Singular spherical maximal operators on a class of two step nilpotent lie groups

LetH ⁿ ≅ℝ²ⁿ ⋉ℝ be the Heisenberg group and letμ _t be the normalized surface measure for the sphere of radiust in ℝ²ⁿ . Consider the maximal function defined byM f=sup _t>0|f*μ _t |. We prove forn≥2 thatM defines an operator bounded onL ^p (H ⁿ ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL ^p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.     

Unimodal betti numbers for a class of nilpotent lie algebras

In this paper we prove that all finite dimens~onal complex nilpotent Lie algebras containing an abelian ideal of codimension 1 have unirnodal Betti numbers.     

Parametrix constructions for right invariant differential operators on nilpotent groups

Algebras of right invariant pseudo-differential operators are constructed on any graded nilpotent group G . We obtain parametrices in such algebras for right invariant differential operators P such that P and its adjoint satisfy the hypoellipticity condition of Rockland .     

Spanning L 2 of a nilpotent Lie group by eigenvectors of invariant differential operators

Let be a non-abelian, connected, simply connected, nilpotent Lie group. We show that the eigenvectors of a finite number of families of left invariant differential operators and their conjugates span a dense subspace of L ² (G). The restriction of the left regular representation to each one of these (left invariant) eigenspaces disintegrates into irreducible unitary representations with multiplicities 0 and 1 only. J. Ludwig and C. Molitor-Braun are supported by the research grant R1F104C09 of the University of Luxembourg.     

Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators

We study a class of Schrödinger operators of the form , where is a nonnegative function singular at 0, that is V(0)=0. Under suitable assumptions on the potential V, we derive sharp lower and upper bounds for the fundamental solution hε. Moreover, we obtain information on the spectrum of the self-adjoint operator defined by Lε in L²(R). In particular, we give a lower bound for the eigenvalues.     

On the fundamental solutions of a class of hyperbolic quartic operators in dimension 3

The fundamental solutions of the linear hyperbolic partial differential operators with constant coefficients of the form are represented by elliptic integrals of the first kind. Mathematics Subject Classification (2000) 35A08, 35A20, 35E05     

The existence of Hall polynomials for x2-bounded invariant subspaces of nilpotent linear operators

We prove the existence of Hall polynomials for $x^2$-bounded invariant subspaces of nilpotent linear operators.     

Computational matrix representation modules for linear operators with explicit constructions for a class of lie operators

A linear mapping from a finite-dimensional linear space to another has a matrix representation. Certain multilinear functions are also matrix-representable. Using these representations, symbolic computations can be done numerically and hence more efficiently. This paper presents an organized procedure for constructing matrix representations for a class of linear operators on finite-dimensional spaces. First we present serial number functions for locating basis monomials in the linear space of homogeneous polynomials of fixed degree, ordered under structured lexicographies. Next basic lemmas describing the modular structure of matrix representations for operators constructed canonically from elementary operators are presented. Using these results, explicit matrix representations are then given for the Lie derivative and Lie-Poisson bracket operators defined on spaces of homogeneous polynomials. In particular, they are comprised of blocks obtained as Kronecker sums of modular components, each corresponding to specific Jordan blocks. At an implementation level, recursive programming is applied to construct these modular components explicitly. The results are also applied to computing power series approximations for the center manifold of a dynamical system. In this setting, the linear operator of interest is parameterized by two matrices, a generalization of the Lie-Poission bracket.     

A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces

Let G be a connected, simply connected real nilpotent Lie group with Lie algebra , H a connected closed subgroup of G with Lie algebra and f a linear form on satisfying f([, ]) = {0} Let _f be the unitary character of H with differential at the origin. Let _f be the unitary representation of G induced from the character _f of H. We consider the algebra (, , f) of differential operators invariant under the action of G on the bundle with basis G/H associated to these data. We show that (, , f) is commutative if and only if _f is of finite multiplicities. This proves a conjecture of Corwin-Greenleaf and Duflo. Mathematics Subject Classification (1991):43A80, 43A85, 22E25, 22E27, 22E30UMR n 7539 du CNRS, Analyse, Géométrie et Applications.UMR n 7586 du CNRS, Théorie des Groupes, Représentations, Applications.     

Explicit solutions for a general class of Markov processes

We derive explicit solutions for the matrix-geometric stationary distribution (and related measures) of a general class of infinite multidimensional Markov processes based on the probabilistic interpretations of the fundamental solution matrices and Bernoulli excursions, leading to explicit expressions for these matrix elements in terms of hypergeometric functions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators

In the framework of [5] we prove regularity of invariant measures for a class of Ornstein-Uhlenbeck operators perturbed by a drift which is not necessarily bounded or Lipschitz continuous. Regularity here means that is absolutely continuous with respect to the Gaussian invariant measure of the unperturbed operator with the square root of the Radon-Nikodym density in the corresponding Sobolev space of order 1.Partially supported by the International Science Foundation (Grant M 38000), the Russian Foundation of Fundamental Research (Grant 94-01-01556), and EC-Science Project SC1^*CT92-0784.Partially supported by the Italian National Project MURST Problemi nonlineari nell'AnalisiPartially supported by the DFG(SFB-256-Bonn, SFB-343-Bielefeld) and EC-Science Project SC1^*CT92-0784.     

On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces

Let be a simply connected connected real nilpotent Lie group with Lie algebra , a connected closed subgroup of with Lie algebra and satisfying . Let be the unitary character of with differential at the origin. Let be the unitary representation of induced from the character of . We consider the algebra of differential operators invariant under the action of on the bundle with basis associated to these data. We consider the question of the equivalence between the commutativity of and the finite multiplicities of . Corwin and Greenleaf proved that if is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases.