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.     

Towards a characterization of generalized twisted field planes

In this work we study semifield planes of orderq ⁿ(q=p ^r ,p prime) with a collineation whose order is ap-primitive divisor ofq ⁿ–1.Research supported in part by NSF Grant No. DMS-9107372Research supported in part by NSF grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI.     

On the computation of the projective surgery obstruction groups

The computation of the projective surgery obstruction groupsL _n ^p (ZG), forG a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. ForG a 2-hyperelementary group, theL _n ^p (ZG) are detected by restriction to certain subquotients ofG, and a complete set of invariants is given for oriented surgery obstructions.Partially supported by NSERC grant A4000.Partially supported by NSF grant DMS-8610730(1) and the Danish Research Council.     

Order preserving nonexpansive operators inL 1

We study the limit behaviour ofT ^k f and of Cesaro averagesA _n f of this sequence, whenT is order preserving and nonexpansive inL ₁ ⁺ . IfT contracts also theL _∞-norm, the sequenceT ⁿ f converges in distribution, andA _n f converges weakly inL _p (1<p<∞), and also inL ₁ if the measure is finite. “Speed limit” operators are introduced to show that strong convergence ofA _n f need not hold. The concept of convergence in distribution is extended to infinite measure spaces. Much of this work was done during a visit of the first author at Ben Gurion University of the Negev in Beer Sheva, supported by the Deutsche Forschungsgemeinschaft.     

Riesz transforms and elliptic PDEs with VMO coefficients

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

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

Sharpening the LYM inequality

The level sequence of a Sperner familyF is the sequencef(F)={f _i (F)}, wheref _i (F) is the number ofi element sets ofF . TheLYM inequality gives a necessary condition for an integer sequence to be the level sequence of a Sperner family on ann element set. Here we present an indexed family of inequalities that sharpen theLYM inequality.Research supported in part by Alexander v. Humboldt-StiftungResearch supported in part by NSF under grant DMS-86-06225 and AFOSR grant OSR-86-0078Research supported in part by NSF grant CCR-8911388Research supported in part by OTKA 327 0113     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain L ⁸ norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L ^q bounds, in the range 2⩽q⩽∞, for L ² - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp L ^q estimates in higher dimensions for a range of exponents q̅_n⩽q⩽∞. The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

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.     

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.     

Bounds for fixed point free elements in a transitive group and applications to curves over finite fields

We investigateV _f , the cardinality of the value set of a polynomialf of degreen over a finite field of cardinalityq. It has been shown that iff is not bijective, thenV _f ≤q−(q−1)/n. Polynomials do exist which essentially achieve that bound. We do prove that if the degree off is prime to the characteristic andf is not bijective, then asymptoticallyV _f ≤(5/6)q. We consider related problems for curves and higher dimensional varieties. This problem is related to the number of fixed point free elements in finite groups, and we prove some results in that setting as well. Both authors partially supported by the NSF.     

Cyclotomic and simplicial matroids

We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μ_n represented by then ^th roots of unity 1, ζ, ζ², …, ζ^n-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then ^th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μ_n in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.     

Some non-uniformly homeomorphic spaces

in this paper we prove that for 0<p, q≤1 the real F-spacesL _q[0, 1] and ℓ _p are not uniformly homeomorphic. The particular casep=q=1 is due to Enflo and our work is motivated by his. Partially supported by NSF grant #DMS-9104040. The author would like to dedicate this paper to Nancy and Bernie Weston on the occasion of their 40th wedding anniversary.     

On the expected number of k-sets

Given a setS ofn points inR ^d , a subsetX of sized is called ak-simplex if the hyperplane aff(X) has exactlyk points on one side. We studyE _d (k,n), the expected number of k-simplices whenS is a random sample ofn points from a probability distributionP onR ^d . WhenP is spherically symmetric we prove thatE _d (k, n)≤cn ^d−1 WhenP is uniform on a convex bodyK⊂R ² we prove thatE ₂ (k, n) is asymptotically linear in the rangecn≤k≤n/2 and whenk is constant it is asymptotically the expected number of vertices on the convex hull ofS. Finally, we construct a distributionP onR ² for whichE ₂((n−2)/2,n) iscn logn. The authors express gratitude to the NSF DIMACS Center at Rutgers and Princeton. The research of I. Bárány was supported in part by Hungarian National Science Foundation Grants 1907 and 1909, and W. Steiger's research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

On local ergodic convergence of semi-groups and additive processes

We prove the local ergodic theorem inL _∞: Let {T _t}_t>0 be a strongly continuous semi-group of positive operators onL ₁. IfT _t is continuous at 0, then ɛ⁻¹∫ ₀ ^F T ₁ ^* f(x)dt→T ₀ ^* f(x) a.e., for everyf∈L _∞. The technique shows how to obtain theL _p local ergodic theorems from theL ₁-contraction case. It applies also to differentiation ofL p additive processes. Then-dimensional case, which is new, is proved by reduction to then-dimensionalL ₁-contraction case, solved by M. Akcoglu and A. del Junco. Research carried out during a sabbatical leave at the Ohio State University.     

Reconstruction of functions from their integrals over<Emphasis Type="Italic">k</Emphasis>-planes

Thek-plane Radon transform assigns to a functionsf(x) on ℝ ⁿ the collection of integralsf(τ)=∫ _τ f over allk-dimensional planesτ. We give a systematic treatment of two inversion methods for this transform, namely, the method of Riesz potentials, and the method of spherical means. We develop new analytic tools which allow to invertf(τ) under minimal assumptions forf. It is assumed thatfεL ^p , 1≤p<n/k, orf is a continuous function with minimal rate of decay at infinity. In the framework of the first method, our approach employs intertwining fractional integrals associated to thek-plane transform. Following the second method, we extend the original formula of Radon for continuous functions on ℝ² tofεL ^p (ℝ ⁿ ) and all 1≤k<n. New integral formulae and estimates, generalizing those of Fuglede and Solmon, are obtained. The work was supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Some geometrical properties of Banach spaces of polynomials

We investigate the asymmetry, gl constants and best factorization estimates of then-dimensional spaces of polynomialsH _p ⁿ =span{e ^ikx;k=1,2,…,n} equipped with theL _p norm for 1≦p≦∞. Supported in part by NSF grant # MCS-8109561.     

The Fourier transform of order statistics with applications to Lorentz spaces

We present a formula for the Fourier transforms of order statistics in ℝ ⁿ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in ℝ ⁿ . Fora ₁≥...≥a _n≥0 andq>0, denote by ℓ _w,q ⁿ then-dimensional Lorentz space with the norm ‖(x ₁,...,x _n)‖=(a ₁(x ₁ ^* ) ^q +...+a _n(x _n ^* ) ^q )^1/q , where (x ₁ ^* ,...,x _n ^* ) is the non-increasing permutation of the numbers |x ₁|,...,|x _n|. We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces intoL _q [10] to prove that, forn≥3 andq≤1, the space ℓ _w,q ⁿ is isometric to a subspace ofL _q if and only if the numbersa ₁,...,a _n form an arithmetic progression. Forq>1, all the numbersa _i must be equal so that ℓ _w,q ⁿ = ℓ _q ⁿ . Consequently, the Lorentz function spaceL _w,q(0, 1) is isometric to a subspace ofL _q if and only ifeither 0<q<∞ and the weightw is a constant function (so thatL _w,q=L_q),or q≤1 andw(t) is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions. Both authors were supported in part by the NSF Workshop in Linear Analysis and Probability held at Texas A&M University in August 1993. The work was done during the first author’s visit to Texas A&M University.     

Bessel identities in the waldspurger correspondence over the real numbers

We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL ₂(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL ₂(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. Research of first author was partially supported by NSF grant DMS-0070762. Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003.