Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions

TL;DR

This paper extends the analysis of maximal operators associated with classical orthogonal expansions like Laguerre, Bessel, and Jacobi, to cases with unrestricted parameters, revealing new frameworks and proving weak type (1,1) estimates.

Contribution

It introduces new orthogonal systems with unrestricted parameters and establishes weak type (1,1) bounds for their maximal operators, expanding harmonic analysis in these contexts.

Findings

Proves weak type (1,1) estimates for maximal operators with unrestricted parameters.

Reveals new orthogonal systems related to classical expansions.

Provides a new proof for the classical Laguerre semigroup maximal operator.

Abstract

Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain parameters of type 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. An important partial result of the paper is a new proof of the weak type (1,1) estimate for the classical multi-dimensional Laguerre semigroup maximal operator.