Singular integrals with rough kernels along real-analytic submanifolds in R n

L ^p mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translated of a real-analytic submanifold in R ⁿ .Work in this paper was done during the second author's visit at the Department of Mathematics, University of Pittsburgh.Supported in part by NSF Grant DMS-9622979.     

Cauchy-riemann operator and decomposition ofL 2 space on ℝ + n+1

We decompose L_α²(ℝ₊ ⁿ⁺¹) into direct sum of closed subspaces which are isomorphic to the monogenic Bergman space. Application on commutators is also discussed. Research supported in part by the NSF Grant DMS-9622890     

Brauer groups are not characterized by Ulm invariants

Two important invariants of a fieldF are its Brauer groupB(F) and its character groupX(F). IfF is countable, these are countable abelian torsion groups, and so are determined by their Ulm invariants. We show here that Ulm’s invariants do not determine Brauer groups or character groups of uncountable fields. An essential tool, which is entirely group theoretic in nature, is a fact about ultraproducts of torsion groups. Supported in part by NSF Grant No. DMS-8500883. Supported in part by NSA Grant No. MDA904-85-H-0014. Supported in part by NSF Grant No. DMS-8500929.     

Homotopy spectral sequences and obstructions

For a pointed cosimplicial spaceX ^•, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX ^•. In this paper,X ^• is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here. Supported in part by NSF Grant DMS-8602432.     

Logarithmic Level Comparison for Small Deviation Probabilities

Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L ² norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm. Wenbo V. Li: Supported in part by NSF Grant DMS-0505805.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

Universal non-completely-continuous operators

A bounded linear operator between Banach spaces is calledcompletely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators fromL ₁ into an arbitrary Banach space, namely, the operator fromL ₁ into ⊆_∞ defined byT ₀(f) = (∫r _n f d μ) _n>-0, wherer _n is thenth Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach spaces. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space. Supported in part by NSF grant DMS-9306460. Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University (supported in part by NSF grant DMS-9311902). Supported in part by NSF grant DMS-9003550.     

Julia and John

Using a recent result of Mañé [Ma] we give a classification of polynomials whose Fatou components are John domains.Supported in part by NSF Grant DMS-8916968     

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     

The X-Ray Transform for a Generic Family of Curves and Weights

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T ^* M. Second author was partly supported by NSF Grant DMS-0400869; third author was partly supported by NSF and a Walker Family Endowed Professorship.     

On the support of harmonic measure for the random walk

We show that harmonic measure for the simple random walk on then ×…×n cube in thed-dimensional lattice is supported on o(n ^d ) vertices. This research was supported in part by NSF grant # DMS-9353149.     

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.     

Wavelet sets in ℝ n

A congruency theorem is proven for an ordered pair of groups of homeomorphisms of a metric space satisfying an abstract dilation-translation relationship. A corollary is the existence of wavelet sets, and hence of single-function wavelets, for arbitrary expansive matrix dilations on L ² (ℝ ⁿ). Moreover, for any expansive matrix dilation, it is proven that there are sufficiently many wavelet sets to generate the Borel structure ofℝ ⁿ. The second author is supported in part by NSF Grant DMS-9401544. The third author was a Graduate Research Assistant at Workshop in Linear Analysis and Probability, Texas A&M University.     

Upper bounds for configurations and polytopes inR d

We give a new upper bound onn ^d(d+1)n on the number of realizable order types of simple configurations ofn points inR ^d , and ofn2d ² n on the number of realizable combinatorial types of simple configurations. It follows as a corollary of the first result that there are no more thann ^d(d+1)n combinatorially distinct labeled simplicial polytopes inR ^d withn vertices, which improves the best previous upper bound ofn ^{cn d/2}.Supported in part by NSF Grant DMS-8501492 and PSC-CUNY Grant 665258.Supported in part by NSF Grant DMS-8501947.     

Two extensions of Lubinsky’s universality theorem

We extend some remarkable recent results of Lubinsky and Levin-Lubinsky from [−1, 1] to allow discrete eigenvalues outside σ _ess and to allow σ _ess first to be a finite union of closed intervals and then a fairly general compact set in ℝ (one which is regular for the Dirichlet problem). Supported in part by NSF grant DMS-0140592 and U.S.-Israel Binational Science Foundation (BSF) Grant No. 2002068.     

On the rate of convergence in the entropic central limit theorem

We study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition.Mathematics Subjects Classification (2000):94A17; 60F05Supported in part by the EU Grant HPMT-CT-2000-00037, The Minkowski center for Geometry and the Israel Science Foundation.Supported in part by NSF Grant DMS-9796221.Supported in part by EPSRC Grant GR/R37210.Supported in part by the BSF, Clore Foundation and EU Grant HPMT-CT-2000-00037.     

A Generalization of a Theorem of Liao

Let X be a metrizable space and let φ：R× X → X be a continuous flow on X. For any given {φt}-invariant Borel probability measure, this paper presents a {φt}-invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the contiImous flow （X, {φt}）. The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non-uniformly hyperbolic systems and skew-product flows.     

Riesz transforms and elliptic PDEs with VMO coefficients

The present paper is concerned withL ^p-theory of the uniformly elliptic differential operator

Bounding the coefficients of a divisor of a given polynomial

We find bounds for the coefficients of a divisorg(X) of a given polynomialf(X).Supported in part by NSF Grant No. DMS-8610730.     

A characterization of Poissonian domains

We give a characterization of Possonian domains inR ⁿ , i.e., those domains for which every bounded harmonic function is the harmonic extension of some function inL ^∞ of harmonic measure. We deduce several properties of such domains, including some results of Mountford and Port. In two dimensions we give an additional characterization in terms of the logarithmic capacity of the boundary. We also give a necessary and sufficient condition for the harmonic measures on two disjoint planar domains to be mutually singular. The author is partially supported by a NSF Postdoctoral Fellowship.