L2-Theory of Riesz Transforms for Orthogonal Expansions

Abstract

We propose a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions. The scheme is supported by numerous examples concerning, in particular, the classical orthogonal expansions in Hermite, Laguerre, and Jacobi polynomials. A general case of expansions associated to a regular or singular SturmLiouville problem is also discussed.

Riesz transforms for the Dunkl harmonic oscillator

“…In [6] the authors developed a unified approach to the theory of Riesz transforms in the setting of multi-dimensional orthogonal expansions with a rather general second order differential operator as the underlying "Laplacian". It is remarkable that this approach also works within the framework of differential-difference operators on R d related to finite reflection groups, the setting intimately connected with the Dunkl theory, which has gained a considerable interest in various fields of mathematics as well as in theoretical physics during the last years.…”

Weak-Type Inequality for Conjugate First Order Riesz-Laguerre Transforms

“…follows from [18]. Now taking into account that U i * f = sup t>0 |e −L 1/2 α t R * i f | then from the Maximal Theorem on Semigroups proved in [24, p. 73] and from (2.1) we get that…”

“…In 2006, Nowak and Stempak [18] proposed a fairly general unified approach to the theory of Riesz transforms and their conjugate in the setting of multi-dimensional orthogonal expansions. In that paper they consider a second order differential operator L self-adjoint with respect to a measure dμ such that it has a decomposition of the form…”

Transferring Boundedness from Conjugate Operators Associated with Jacobi, Laguerre, and Fourier–Bessel Expansions to Conjugate Operators in the Hankel Setting

“…A conjugation associated to Laguerre polynomials was introduced firstly by Muckenhoupt [19]. In [22] conjugate function in the Laguerre setting is defined as follows.…”

“…Then R B α,1 maps continuously L 2 2α+1 (0, 1) into itself. Recently, Ciaurri and Stempak [8] and Nowak and Stempak [22] considered a new conjugation in the Fourier-Bessel setting. For an appropriate function f , the ideas in [6] and [22] suggest to define, in our setting, the conjugate R B…”