Weighted norm inequalities of some singular integrals associated with Laguerre expansions

2024/11/28 by The Anh Bui, Bui, The Anh
Mathematics · #Mathematical Inequalities and Applications #Approximation Theory and Sequence Spaces #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2411.19404

Abstract

Let ν=(ν₁,…,ν_n)∈ (-1,∞)ⁿ, n≥ 1, and let L_ν be a self-adjoint extension of the differential operator L_ν:= ∑_i=1ⁿ [-(∂²)/(∂ x_i²) + x_i² + (1)/(x_i²)(ν_i² - (1)/(4))] on C_c^∞(ℝ₊ⁿ) as the natural domain. In this paper, 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 \mathcal L_ν. In the special case of the Riesz transform, the paper completes the description of the Riesz transform for the full range of ν∈ (-1,∞)ⁿ which significant improves the result in [J. Funct. Anal. 244 (2007), 399--443] for ν_i≥ -1/2, ν_i∉ (-1/2,1/2) for i=1,…, n.

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