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Weighted norm inequalities of some singular integrals associated with Laguerre expansions

The Anh Bui

Vol. 7 (2025), No. 3, 771–805
Abstract

Let ν = (ν1,,νn) (1,)n, n 1, and let ν be a self-adjoint extension of the differential operator

Lν := i=1n[ 2 xi2 + xi2 + 1 xi2(νi2 1 4)]

on Cc(+n) as the natural domain. We investigate the weighted estimates of singular integrals in the Laguerre setting including the maximal function, the Riesz transform and the square functions associated to the Laguerre operator ν. In the special case of the Riesz transform, we complete the description of the Riesz transform for the full range of ν (1,)n which significantly improves a result of Nowak and Stempak (J. Funct. Anal. 244:2 (2007), 399–443) for νi 1 2,νi(1 2, 1 2) for i = 1,,n.

Keywords
Laguerre function expansion, heat kernel, Riesz transform, square function, maximal function
Mathematical Subject Classification
Primary: 42B20, 42B25
Milestones
Received: 20 April 2024
Revised: 13 April 2025
Accepted: 27 June 2025
Published: 4 September 2025
Authors
The Anh Bui
School of Mathematical and Physical Sciences
Macquarie University
Sydney, NSW
Australia