By Pierre Deligne

The first a part of this monograph is dedicated to a characterization of hypergeometric-like services, that's, *twists* of hypergeometric services in *n*-variables. those are taken care of as an (*n*+1) dimensional vector area of multivalued in the community holomorphic services outlined at the house of *n*+3 tuples of specific issues at the projective line *P* modulo, the diagonal element of vehicle *P*=*m*. For *n*=1, the characterization should be considered as a generalization of Riemann's classical theorem characterizing hypergeometric services by way of their exponents at 3 singular points.

This characterization allows the authors to check monodromy teams reminiscent of varied parameters and to turn out commensurability modulo internal automorphisms of *PU*(1,*n*).

The ebook comprises an research of elliptic and parabolic monodromy teams, in addition to hyperbolic monodromy teams. the previous play a task within the evidence magnificent variety of lattices in *PU*(1,2) built because the primary teams of compact complicated surfaces with consistent holomorphic curvature are in truth conjugate to projective monodromy teams of hypergeometric features. The characterization of hypergeometric-like services by means of their exponents on the divisors "at infinity" allows one to end up generalizations in *n*-variables of the Kummer identities for *n*-1 concerning quadratic and cubic adjustments of the variable.

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**Commensurabilities among Lattices in PU (1,n).**

The 1st a part of this monograph is dedicated to a characterization of hypergeometric-like features, that's, twists of hypergeometric services in n-variables. those are taken care of as an (n+1) dimensional vector house of multivalued in the neighborhood holomorphic services outlined at the house of n+3 tuples of exact issues at the projective line P modulo, the diagonal part of automobile P=m.

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For x0 ∈ S, the set {rm : m ∈ Zn } is linearly independent. 2, we have |bm | < ∞, m∈Zn where 1 T →∞ T bm = lim T 1 n−1 DR2 (x0 )eirm R dR = C , n rm 22 C1. An introduction to multiple Fourier series for C = 0. However, it is evident that m∈Zn 1 = n rm m∈Zn 1 = +∞. 13) holds. 6 (M. Riesz convexity theorem) Let x0 ∈ Rn and t > 0. We deﬁne 0 α Aα (t) = tα D√ (x0 ) = (t − |m|2 )α eim·x . t √ |m|< t Suppose that V (t), W (t) are both positive increasing functions on (0, +∞) and let Uθ (t) = (V (t))1−θ (W (t))θ , for 0 ≤ θ ≤ 1.

Bochner-Riesz means of multiple Fourier integral Proof. Assume lim R→∞ α BR (f ) − f p =0 α (f ) for every f ∈ Lp (Rn ). Thus, we obtain that BR p has bound with respect to R. It follows from the uniform boundedness theorem in functional analysis that α (f ) p ≤ Cp f p . BR Conversely, we suppose that there exists a constant Cp such that the inequality α BR (f ) p ≤ Cp f p holds for any f ∈ Lp (Rn ). Notice that the space D(Rn ), the class of inﬁnitely diﬀerential functions which have compact support, is dense in Lp (Rn ).

2 1 −β 2 dt 1 t fx ( ) − s tβ− 2 Jn− 1 −β (t)dt 2 R ∞ 1 t fx ( ) − s tβ− 2 Jn− 1 −β (t)dt. 9) 2 R 1 as t → 0, we immediately have 1 1 − s tβ− 2 Jn− 1 −β (t)dt = O 2 0 fx t R − s tn−1 dt . 4), we easily have 1 fx 0 t R 1 − s tβ− 2 Jn− 1 −β (t)dt = o(1), 2 as R → ∞. From the equation Jν (t) = 2 νπ π cos t − − + O t−3/2 , πt 2 4 as t → +∞, we have ∞ t R fx 1 ∞ = 1 fx 1 − s tβ− 2 Jn− 1 −β (t)dt 2 t R ∞ +O 1 fx n−β 2 cos t − 2 π dt −s π t1−β dt t − s 2−β , R t where the second term can be estimated as ∞ 1 fx t R −s dt t2−β t = t−(2−β+n−1) 0 ∞ +O 1 fx t−(n+2−β) τ ∞ − s τ n−1 dτ 1 R t τ fx − s τ n−1 dτ dt .