Analysis, et Cetera. Research Papers Published in Honor of by Paul H. Rabinowitz, Eduard Zehnder

By Paul H. Rabinowitz, Eduard Zehnder

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1). We also note that E(x,y]-k) = (-l)nE(x,y;k). 4) Next introduce the operator E+(k) : B(Rn) for any k E C + \ { 0 } by —> S * ( R n ) , defined £ + ( * ) = \(2nrk»-*T+(k)*JT+(k). 5) Here T+(&) : B(Rn) —> Χ 2 ( 5 η _ 1 ) is the trace operator introduced in Section 4, and J is the unitary operator in L 2 ( S n _ 1 ) defined by J : φ(ω) —► φ(—ω). 1. The operator E+(k) (any k G C + \ { 0 } ; . (Rn), E+(k)f= f n has the following eik^E(x,y;k)eik^f(y)dy. 6) B(Rn), WE^^fWß^^-'lkr'WfWB. 1). We also have (iv) The function C + \ { 0 } 3 k -► E+(k) e is analytic in C + and is weakly* continuous C{B,B*) in C + \ { 0 } .

3. 1) for any k G C + \ { 0 } . We start with some general remarks. Given two Banach spaces X and y , we denote by C(X,Y) the Banach space of continuous linear operators T : X -> Y. We write C(X) for C(X,X). 6), is an operator-valued function with values in C(L2ÇRn)). It is analytic in C + and admits no continuous extension to any point on the real axis. However, it is well known t h a t if one modifies the definition of G(k), considering it as operatorvalued function with values in £ ( L 2 , 3 ( R n ) , L 2 ' ~ s ( R n ) ) , then for any s > 1/2, G(k) admits continuous boundary values on R \ { 0 } .

2 2 —7773(777? + m ? ) L(L>+ + D-) = {1,7711,7772,7713,73,771! + 7772, + 777173,7713(771271 - 777272) + 27273} providing the embedding of T2 into P 7 . The vector fields XQ1 and XQ4 are the holomorphic vector fields ofT2 and each of them is doubly tangent to D+ and D- at two points tf1 on each curve as depicted in Figure 4. R e m a r k . If one sets L(D+ + D _ ) = { y 0 , . . }, then the birational maps specified in the coherence condition are of the simple form T : (777,7) Π ( — , . . , — \Vi Vi 35 PAINLEVÉ SOLUTIONS C Γ2 C Figure 4 D+ + P D- with yi = 73 about the points gi and yi = m\ about the points £;.

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