Estimates in the corona theorem and ideals ofH ∞ : A problem of T. Wolff

The main result of the paper is that there exist functionsf ₁,f ₂,f inH ^∞ satisfying the “corona condition”

The coarse Lipschitz geometry of \ell_p\oplus\ell_q

We show that for 1 < p < ∞ with p ≠ 2 the space L _p (0,1) is not uniformly homeomorphic to . We also show that if 1 < p < 2 < q < ∞ the space has unique uniform structure, answering a question of Johnson, Lindenstrauss and Schechtman (Geom. Funct. Anal. 6:430–470, 1996). The first author was supported by NSF grant DMS-0555670 and the second author was supported by NSF grant DMS-0701097.     

Quantum Diffusion for the Anderson Model in the Scaling Limit

We consider random Schr?dinger equations on for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψ_t the solution with initial data ψ₀. The space and time variables scale as with 0 < κ < κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ_t converges weakly to a solution of a heat equation in the space variable x for arbitrary L ² initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ. This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved. Submitted: April 18, 2006. Accepted: October 12, 2006. László Erdős: Partially supported by NSF grant DMS-0200235 and EU-IHP Network ‘Analysis and Quantum’ HPRN-CT-2002-0027. Manfred Salmhofer: Partially supported by DFG grant Sa 1362/1-1 and an ESI senior research fellowship. Horng-Tzer Yau: Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.     

The subindependence of coordinate slabs inl p n balls

It is proved that if the probabilityP is normalised Lebesgue measure on one of thel _p ⁿ balls in R ⁿ , then for any sequencet ₁ , t ₂ , …, t _n of positive numbers, the coordinate slabs {|x _i |≤t _i } are subindependent, namely, . A consequence of this result is that the proportion of the volume of thel ₁ ⁿ ball which is inside the cube[−1, t] ⁿ is less than or equal tof _n (t)=(1−(1−t) ⁿ ) ⁿ . It turns out that this estimate is remarkably accurate over most of the range of values oft. A reverse inequality, demonstrating this, is the second major result of the article. Supported in part by NSF DMS-9257020. Supported by a grant from Public Benefit Foundation Alexander S. Onassis. This work will form part of a Ph.D. thesis written by the second-named author.     

Asymptotic formulas for the hyperġeometric function2 F 1 of matrix argument,useful in multivariate analysis

Summary LetS _i have the Wishart distributionW _p(∑_i,n_i) fori=1,2. An asymptotic expansion of the distribution of for large n=n₁+n₂ is derived, when∑ ₁∑ ₂ ⁻¹ =I+n^−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function₂ F ₁, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S ₂(S ₁+S ₂)⁻¹| under fixed∑ ₁∑ ₂ ⁻¹ , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for₁ F ₁ were investigated by Sugiura [7].     

Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

Index sets of computable structures

The index set of a computable structure is the set of indices for computable copies of . We determine complexity of the index sets of various mathematically interesting structures including different finite structures, ℚ-vector spaces, Archimedean real-closed ordered fields, reduced Abelian p-groups of length less than ω², and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be m-complete Π _n ⁰ , d-Σ _n ⁰ , or Σ _n ⁰ , for various n. In each case the calculation involves finding an optimal sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Π_n, d-Σ_n, or Σ_n) yields a bound on the complexity of the index set. Whenever we show m-completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey’s theory. Supported by the NSF grants DMS-0139626 and DMS-0353748. Supported by the NSF grant DMS-0502499 and by the Columbian Research Fellowship of the George Washington University. Supported by the NSF grant DMS-0353748. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 538–574, September–October, 2006.     

Proof of a hypercontractive estimate via entropy

Consider the probability spaceW={−1, 1} ⁿ with the uniform (=product) measure. Letf: W →R be a function. Letf=Σf _IX_I be its unique expression as a multilinear polynomial whereX _I=Π _i∈I x _i. For 1≤m≤n let =Σ_|I|=m f _IX_I. LetT _ɛ (f)=Σf _Iɛ^|I| X _I where 0<ɛ<1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that

Optimal interpolation of convergent algebraic series

Let , –1<x ₁<...<x _n <1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery . This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003).     

A bilinear oscillatory integral along parabolas

We establish an norm estimate for a bilinear oscillatory integral operator along parabolas incorporating oscillatory factors . The first author was partially supported by NSF grant of China grant 10671079. The second author was supported by NSF grant DMS-0456976.     

Word maps have large image

An elementw in the free group onr letters defines a mapf _w,G:G^v→G for each groupG. In this note, we show that wheneverw≠1 andG is a semisimple algebraic group,f _w,G is dominant. As an application, we show that for fixedw and Γ _i a sequence of pairwise non-isomorphic finite simple groups,

Polynomial Szemerédi theorems for countable modules over integral domains and finite fields

Given a pair of vector spacesV andW over a countable fieldF and a probability spaceX, one defines apolynomial measure preserving action ofV onX to be a compositionT o ϕ, where ϕ:V→W is a polynomial mapping andT is a measure preserving action ofW onX. We show that the known structure theory of measure preserving group actions extends to polynomial actions and establish a Furstenberg-style multiple recurrence theorem for such actions. Among the combinatorial corollaries of this result are a polynomial Szemerédi theorem for sets of positive density in finite rank modules over integral domains, as well as the following fact:Let be a finite family of polynomials with integer coefficients and zero constant term. For any α>0, there exists N ∈ ℕ such that whenever F is a field with |F|≥N and E ⊆F with |E|/|F|≥α, there exist u∈F, u≠0, and w∈E such that w+ϕ(u)∈E for all ϕ∈ . The first two authros are supported by NSF, grant DMS-0070566 and DMS-0245350. The second author was supported by the A. Sloan Foundation. The third author is supported by NSF, grant DMS-0200700.     

Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

We investigate boundedness of the evolutione ^itH in the sense ofL ²(ℝ³→L ²(ℝ³) as well asL ¹(ℝ³→L ^∞(ℝ³) for the non-selfadjoint operator where μ>0 andV ₁, V₂ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)–A4) below, but without imposing any restrictions on the edges±μ of the essential spectrum. Our goal is to develop an “axiomatic approach,” which frees the linear theory from any nonlinear context in which it may have arisen. This work was initiated in June of 2004, while the first author visited Caltech, and he wishes to thank that institution for its hospitality and support. The first author was partially supported by the NSF grant DMS-0303413. The second author was partially supported by a Sloan fellowship and the NSF grant DMS-0300081. The authors thank Avy Soffer for his interest in this work.     

Harmonic measure of curves in the disk

A powerful tool for studying the growth of analytic and harmonic functions is Hall's Lemma, which states that there is a constantC>0 so that the harmonic measure of a subsetE of the closed unit disk evaluated at 0 satisfies whereE _rad is the radial projection ofE onto . FitzGerald, Rodin and Warschawski proved that ifE is a continuum in whose radial projection has length at most π then (*) is true withC=1, and they asked how large the length, |E _rad|, can be in order for their result to be valid. We prove that (*) holds withC=1 for every continuum satisfying and θc cannot be replaced by a larger number. Fuchs asked for the largest constantC so that (*) holds for allE. We show that for every continuum , (*) holds withC=C _2π≅.977126698498665669…, whereC _2π is the harmonic measure of the two long sides of a 3∶1 rectangle evaluated at the center. There are Jordan curves for which equality holds in (*) withC=C _2π. The authors are supported in part by NSF grants DMS-9302823 and DMS-9401027, and while at MSRI by NSF grant DMS-9022140.     

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ³ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceM _k of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ³) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL ²-nullspace, we prove thatM _k is locally the quotient of a real analytic manifold of dimension 3k–6 by a finite group (i.e. a real analytic orbifold), fork 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofM _k is independent of the genus of the underlying punctured Riemann surface to which is conformally equivalent. These results also apply to hypersurfaces of H ⁿ⁺¹ with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.Research of the first author supported in part by NSF grant # DMS9404278 and an NSF Postdoctoral Fellowship, of the second auther by NSF Young Investigator Award, a Sloan Foundation Postdoctoral Fellowship and NSF grant # DMS9303236, and of the third author by NSF grant # DMS9022140 and an NSF Postdoctoral Fellowship.     

Free abelian covers, short loops, stable length, and systolic inequalities

We explore the geometry of the Abel–Jacobi map f from a closed, orientable Riemannian manifold X to its Jacobi torus . Applying M. Gromov’s filling inequality to the typical fiber of f, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The procedure works, provided the lift of the fiber is non-trivial in the homology of the maximal free abelian cover, , classified by f. We show that the finite-dimensionality of the rational homology of is a sufficient condition for the homological non-triviality of the fiber. When applied to nilmanifolds, our “fiberwise” inequality typically gives stronger information than the filling inequality for X itself. In dimension 3, we present a sufficient non-vanishing condition in terms of Massey products. This condition holds for certain manifolds that do not fiber over their Jacobi torus, such as 0-framed surgeries on suitable links. Our systolic inequality applies to surface bundles over the circle (provided the algebraic monodromy has 1-dimensional coinvariants), even though the Massey product invariant vanishes for some of these bundles. A. I. Suciu was supported by the National Science Foundation (grant DMS-0105342).     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities

We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is whenfεL ²(Ω) and whenfεH ¹(Ω). The convergence rate in the energy norm is in the first case and in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefεH ^m (Ω) is also discussed. The work of the first author was partially supported by NSF under grant DMS-96-00133     

On the Classification of Cartan Actions

We study higher-rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by with k ≥ 3 whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic -actions for k ≥ 3. The first author is supported in part by NSF grants DMS-0140513. The second author is supported in part by NSF grant DMS-0203735. Received: July 2005 Revision: March 2006 Accepted: March 2006