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By Paul H. Rabinowitz, Eduard Zehnder

ISBN-10: 0125742495

ISBN-13: 9780125742498

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13) is nothing else but the map: L^S"-1) 3φ^ Τ+ψγφ G B*(Rn). 1). This set is a closed subspace of B*(Rn) (limits in L? 1) are solutions). 13) is a closed subspace of B*(Hn). 11) proved before. 16) hold. These estimates will follow from the next theorem. 5. 26) φ(ω)β(χ,ω;1ζ)<1ω where φ G L 2 ( S n _ 1 ) . Set: et = e x p ( ± i ( n - 1 ) π / 4 ) . 27) expansions hold in the B* u{x) = {^η-1)'^1β-ί^\φ{χ/\χ\) sense, + v(x) where eîk^v G i ? * ( R n ) (see Section 2 for the definition (ii) For k G R + , (129) where v G B*(Rn).

These estimates are of independent interest and are useful in other problems such as those arising in scattering theory of Schrödinger operators. The estimates refine and extend (for the Laplace operator) results established in [4]. ) The plan of the paper is as follows. In Section 2 we give some preliminary results and definitions. The main resolvent estimates in the upper half-plane are established in Section 3. 2) by the Phragmen-Lindelöf principle. In Section 4 the main representation theorem is established.

Steps 1 to 6 guarantee that the affine variety A completes into an Abelian variety by means of the functions of L(D) constructed in step 4. The proof of this fact is a consequence of the following lemma. L e m m a . Let ZQ φ 0 be the affine part of P n with coordinates z = (zo5 · · · ? ^n) and kt A be an irreducible protective variety with smooth, n — k dimensional affine part A: k A=f]{zePn, Pi(z) = 0} = (An{z0^o})u(An{z0 Consider the = o}) embedding zeAcPn^yePN, by means of homogeneous polynomials in z.

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Analysis, et Cetera. Research Papers Published in Honor of Jürgen Moser's 60th Birthday by Paul H. Rabinowitz, Eduard Zehnder


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