 η 2 )]dξ dξ 2 ϕ(ξ, ξ 2 ) = (πi) 2 Γ Γ 2 (ξ η )(ξ 2 η 2 ) [ζ η + ζ 2 η 2 ξ ζ ξ 2 ζ 2 + (ζ η )(ζ 2 η 2 ) (ξ ζ )(ξ 2 ζ 2 ) ]dξ dξ 2 dξ dξ 2 ϕ µ t Γ Γ 2 ξ η 2 ξ 2 η 2 + ϕ dξ dξ 2 µt 2 Γ Γ 2 ξ η ξ 2 η 2 2 dξ dξ 2 + ϕ µ t t 2 Γ Γ 2 ξ η 2 ξ 2 η 2 2 N 3 ϕ µ t µ ÎÖ Ù º Æ Γ k Đ ξ k η k 0, «ξ k η k Ë Û k =, 2.
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