Related papers: Difference-elliptic operators and root systems
We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat…
We propose a new technique to generate reasonable systems of partial differential equations (PDE) that could be potential candidates for depicting models in natural sciences related to quasi-linear equations. Such systems appear within…
Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the…
The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions are studied making use of the De Rham-Hodge-Skrypnik differential complex. The…
The ring $\text{Diff}_{\mathbf{h}}(n)$ of $\mathbf{h}$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\mathbf{h}$-deformed…
We prove uniform $L^p$ estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding resul of [3] in the case of Laplace-- Beltrami operators on…
This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior…
We study a family of mutually commutative difference operators associated with the affine root systems. These operators act on the space of meromorphic functions on the Cartan subalgebra of the affine Lie algebra. We show that the space…
We define two isomorphic algebras of differential operators: the first algebra consists of ordinary differential operators and contains the hypergeometric differential operator, while the second one consists of partial differential…
Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…
We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a…
In this work we obtain continuity results on H\"older spaces for operators belonging to a Weyl-H\"ormander calculus for metrics such that the class of the associated operators contains, in particular, certain hypoelliptic laplacians. With…
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix…
We define the formal affine Demazure algebra and formal affine Hecke algebra associated to a Kac-Moody root system. We prove the structure theorems of these algebras, hence, extending several result and construction (presentation in terms…
The purpose of this paper is to study algebras of singular integral operators on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the composition of Calder\'on-Zygmund operators with different homogeneities, such as…
This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic…
Based on the notion of the resolvent and on the Hilbert identities, this paper presents a number of classical results in the theory of differential operators and some of their applications to the theory of automorphic functions and number…
In a recent article, Arendt and ter Elst have shown that every sectorial form is in a natural way associated with the generator of an analytic strongly continuous semigroup, even if the form fails to be closable. As an intermediate step…
This work continues the research of generalized Heisenberg algebras connected with several orthogonal polynomial systems. The realization of the annihilation operator of the algebra corresponding to a polynomial system by a differential…
We introduce the so-called Clifford-Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential…