Abstract
Generalizing results of Rossi and Vergne for the holomorphic discrete series on symmetric domains, on the one hand, and of Chailuek and Hall for Toeplitz operators on the ball, on the other hand, we establish existence of analytic continuation of weighted Bergman spaces, in the weight (Wallach) parameter, as well as of the associated Toeplitz operators (with sufficiently nice symbols), on any smoothly bounded strictly pseudoconvex domain. Still further extension to Sobolev spaces of holomorphic functions is likewise treated.
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Notes
This modification requires the following fact, which is proved in the same way as Proposition 5 in [15]: if \(A(z)\), \(B(z)\) are holomorphic families of \(\Psi \)DOs of order \(-\frac{z}{2}\) and \(z\), respectively, on \({\mathbf C}\), in the sense of §2.3 in [15], then \(A(z)B(z)\) is a holomorphic family of \(\Psi \)DOs on \({\mathbf C}\) of order \(\frac{z}{2}\).
See Range [19], Lemma V.3.4, which even gives a more refined variant where one of the zeta coordinates is additionally made to coincide with the imaginary part of the (smooth globalization of the) Levi polynomial.
That is, it coincides with the restriction to \(\alpha >-1\) of some function holomorphic in a neighborhood of the interval \((-1,\infty )\), which has analytic continuation to a neighborhood of \(\alpha _0\) in the usual sense.
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Research supported by GA ČR Grant No. 201/12/0426 and Barrande Project MEB021108.
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Communicated by Richard Rochberg.
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Bommier-Hato, H., Engliš, M. & Youssfi, EH. Analytic Continuation of Toeplitz Operators. J Geom Anal 25, 2323–2359 (2015). https://doi.org/10.1007/s12220-014-9515-0
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DOI: https://doi.org/10.1007/s12220-014-9515-0