Harmonic analysis operators related to symmetrized Jacobi expansions

Following a symmetrization procedure proposed recently by Nowak and Stempak, we consider the setting of symmetrized Jacobi expansions. In this framework we investigate mapping properties of several fundamental harmonic analysis operators, including Riesz transforms, Poisson semigroup maximal operator, Littlewood–Paley–Stein square functions and multipliers of Laplace and Laplace–Stieltjes transform type. Our paper delivers also some new results in the original setting of classical Jacobi expansions.     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

Riesz transforms for Jacobi expansions

We define Riesz transforms and conjugate Poisson integrals associated with multi-dimensional Jacobi expansions. Under a slight restriction on the type parameters, we prove that these operators are bounded in L ^p , 1 < p < ∞, with constants independent of the dimension. Our tools are suitably defined g-functions and Littlewood-Paley-Stein theory, involving the Jacobi-Poisson semigroup and modifications of it. Research of both authors supported by the European Commission via the Research Training Network “Harmonic Analysis and Related Problems”, contract HPRN-CT-2001-00273-HARP. The first-named author was also supported by MNiSW Grant N201 054 32/4285.     

Uncertainty principles for Jacobi expansions

In this paper an uncertainty principle for Jacobi expansions is derived, as a generalization of that for ultraspherical expansions by Rösler and Voit. Indeed a stronger inequality is proved, which is new even for Fourier cosine or ultraspherical expansions. A complex base of exponential type on the torus related to Jacobi polynomials is introduced, which are the eigenfunctions both of certain differential-difference operators of the first order and the second order. An uncertainty principle related to such exponential base is also proved.     

The convolution structure for Jacobi function expansions

The product ϕ _λ ^(α,β) (t₁)ϕ _λ ^(α,β) (t₂) of two Jacobi functions is expressed as an integral in terms of ϕ _λ ^(α,β) (t₃) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.     

The convolution structure for Jacobi function expansions

The product ? _λ ^(α,β) (t₁)? _λ ^(α,β) (t₂) of two Jacobi functions is expressed as an integral in terms of ? _λ ^(α,β) (t₃) with explicit non-negative kernel, when α≧β≧?1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.     

Biorthogonal expansions for symmetrizable operators

We give an extension of a classical result due to Krein on biorthogonal expansions of compact operators which are symmetrizable with respect to a nondegenerate positive operator. Our approach makes essential use of the spectral expansion of an appropriate compact selfadjoint operator, the existence of which is due to Dieudonné.     

Restricting Hecke-Siegel operators to Jacobi modular forms

We compute the action of Hecke operators on Jacobi forms of “Siegel degree” n and m×m index M, provided 1?j?n−m. We find they are restrictions of Hecke operators on Siegel modular forms, and we compute their action on Fourier coefficients. Then we restrict the Hecke-Siegel operators T(p), Tj(p²) (n−m<j?n) to Jacobi forms of Siegel degree n, compute their action on Fourier coefficients and on indices, and produce lifts from Jacobi forms of index M to Jacobi forms of index M^′ where detM|detM^′. Finally, we present an explicit choice of matrices for the action of the Hecke operators on Siegel modular forms, and for their restrictions to Jacobi modular forms.     

A commutator approach to absolute continuity for unbounded Jacobi operators

This paper uses commutator equations to study the absolute continuity of spectral measures associated with certain subclasses of unbounded self-adjoint Jacobi matrix operators determined by properties of the diagonal and subdiagonal sequences. If the diagonal sequence is the zero sequence, properties of the difference sequence of the subdiagonal determine the choice of a bounded operator for the commutator equation. The structure of the resulting commutator leads to results on absolute continuity.     

Infinite-dimensional analysis related to generalized translation operators

We give an extensive generalization of the white-noise analysis (in the Gaussian and non-Gaussian case) in which the role of translation operators is played by a fixed family of generalized translation operators. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 364–409, March, 1997.     

Harmonic functions associated to Dunkl operators

We prove some potential theoretical properties of harmonic functions associated to Dunkl operators. We solve the corresponding Dirichlet problem and establish the related Harnack principle and normality criteria.     

Harmonic functions associated to Dunkl operators

We prove some potential theoretical properties of harmonic functions associated to Dunkl operators. We solve the corresponding Dirichlet problem and establish the related Harnack principle and normality criteria.     

Some stability estimates for the symmetrized first eigenfunction of certain elliptic operators

We prove two stability-type estimates involving the Schwarz rearrangement of the normalized first eigenfunction u ₁?>?0 of certain linear elliptic operators whose first eigenvalue λ₁ is close to the lowest possible one (i.e., ${\lambda_1^\star}$ , the first eigenvalue of the Dirichlet Laplacian in a suitable ball). In particular, we prove that if ${\lambda_1\approx \lambda_1^\star}$ then the L ^∞-distance between the rearrangement ${u_1^\star}$ and the normalized first eigenfunction of the Dirichlet Laplacian corresponding to ${\lambda_1^\star}$ is less than a suitable power of the difference ${\lambda_1-\lambda_1^\star}$ times a universal constant. We also show that the L ^∞-distance between the first eigenfunction of the Dirichlet Laplacian in a ball whose first eigenvalue equals λ₁ and the rearrangement ${u_1^\star}$ can be controlled with a power of the value assumed by ${u_1^\star}$ on the boundary of that ball.