On quaternion Cauchy-Szeg kernel and related boundary value problem - PowerPoint PPT Presentation

On quaternion Cauchy-Szegö kernel and related boundary value problem Irina Markina, Der Chen Chang, Wei Wang University of Bergen, Norway Georgetown University, Washington D.C, USA Zhejiang University, PR China On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 1/39

Upper half plain and Cauchy kernel y = Im z U = R × R + ⊂ C x = Re z Any holomorphic function in U , continuous up to the boundary bU , is defined by the values at the boundary b U . � + ∞ f b ( t ) 1 f ( z ) = t − zdt. 2 πi −∞ On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 2/39

Siegel u.h.sp and Cauchy-Szegö kernel Im z 1 U ⊂ C 2 i R 3 ∼ = Re z 1 × C U = { ( z 1 , z 2 ) ∈ C 2 | Im z 1 > | z 2 | 2 } � | F ( z + iǫ ) | 2 dµ ( z ) F ∈ H 2 ( U ) : holomorphic, � F 2 � H 2 = sup ε> 0 b U ε → 0 F ( z + iǫ ) = F b exists in L 2 ( b U ) for z ∈ b U ; • lim • Set B ( b U ) of limits F b ’s is a closed subset of L 2 ( b U ) ; • � F 2 � H 2 ( U ) = � F b � L 2 ( b U ) . On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 3/39

Siegel u.h.sp and Cauchy-Szegö kernel U = { ( z 1 , z 2 ) ∈ C 2 | Im z 1 > | z 2 | 2 } � | F ( z + iǫ ) | 2 dµ ( z ) F ∈ H 2 ( U ) : holomorphic, � F 2 � H 2 = sup ε> 0 b U Function F from the Hardy space H 2 ( U ) is completely defined by its boundary values F b . � F b ( w ) S ( z, w ) dw, F ( z ) = z ∈ U b U • z �→ S ( z, w ) is holomorphic in U for all w ∈ U • S ( z, w ) = S ( w, z ) � � − 2 1 i • S ( z, w ) = 2 ( ¯ w 2 − z 2 ) − z 1 ¯ w 1 4 π 2 On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 4/39

Projection operator The projection operator C : L 2 ( b U ) → B ( b U ) assoiates L 2 ( b U ) ∋ f �→ Cf = F b for some F ∈ H 2 ( U ) by making use of the Cauchy-Szegö kernel we obtain � Cf ( z ) = lim S ( z + εi, w ) f ( w ) dβ ( w ) ε → 0 b U by limit in L 2 ( b U ) for any z ∈ b U . S TEIN , E. M. , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals ,1993. Goal: to construct analogues of S ( z, w ) for the quaternion Siegel u.h.sp U On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 5/39

Space of quatenions Q q = ( x 1 + i x 2 + j x 3 + k x 4 ) = ( x 1 , − → u ) ∈ R 4 σ = ( y 1 + i y 2 + j y 3 + k y 4 ) = ( y 1 , − → v ) ∈ R 4 q + σ = ( x 1 + y 1 , − → u + − → v ) , qσ = ( x 1 y 1 − − → u · − → v , x 1 − → v + y 1 − → u + − → u × − → v ) q = ( x 1 , −− → | q | 2 = q ¯ ¯ u ) , q, qσ = ¯ σ ¯ q q ¯ q − 1 = | q | 2 Space Q is a normed division algebra On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 6/39

Right quaternion space Let V be a right vector space over Q V × Q → V, where ( v, σ ) �→ vσ. Let � v 1 , v 2 � be an Hermitian product on V σ ∈ Q � v 1 σ, v 2 � = ¯ σ � v 1 , v 2 � , � v 1 , v 2 σ � = � v 1 , v 2 � σ, ( V, � v � ) is a Hilbert space w.r.t. � v � = � v, v � 1 / 2 for any L ( v ) = � v L , v � , v ∈ V bounded right linear functional: L ( vσ ) = L ( v ) σ On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 7/39

Siegel upper half space in Q n n � V = Q n = ( q 1 , . . . , q n ) = q, Let � p, q � = p l q l ¯ l =1 U = { ( q 1 , q 2 , . . . , q n ) = ( q 1 , q ′ ) ∈ Q n | Re q 1 > | q ′ | 2 } The space U is invariant under transformations. • Translation: → � q 1 + p 1 + 2 � p ′ , q ′ � , q ′ + p ′ � τ p : ( q 1 , q ′ ) �− , for p = ( p 1 , p ′ ) ∈ b U . On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 8/39

Siegel upper half space in Q n U = { ( q 1 , q ′ ) ∈ Q n | Re q 1 > | q ′ | 2 } • Rotations: R a : ( q 1 , q ′ ) �− → ( q 1 , a q ′ ) for a ∈ Sp( n − 1) R σ : ( q 1 , q ′ ) �− → ( σq 1 σ, q ′ σ ) for σ ∈ Q , | σ | = 1 • Dilations: δ r : ( q 1 , q ′ ) �− → ( r 2 q 1 , rq ′ ) , r > 0 . On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 9/39

Holomorphic functions in C f : Ω ⊂ C → C , f = u + iv , z = x + iy is holomorphic if f ( z 0 + h ) − f ( z 0 ) f ′ ( z 0 ) = lim or h h → 0 ¯ or ∂ z f = 0 ← → ∂ x u = ∂ y v, ∂ y u = − ∂ x v ∞ � a n ( z − z 0 ) n f ( z ) = n =0 On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 10/39

Regular functions in Q Let f : Ω ⊂ Q → Q , if we require that h → 0 h − 1 � f ( q 0 + h ) − f ( q 0 ) � lim exists for all q ∈ Q then f ( q ) = aq + b is an affine function On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 11/39

Regular functions in Q Let f : Ω ⊂ Q → Q , if we require that ∞ � P n ( q − q 0 ) n f ( q ) = n =0 exists for all q 0 ∈ Q with some polynomials P n then it equivalent to requirement that f ( q ) is analytic with respect of 4 real variables, defined by q . On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 12/39

Regular functions in Q Let f : Ω ⊂ Q → Q . A C 1 (Ω) function f = f 1 + i f 2 + j f 3 + k f 4 is (left) regular if it satisfies Cauchy-Riemann-Fueter equations ¯ for all ∂ q f ( q ) = 0 , q ∈ Ω , where q = x 0 + i x 1 + j x 2 + k x 3 = z 0 + z 1 j, ∂ q = ∂ x 0 + i ∂ x 1 + j ∂ x 2 + k ∂ x 3 = ¯ ¯ ∂ z 0 + ¯ ∂ z 1 j On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 13/39

Regular functions in U The space of regular functions on U is invariant under mentioned above transformations of U . Namely, if f is regular on U , then f ( τ p ( · )) , p ∈ b U ; σf ( R σ ( · )) , σ ∈ Q , | σ | = 1 , f ( R a ( · )) , a ∈ Sp( n − 1); and f ( δ r ( · )) are all regular on U . On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 14/39

Hardy space H 2 ( U ) Re q 1 U ⊂ Q n b U + ε e e Q n − 1 × Im Q 1 � L 2 ( B U ) , � f, g � L 2 = f ( q ) g ( q ) dβ ( q ) . b U Hardy space H 2 ( U ) is all regular functions in U � | F ( q + ε e ) | 2 dβ ( q ) < ∞ � F � 2 H 2 ( U ) = sup ε> 0 b U It is right quaternion Hilbert space, invariant under the above mentioned transformations On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 15/39

Boundary values THEOREM 1. Suppose that F ∈ H ( b U ) . Then • There exists lim ε → 0 F ( q + ε e ) = F b ( q ) in L 2 ( b U ) • � F b � L 2 ( b U ) = � F � H 2 ( U ) , • The space B ( b U ) of all boundary values is a closed subspace of L 2 ( b U ) . • C ( f )( q ) = � QH S ( h − 1 ∗ q ) f ( h ) d ( h ) , q, h ∈ QH where we used the identification b U ∼ = QH . On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 16/39

Siegel upper half plain in C x = Re z U = R 2 + ⊂ C y = Im z = b U ( R , +) : U → U ( a, b ) �→ ( a, b + r ) Identification ( R , +) ∼ = Im z = ( b U ) by the action at (0 , 0) : ( R , +) ∋ r �→ r ∈ b U On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 17/39

Siegel upper half space in C 2 Re z 1 U ⊂ C 2 R 3 ∼ = Im z 1 × C U = { ( z 1 , z 2 ) ∈ C 2 | Re z 1 > | z 2 | 2 } H 1 = { ( t, ω ) ∈ R × C with ∗} ( H 1 , ∗ ) : U → U ( z 1 + | ω | 2 + it + 2 � ω, z 2 � , z 2 + ω ) ( z 1 , z 2 ) �→ Identification ( H 1 , ∗ ) ∼ = ( b U ) : H 1 ∋ ( t, ω ) �→ ( | ω | 2 + it, ω ) ∈ b U On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 18/39

Quaternion Heisenberg group QH = ( R 3 × R 4 ∼ = Im Q × Q , ∗ ) ( x, v ) ∗ ( y, w ) = ( x + y + 2 Im � v, w � , v + w ) ( x,v ) b U ∋ ( q 1 , q ′ ) �→ ( q 1 + | v | 2 + ix 1 + jx 2 + kx 3 + 2 � v, q ′ � , q ′ + v ) ∈ b U = R 3 × R 4 b U ∼ = QH ∼ via action on (0 , 0) ∈ b U On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 19/39

The Cauchy-Szegö kernel S ( q, p ): U × U → Q is unique function satisfying 1. S ( · , p ) ∈ H 2 ( U ) for each p ∈ U . S b ( q, p ) is defined for each p ∈ U and for a.a. q ∈ b U . 2. The kernel S is symmetric: S ( q, p ) = S ( p, q ) . S b ( q, p ) is defined for each q ∈ U and a.a. p ∈ b U . 3. The kernel S satisfies the reproducing property for F ∈ H 2 ( U ) � S b ( q, Q ) F b ( Q ) dβ ( Q ) , F ( q ) = q ∈ U b U On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 20/39

Symmetries of the kernel S ( τ Q ( q ) , τ Q ( p )) = S ( q, p ) , S ( R a ( q ) , R a ( p )) = S ( q, p ) , σS ( R σ ( q ) , R σ ( p )) σ = S ( q, p ) , S ( δ r ( q ) , δ r ( p )) r 4 n +6 = S ( q, p ) . for q, p ∈ U , Q ∈ b U , a ∈ Sp( n − 1) , σ ∈ Q , | σ | = 1 , and r > 0 . On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 21/39

Main theorem The Cauchy-Szegö kernel is given by � � n � p ′ k q ′ S ( q, p ) = s q 1 + p 1 − 2 ¯ , p, q ∈ U k k =2 where ∂ 2 n φ s ( φ ) = c n | φ | 4 , φ = x 1 + x 2 i + x 3 j + x 4 k ∈ Q . ∂x 2 n 1 On quaternion Cauchy-Szeg¨ o kernel and related boundary value problem – p. 22/39