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Sidon sets and Riesz products

Jean Bourgain — 1985

Annales de l'institut Fourier

Let G be a compact abelian group and Γ the dual group. It is shown that if Δ Γ is a Sidon set, then the interpolating measures on Λ can be obtained as mean of Riesz products. If Λ is a Sidon set tending to infinity, Λ is of first type. Our approach yields in fact elementary proofs of certain characterizations of Sidonicity obtained in G. Pisier, C.R.A.S., Paris Ser. A, 286 (1978), 1003–1006 – Math. Anal. and Appl., Part B, Advances in Math., Suppl. Sts. vol. 7, 685-726 – preprint, using random Fourier...

On finitely generated closed ideals in H ( D )

Jean Bourgain — 1985

Annales de l'institut Fourier

Assume f 1 , ... , f N a finite set of functions in H ( D ) , the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function f in H ( D ) to belong to the norm-closure of the ideal I ( f 1 , ... , f N ) generated by f 1 , ... , f N , namely the property | f ( z ) | α ( | f 1 ( z ) | + ... + | f N ( z ) | ) for z D for some function α : R + R + satisfying lim t 0 α ( t ) / t = 0 . The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property | f ( z ) | C max 1 j N | f j ( z ) | for z D for...

Translation invariant forms on L p ( G ) ( 1 < p < )

Jean Bourgain — 1986

Annales de l'institut Fourier

It is shown that if G is a connected metrizable compact Abelian group and 1 < p < , any (possibly discontinuous) translation invariant linear form on L p ( G ) is a scalar multiple of the Haar measure. This result extends the theorem of G.H. Meisters and W.M. Schmidt (J. Funct. Anal. 13 (1972), 407-424) on L 2 ( G ) . Our method permits in fact to consider any superreflexive translation invariant Banach lattice on G , which is the adopted point of view. We study the representation of an element f of this invariant lattice...

A spectral gap theorem in SU ( d )

Jean BourgainAlex Gamburd — 2012

Journal of the European Mathematical Society

We establish the spectral gap property for dense subgroups of SU ( d ) ( d 2 ) , generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU ( 2 ) .

Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case

Jean BourgainAynur Bulut — 2014

Journal of the European Mathematical Society

We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in d to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in 3 . The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation,...

Sur les séries de Fourier des fonctions continues unimodulaires

Jean BourgainJean-Pierre Kahane — 2010

Annales de l’institut Fourier

Les applications continues du cercle T dans T ont des séries de Fourier intéressantes  : le théorème établi ici dit que si les coefficients de Fourier a ( n ) sont de carré sommable avec certains poids pour n > 0 , il en est de même pour n < 0 . C’est encore vrai pour V M O , mais faux pour les applications mesurables bornées.

Control for Schrödinger operators on 2-tori: rough potentials

Jean BourgainNicolas BurqMaciej Zworski — 2013

Journal of the European Mathematical Society

For the Schrödinger equation, ( i t + ) u = 0 on a torus, an arbitrary non-empty open set Ω provides control and observability of the solution: u t = 0 L 2 ( 𝕋 2 ) K T u L 2 ( [ 0 , T ] × Ω ) . We show that the same result remains true for ( i t + - V ) u = 0 where V L 2 ( 𝕋 2 ) , and 𝕋 2 is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V C ( 𝕋 2 ) and conjectured for V L ( 𝕋 2 ) . The higher dimensional generalization remains open for V L ( 𝕋 n ) .

A new function space and applications

Jean BourgainHaïm BrezisPetru Mironescu — 2015

Journal of the European Mathematical Society

We define a new function space B , which contains in particular BMO, BV, and W 1 / p , p , 1 < p < . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving L p norms of integer-valued functions in B . We introduce a significant closed subspace, B 0 , of B , containing in particular VMO and W 1 / p , p , 1 p < . The above mentioned estimates imply in particular that integer-valued functions belonging to B 0 are necessarily constant. This framework provides a “common roof” to various,...

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