Almost Everywhere Convergence of Bochner-Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group H_n and the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclideandimension of H_n. We prove convergence a.e. of the Bochner-Rieszmeans as r 0 for > 0and for all f L^p(H_n), provided that . Our proof is based on explicit formulas for the operators with a C, defined on the dual ofH_n by , which may be of independent interest. Here is given by for all (z,u) H_n. 2000 Mathematical Subject Classification: 22E30, 43A80.     

Fock Space and the Poisson Process

Using the Wiener–Poisson isomorphism, we show that if(F_t)_{0 t 1} is a real, bounded, predictable process adaptedto the filtration of a compensated Poisson process (X_t)_{0 t 1}, and if is the operator corresponding to multiplication by , then for any regular self-adjoint quantum semimartingale , the essentially self-adjoint quantumsemimartingale satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process toa measure space (M, M, µ) as an isometry I: L² (M, M,µ) L²(, F, P) and give a new construction of the generalisedWiener–Poisson isomorphism W_I: F₊ (L²(M)) L² (, F, P)using exponential vectors. Using C^*-algebra theory, given anymeasure space we construct a canonical generalised Poisson process.Unlike other constructions, we make no a priori use of Poissonmeasures. 2000 Mathematics Subject Classification 60G20, 60G35,46L53, 81S25.     

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V_* andW_*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals₁, ₂ V_* and ₁, ₂ W_* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V^** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.     

On the Fredholm Alternative for the p-Laplacian in One Dimension

We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L¹(0,a) is a given function. The number _k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Let_{p p} x/a) denote the eigenfunction associated with ₁; then _p(k_p x/a) is the eigenfunction associated with _k. We show the existenceof solutions to (P) in the following cases. (i) When k=1 and f satisfies the orthogonality condition the set of solutions is bounded. (ii) If k=1 and f_t L¹(0,a) is a continuous family parametrizedby t [0,1], with then there exists some t^* [0,1] such that (P) has a solutionfor f = f_t^*. Moreover, an appropriate choice of t_* yields asolution u with an arbitrarily large L¹(0,a)-norm which meansthat such f cannot be orthogonal to _pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto _p(k p x/a) on the subintervals , again, the set of solutions is bounded. is a continuous family satisfying either or another related condition, then there exists some t^* [0,1]such that (P) has a solution for f = f_t*. Prüfer's transformation plays the key role in our proofs.2000 Mathematical Subject Classification: primary 34B16, 47J10;secondary 34L40, 47H30.     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

Moduli of McKay quiver representations I: the coherent component

For a finite abelian group G GL (n, ), we describe the coherent component Y of the moduli space of-stable McKay quiver representations. This is a not-necessarily-normaltoric variety that admits a projective birational morphism obtained by variation of GeometricInvariant Theory quotient. As a special case, this gives a newconstruction of Nakamura's G-Hilbert scheme Hilb^G that avoidsthe (typically highly singular) Hilbert scheme of |G|-pointsin . To conclude, we describe the toric fan of Y and hence calculate the quiver representationcorresponding to any point of Y.     

Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2

Let F:Cⁿ Cⁿ be a holomorphic map, F^k be the kth iterate ofF, and p Cⁿ be a periodic point of F of period k. That is,F^k(p) = p, but for any positive integer j with j < k, F^j(p) p. If p is hyperbolic, namely if DF^k(p) has no eigenvalue ofmodulus 1, then it is well known that the dynamical behaviourof F is stable near the periodic orbit = {p, F(p),..., F^k–1(p)}.But if is not hyperbolic, the dynamical behaviour of F near may be very complicated and unstable. In this case, a veryinteresting bifurcational phenomenon may occur even though may be the only periodic orbit in some neighbourhood of : forgiven M N\{1}, there may exist a C^r-arc {F_t: t [0,1]} (wherer N or r = ) in the space H(Cⁿ) of holomorphic maps from Cⁿinto Cⁿ, such that F₀ = F and, for t (0,1], F_t has an Mk-periodicorbit _t with as t 0. Theperiod thus increases by a factor M under a C^r-small perturbation!If such an F_t does exist, then , as well as p, is said to beM-tupling bifurcational. This definition is independent of r. For the above F, there may exist a C^r-arc in H(Cⁿ), with t [0,1], such that and, for t (0,1], has two distinct k-periodic orbits _t,1 and _t,2 with d(_t,i, ) 0 as t 0 for i = 1,2. If such an does exist, then , as well as p, is said to be 1-tupling bifurcational. In recent decades, there have been many papers and remarkableresults which deal with period doubling bifurcations of periodicorbits of parametrized maps. L. Block and D. Hart pointed outthat period M-tupling bifurcations cannot occur for M >2 in the 1-dimensional case. There are examples showing thatfor any M N, period M-tupling bifurcations can occur in higher-dimensionalcases. An M-tupling bifurcational periodic orbit as defined here actsas a critical orbit which leads to period M-tupling bifurcationsin some parametrized maps. The main result of this paper isthe following. Theorem. Let k N and M N, and let F: C² C² be a holomorphicmap with k-periodic point p. Then p is M-tupling bifurcationalif and only if DF^k(p) has a non-zero periodic point of periodM. 1991 Mathematics Subject Classification: 32H50, 58F14.     

Weyl-Titchmarsh M-Function Asymptotics for Matrix-valued Schrodinger Operators

We explicitly determine the high-energy asymptotics for Weyl–Titchmarshmatrices corresponding to matrix-valued Schrödinger operatorsassociated with general self-adjoint m x m matrix potentials, where m N. More precisely,assume that for some N N and x₀R, for all c>x₀, and that x x₀ is a right Lebesgue point ofQ^(N–1). In addition, denote by I_m the mxm identity matrixand by C the open sector in thecomplex plane with vertex atzero, symmetry axis along the positive imaginary axis, and openingangle , with 0 < < . Then we prove the following asymptoticexpansion for any point M_+(z,x) of the unique limit point ora point of the limit disk associated with the differential expression in and a Dirichlet boundary condition at x=x₀: The expansion is uniform with respect to arg(z)for |z| in C and uniform in x as long as x varies in compactsubsets of R intersected with the right Lebesgue set of Q^(N–1).Moreover, the m x m expansion coefficients m_+,k(x) can be computedrecursively. Analogous results hold for matrix-valued Schrödinger operatorson the real line. 2000 Mathematics Subject Classification: 34E05,34B20, 34L40, 34A55.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

Asymptotic Behavior of Solutions of Differential Equations with Variable Delays

The authors consider the system of forced differential equationswith variable delays whereB_j(t) is a continuous n x n matrix on R⁺, F C(R⁺, Rⁿ) and C(R⁺, R⁺). Using Razumikhin-type techniques and Liapunov'sdirect method, they establish conditions to ensure the ultimateboundedness and the global attractivity of solutions of (*),and when F(t) = 0, the asymptotic stability of the zero solution.Under those same conditions, they also show that is a necessary and sufficient condition for allof the above properties to hold. 1991 Mathematics Subject Classification:34K15, 34C10.     

On Lp boundedness of wave operators for 4-dimensional Schrodinger operators with threshold singularities

Let H=–+V(x) be a Schrödinger operator on L²(R⁴),H₀=–. Assume that |V(x)|+| V(x)|C x ^– for some>8. Let be the wave operators. It is known that W_± extend to bounded operators in L^p(R⁴)for all 1p, if 0 is neither an eigenvalue nor a resonance ofH. We show that if 0 is an eigenvalue, but not a resonance ofH, then the W_± are still bounded in L^p(R⁴) for all psuch that 4/3<p<4.     

On Transfer Operators for Continued Fractions with Restricted Digits

For any non-empty subset I of the natural numbers, let _I denotethose numbers in the unit interval whose continued fractiondigits all lie in I. Define the corresponding transfer operator for , where Re (rß) = _I is the abscissa of convergence of the series . When acting on a certain Hilbert space H_I, rß, weshow that the operator L_I, rß is conjugate to an integraloperator K_I, rß. If furthermore rß is real,then K_I, rß is selfadjoint, so that L_I, rß: H_I, rß H_I, rß has purely real spectrum.It is proved that L_I, rß also has purely real spectrumwhen acting on various Hilbert or Banach spaces of holomorphicfunctions, on the nuclear space C [0, 1], and on the Fréchetspace C [0, 1]. The analytic properties of the map rß L_I, rßare investigated. For certain alphabets I of an arithmetic nature(for example, I = primes, I = squares, I an arithmetic progression,I the set of sums of two squares it is shown that rß L_I, rß admits an analytic continuation beyond thehalf-plane Re rß > _I. 2000 Mathematics SubjectClassification 37D35, 37D20, 30B70.     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

Trace Inequalities of Sobolev Type in the Upper Triangle Case

We give a new non-capacitary characterization of positive Borelmeasures µ on Rⁿ such that the potential space I*L^p isimbedded in L^q(dµ) for $1qp+, that is, the trace inequality holds, for Riesz potentials I = (- )². A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere Sⁿ is studied,as well as the holomorphic case for Hardy–Sobolev spaces in the ball. 1991 MathematicsSubject Classification: primary 31C15, 42B20; secondary 32A35.     

Cohomology of Quantized Function Algebras at Roots of Unity

Let G be a simply-connected, semisimple algebraic group overk, an algebraically closed field of characteristic zero. Let O[G] be the quantized function algebra of G at a primitivelth root of unity , and let be the ‘restricted’ quantized function algebra at, a finite-dimensional k-algebra obtained from O[G] by factoringout a centrally generated ideal. It is known that is a Hopf algebra. We study the cohomology ring, a graded commutative algebra, and, for any finite-dimensional -module M, the -module . We prove that for sufficiently large l there isan isomorphism of graded algebras where each X_i is homogeneous of degree $2$, and $2N$ equalsthe number of roots associated to G. Moreover we show that inthis case is a finitely generated -module. We also show under much less restrictive conditions on l that continues to be a finitely generated graded commutativealgebra over which is a finitely generated module. 1991 Mathematics Subject Classification: 16W30,17B37, 17B56.     

Algebraic K-Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H₁(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H₂(; Z) injects in and in , where A is as before, and ₂ is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.     

Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(O_S)where O_S is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of O_S*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and O_S thecorresponding S-arithmetic ring. Let Q be a subgroup of O_S*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:

(1) O_S Q is of type FP_n;

(2) O_S is n-tameas a ZQ-module;

(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.

If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:

(1) the group O_S Q is not of type FP_r+1};

(2) if Qhas maximum rank r = |S| –1 then the group O_S Q is oftype FP_r.

In particular, is of type FP_{|S| –1} but not of type FP_|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.     

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x ₀; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u₀ (x) on . Here is a regular and bounded domain in Rⁿ, with n 1, and₀ and ₁ endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u₀ H^1/2 , u₀ = 0 on ₀. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).     

Projection Methods for Discrete Schrodinger Operators

Let H be the discrete Schrödinger operator acting on l² Z⁺, where the potential v is real-valued and v(n) 0 as n . Let P be the orthogonal projection onto a closedlinear subspace l² Z⁺). In a recent paper E. B. Davies definesthe second order spectrum Spec₂(H, ) of H relative to as theset of z C such that the restriction to of the operator P(H- z)²P is not invertible within the space . The purpose of thisarticle is to investigate properties of Spec₂(H, ) when islarge but finite dimensional. We explore in particular the connectionbetween this set and the spectrum of H. Our main result providessharp bounds in terms of the potential v for the asymptoticbehaviour of Spec₂(H, ) as increases towards l² Z⁺). 2000 MathematicsSubject Classification 47B36 (primary), 47B39, 81-08 (secondary).     

On the Index of Vector Fields Tangent to Hypersurfaces with Non-Isolated Singularities

Let F be a germ of a holomorphic function at 0 in Cⁿ⁺¹, having0 as a critical point not necessarily isolated, and let be a germ of a holomorphic vectorfield at 0 in Cⁿ⁺¹ with an isolated zero at 0, and tangent toV := F^–1(0). Consider the O_V,0-complex obtained by contractingthe germs of Kähler differential forms of V at 0 (0.1) with the vector field X:=|_Von V: (0.2)