Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions
Artikel i vetenskaplig tidskrift, 2017

Classical settings of discrete and continuous orthogonal expansions, like those of Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain type parameters that are usually restricted to a half-line, or a product of half-lines if higher dimensions are considered. Following earlier research done by Hajmirzaahmad, we deal in this paper with Laplacians in the above-mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the classical ones, but are different. This reveals new frameworks related to classical orthogonal expansions and thus new potentially rich research areas, at least from the harmonic analysis perspective. To support the last claim we focus on maximal operators of multi-dimensional Laguerre, Bessel and Jacobi semigroups, with unrestricted type parameters, and prove that they satisfy weak type (1, 1) estimates with respect to the appropriate measures. Generally, these measures are not (locally) finite, which makes a contrast with the classical situations and generates new difficulties. A significant result of the paper is a new proof of the weak type (1, 1) estimate for the classical multi-dimensional Laguerre semigroup maximal operator.

Hankel transform

Laguerre expansions

Jacobi expansions

Continuous Fourier Bessel expansions

Författare

Adam Nowak

Institute of Mathematics of the Polish Academy of Sciences

Peter Sjögren

Chalmers, Matematiska vetenskaper

Göteborgs universitet

Tomasz Z. Szarek

Institute of Mathematics of the Polish Academy of Sciences

Advances in Mathematics

0001-8708 (ISSN) 1090-2082 (eISSN)

Vol. 318 307-354

Ämneskategorier

Matematik

Fundament

Grundläggande vetenskaper

DOI

10.1016/j.aim.2017.07.026

Mer information

Skapat

2017-10-08