Download Automorphic Pseudodifferential Analysis and Higher Level by André Unterberger PDF

By André Unterberger

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2002.

The topic of this ebook is the learn of automorphic distributions, during which is intended distributions on R2 invariant less than the linear motion of SL(2,Z), and of the operators linked to such distributions below the Weyl rule of symbolic calculus.

Researchers and postgraduates attracted to pseudodifferential analyis, the speculation of non-holomorphic modular types, and symbolic calculi will enjoy the transparent exposition and new effects and insights.

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Extra resources for Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi

Example text

28), we emphasized that ~~ is meaningful. 25). e) = a(a + 1) ... :t' t) dt . 13j of the Gamma function on vertical lines, the same (even better, actually) 4. The structure of automorphic distributions 35 inequality holds for (((±iA))-l. 27) 1-1\2 with E = 0 or 1, so that, saving an extra factor at the end, and using an integration by parts, we are done for hand 12 . ~ and write (r 1: IW- 1 I :<::; C N (:~ to 1: :<::; CN In 1m (1 r (F1 1 h) IW- 1 (~, t) I dt m I~ 1 (1 +t + ~:) 2 + Inl)-if t1 00 m=O k,£ &) r(f!

53) is a special case of a class of automorphic functions introduced by Selberg [43J (following the use of Poincan§'s series in the holomorphic case), and used by many authors in the field ([15, 20J or [62, Section 20]). A still special case is given by the family of functions (depending on 1/ E C with Re 1/ < -1 and on an integer N) Chapter 1. 61 ) o In the next section, we shall make the decomposition of the Bezout distribution into homogeneous components explicit, at the same time proving that not only ~£, with C ~ 1, but also ~ itself makes sense as a tempered distribution.

6 and observe that, when acting on even distributions, any of the operators 2i7f£, 9 or £(s) preserves the type (even or odd) of distributions. 42) has the type specified by j: it is of even type if j = 0, of odd type if j = 1. 7 and remembering that 9 £ = -£ g, one sees that, when acting on distributions of the type specified by j, £' X C+ II; + 112) 9 £' (1 + II; - 112) = 7f-Vl r (1+Vl +V212i1f£+2j) r (1+Vl-V242i1f£+2j) £ r e-Vl-V242i1f£+2j) r e- Vl +v212i1f£+2j) (1 + 111 2 +112) g£ (1 + 111 -112) .

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