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Consider the2D defocusing cubic NLSiu t+Δu−u|u|2=0 with Hamiltonian $$\smallint \left( {\left| {\nabla \phi } \right|^2 + \tfrac{1}{2}\left| \phi \right|^4 } \right)$$ . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing |φ|4 by |φ|4 :, is an invariant measure for the appropriately modified equationiu t + Δu‒ [u|u 2−2(∫|u|2 dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data ϕ the solutionu, u(0)=ϕ, satisfiesu(t)−e itΔφ ∈C Hs (ℝ), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).

Consider 1D nonlinear Schrodinger equation (0.1) $$iu_t - u_{xx} + V(x)u + \varepsilon \frac{{\partial {\rm H}}}{{\partial \bar u}} = 0$$ and nonlinear wave equation (0.2) $$y_{tt} - y_{xx} + \rho y + \varepsilon F'(y) = 0$$ under Dirichlet boundary conditions. We assume hereH(u,ū) andF(y) polynomials. It is proved that for “typical” periodic potentialV in (0.1) and typical ρ∈R in (0.2) the following is true. Letu(0) (resp.y(0),y′(0)) be smooth initial data fort=0. Then the corresponding solutionu(t) of (0.1) (resp.y(t) of (0.2)) will be ɛ1/2-close to the unperturbed solution (with appropriate frequency adjustment), for times |t

The general problem discussed here is the persistency of quasi-periodic solutions of linear or integrable equations after Hamiltonian perturbation. This subject is closely related to the well-known "KAM-theory" of invariant tori in smooth dynamical systems. The main source of new difficulties is the fact that one considers here (finite dimensional) tori in an infinite dimensional phase space. For instance, already in the study of time periodic solutions, the fact that the phase space is infinite dimensional leads to small divisor problems. Now the standard KAM-technology may be reworked in a form applicable to treat persistency problems in space dimension (1D) in the case of Dirichlet boundary conditions say, where in particular the normal frequencies of the problem are well-separated. Results along this line were obtained by S. Kuksin [Ki] and C. E. Wayne [W] among other works. We mentioned [Ki] as the main expository reference for the KAM approach. Considering PDE's in iD with periodic boundary conditions and especially in space dimension > 1, a significant new problem arises due to the presence of clusters of normal frequencies.

The aim of this report is to describe some recent research in the area of nonlinear evolution equations. The choice of the topics is largely influenced by the author’s own interests and it is in no way a complete survey of this field, which would be nearly impossible to achieve in a single exposition. Some very outstanding achievements in recent years such as, for instance, the work of Christodoulou and Klainerman [C-K] on global solutions of the Einstein equations will not be discussed here.

The purpose of this expose is to present a summary of some new developments in the theory of Hamiltonian nonlinear evolution equations, more specifically, nonlinear Schrodinger equations. The themes and methods discussed are closely related to classical mechanics. The first topic is the existence of an invariant measure for the flow. This invariant measure is the (properly normalized) Gibbs measure from statistical mechanics and we establish wellposedness of the equation on its support. Those investigations are closely related to the paper [L-R-S]. Results in this direction are obtained in 1D (in the focusing and defocusing case) and in the 2D defocusing case. The second topic concerns the persistency of time periodic and quasi-periodic solutions for Hamiltonian perturbations of linear and integrable equations. We follow a method, the so-called Liapounov-Schmidt decomposition, originating from the works of [C-Wl, 2], rather than the KAM procedure (cf. [Kuk1]). The main advantage of this technique is the fact that it overcomes certain limitations of the KAM scheme, which is necessary to deal in particular with the problems in space-dimension D ≥ 2. This work is a new approach to KAM problems, also in finite dimensional phase space. Persistency results for PDE’s are obtained in the time periodic case in arbitrary dimension and for quasi-periodic solutions when D ≤ 2. The small divisor problems appearing when inverting the linearized operators are related to the works of Frolich and Spencer on the Anderson model.

The aim of the Halasz-Montgomery inequality is to derive distributional properties for Dirichlet polynomials $$ F(t) = \sum\limits_{n\sim M} {{a_n}{n^{it}}\,\left( {\left| t \right| < T} \right)} $$ (1.1) (n ~ m = n proportional to M) from properties of the ζ-function or its partial sums. It is based on the simple Hilbert space inequality $$ \sum\limits_{r \leqslant R} {\left| {\left\langle {\xi, {\varphi_r}} \right\rangle } \right|} \leqslant \left\| \xi \right\|{\left( {\sum\limits_{r,s \leqslant R} {\left| {\left\langle {{\varphi_r},{\varphi_s}} \right\rangle } \right|} } \right)^{1/2}} $$ (1.2) .

In this paper we develop a technique to improve on [M]’s \(\tfrac{1}{8}\) result for the boundedness on L 4(ℝ3) of cone multipliers $$ {m_\alpha }\left( {{x_1},{x_2},{x_3}} \right) = \phi \left( {{x_3}} \right)\left( {1 - \frac{{\sqrt {x_1^2 + x_2^2} }} {{{x_3}}}} \right)_ + ^\alpha $$ with o ∈ C0 ∞(l, 2). More precisely, we get this property for certain values of α < \(\tfrac{1}{8}\). There is a similarity in approach with estimates for the Bochner-Riesz problem in the case of the ball. Our argument shows also that if μ is a measure supported by \( {\Gamma_{(1)}} = \left\{ {x \in \left. {{\mathbb{R}^3}} \right|\left| {{x_3}} \right| = \sqrt {x_1^2 + x_2^2}, 1 < {x_3} < 2} \right\} \) and ρ = 0 on a neighborhood of the cone Γ, then if \( \frac{{d\mu }} {{d\sigma }} \in {L^2}\left( \sigma \right),\sigma = \) surface measure of T, one may bound ||(μ * μ) ρ|| p for certain p < 2. This fact and especially an understanding for what surfaces this phenomenon holds, seems of independent interest.

This paper studies the classical problem of determining the maximum growth of the Taylor series majorant of holomorphic functions with bound 1 in the unit disk, as a function of the radius of the Taylor series. We determine for the first time the asymptotic behavior of this maximum as the radius tends to 1, with a very small error term. The proof uses full asymptotic expansions of certain oscillatory integrals and the discrete Hilbert transform.

and phase space L2(1r). It has been conjectured by a number of authors (cf. [FGP], [Bel]) that for typical parameter values for a, b, the wave function t satisfying equations (0.1), (0.2) will be almost periodic in time; hence f(t) (the Fourier transform) remains localized. We will prove this here, assuming the parameter S in (0.2) small. Thus we have: THEOREM. For K stnall and (a,b) o?>tside a set of small rneasure ( > O for K > 0), the following holds. Let 4! = W(t, x) solve (0 1), (0 2) and let to = 4!(0, x) be a suJgciently smooth function on T. Then @ is an alrnost periodic function of timbe, say as an Hl(T)-valued map. In particular supt ll@(t)llHl

This paper is a follow up of [B1]. It is shown that the sequence of squares {n 2|n=1, 2, ...} contains Λ(p)-subsets of “maximal density”, for any givenp>4. The proof is based on the probabilistic method developed in [B1] and a precise estimate of the Λ(p)-constant for the sequence of squares itself. Analogues of this phenomenon are obtained for other arithmetic sets, such as the sequence ofkth powers {n k |n=1, 2, ...} or the sequence of prime numbers. Sections 2 and 3 of the paper are of independent interest to orthogonal system theory.

converge almost surely for N -+ co, assuming f a function of class L~(~, ~). Here and in the sequel, one denotcs by ~ a probability measure and by T a measure-preserving automorphism. The natural problem of developing the L~-thcory for p < 2 was studied in [B~] and a partial result was obtained. We continue this line of investigation here. The approach used in [Ba] , [Ba] relies on a method which may be summarized as follows: a) Reduction of the general problem to statements about the shift S on Z, which are of a " finite " and " quantitative " nature (in the sense of inequalities involving finitely many iterates of the transformation). b) Proof of certain maximal function inequalities, relative to the shift, by Fourier Analysis methods. c) Use of the " major arc " description of the relevant exponential sums, similar to that in the Hardy-Littlewood circle method.

It is proved that given e>0, there is δ(e)>0 such that ifS is a measurable set of [0,N], |S|>eN, then there is a triplex, x+h, x+h2 inS withh satisfyingh>δ(e)N1/2. The argument is related to [B] and uses the behavior of certain non-linear convolution-type operators. The method can be adapted in a variety of situations. For instance, it can be used to prove the analogue of the previous statement with the square replaced by another power,h3,h4 etc.

It is shown that form≠n the polydisc algebrasA(D m) andA(D n) are not isomorphic as Banach spaces. More precisely, there is no linear embedding of the dual spaceA(D n)* intoA(D m)* form

It is shown that for Banach spaces the Radon-Nikodym property and the Bishop-Phelps property are equivalent. Using similar techniques, we prove that ifC is a bounded, closed and convex subset of a Banach space such that every nonempty subset ofC is dentable, then the strongly exposing functionals ofC form a denseGδ-subset of the dual.

It is shown that the class of separableℒ1-spaces with the Radon-Nikodym property only admitsL {sp1} as universal element.

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc. 22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L1 semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l1. As a consequence of the proof, it is shown that if X is a subspace of L1, either L1 embeds in X or l1 embeds in L1X. The simpler result that L1 does not semi-embed in c0 is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.

It is shown that the disc algebra has no Besselian basis. In fact, concrete minorations on certain Lebesgue functions are obtained. A consequence is the nonisomorphism of the disc algebra and the space of uniformly convergent Fourier series on the circle. 0. Introduction. The purpose of this paper is to prove the following result on the disc algebra A. THEOREM. Assume m is a positive integer and ((>k)l k||oo k(Z)Xk > p(M)logn. 1 c( lak 12)1/2 (resp. 11 E akxk 11 < C( lak 12)1/2). Notice that the space A (resp. L1/Hl) has a basis (see [B]). The above theorem permits a local version of Corollary 1 in terms of Snitedimensional complemented subspaces of these spaces. Denote by U the completion of the space of analytic polynomials on the circle (or on the disc) under the norm IIPIIU = SUP IIP * Dm 1100 m where Dm(0) = E0

It is shown that the spaces of homogenous polynomials on the complex 2-ball are uniformly isomorphic tol∞-spaces. The argument is based on explicit constructions and the decomposition method. A new construction is given of bases in the spaceAN of monomials 1,z,z2, ...,zN−1 on the disc (due to Bochkarev [Boc]). Also using decomposition methods, the existence of a base in the ball algebra is obtained.

It is shown that ifT :L 1 →L 1 is a bounded linear operator failing the Dunford-Pettis property, thenT fixes a copy of ⊕ll1 (l 2).

Using the duality between Dunford-Pettis operators onL1 and Pettis-Cauchy martingales, we prove that the Dunford-Pettis operators fromL1 intoL1 form a lattice. We show also that a Banach spaceX has the Radon-Nikodým property provided the Dunford-Pettis members of ℒ(L1,X) are representable. The lifting of dual valued Dunford-Pettis operators is investigated. Some remarks are included.

Assume A=I~n\E a domain in F, n, where E is a compact set. Denote o~(A,A,x) the harmonic measure for A of A, evaluated at x ~ R d. According to 0ksendal 's theorem [O], o E= o(A, . ,x) is singular with respect to d-dimensional Lebesgue measure. For d > 2 and general domains, this result seemed to be so far the only known localization property. Recently for d=2 , it has been shown by P. Jones and T. Wolff [J-W] that S(o9~) has dimension at most 1. This result completes previous work due to N.G. Makarov and L. Carleson (see I-M] and [C2]). Let g be the Green's function of A ~ r with pole at some point of C * = r {o9}. In both the Carleson and Jones-Wolff arguments, the integral ~g , ~g ~nn log ~nn as boundary