For the bilinear Hilbert transform given by H fg(x) = p. v. Z f(x \Gamma y)g(x + y) dy y we announce the inequality kH fgk p3 K p1 ;p 2 kfk p1 kgk p2 , provided 2 ! p 1 ; p 2 ! 1, 1=p 3 = 1=p 1 + 1=p 2 and 1 ! p 3 ! 2. We announce a partial resolution to long standing conjectures concerning

Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying | α β − 1

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a ∗-structure, conjugate-linear on the hom-sets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also

Let F be a totally real field. In this paper we study the Ramanujan Conjecture for Hilbert modular forms and the Weight-Monodromy Conjecture for the Shimura varieties attached to quaternion algebras over F. As a consequence, we deduce, at all finite places of the field of definition, the full

this paper. For simplicity, we adopt the following convention: X denotes a real unitary space, a, b, r denote real numbers, s 1 , s 2 , s 3 denote sequences of X , R 1 , R 2 , R 3 denote sequences of real numbers, and k, n, m denote natural numbers. The scheme Rec Func Ex RUS deals with a real

Abstract. Assuming the Bloch-Kato Conjecture, we determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n

that letter). Consider the real sequence space `2 = {f: N+ → R: Σ∞i=1f(i)2 < ∞}, where N+ stands for N \ {0}. Let 〈en: n ∈ N+ 〉 be the standard orthonormal base of `2, i.e., en(n) = 1 and en(m) = 0 for m 6 = n. It follows from Satz 15 in [3] that there are two unbounded self-adjoint operators X and Y

prove that the poset Pn is a simplicial complex on the vertex set [3,n]. We study the f-vector, the f-polynomial, the reduced Euler characteristic, the Möbius function, the h-vector and the h-polynomial of Pn. We also derive the zeta polynomial of Pn and give the formula for the number of the chains

Shimura proved the following fundamental result (see [37, Theorem 4.3]) on the critical values of the standard L-function attached to a holomorphic Hilbert modular form. Theorem 1.1 (Shimura). Let f be a primitive holomorphic Hilbert modular cusp form of type (k, ψ) over a totally real number field

Abstract. Let F be a real quadratic field with ring of integers Ø and with class number 1. Let Γ be a congruence subgroup of GL2(Ø). We describe a technique to compute the action of the Hecke operators on the cohomology H 3 (Γ ; C). For F real quadratic this cohomology group contains the cuspidal