TY - JOUR

T1 - Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces

AU - Bui, The Anh

AU - Duong, Xuan Thinh

PY - 2017

Y1 - 2017

N2 - Consider the space X = (0,∞) equipped with the Euclidean distance and the measure dμα(x) = xαdx where α ∈ (−1,∞) is a fixed constant and dx is the Lebesgue measure. Consider the Laguerre operator (Formula Presented) on X. The aim of this article is threefold. Firstly, we establish a Calderón reproducing formula using a suitable distribution of the Laguerre operator. Secondly, we study certain properties of the Laguerre operator such as a Harnack type inequality on the solutions and subsolutions of Laplace equations associated to Laguerre operators. Thirdly, we establish the theory of the weighted homogeneous Besov and Triebel-Lizorkin spaces associated to the Laguerre operator. We define the weighted homogeneous Besov and Triebel-Lizorkin spaces by the square functions of the Laguerre operator, then show that these spaces have an atomic decomposition. We then study the fractional powers L−γ, γ > 0, and show that these operators map boundedly from one weighted Besov space (or one weighted Triebel-Lizorkin space) to another suitable weighted Besov space (or weighted Triebel-Lizorkin space). We also show that in particular cases of the indices, our new weighted Besov and Triebel-Lizorkin spaces coincide with the (expected) weighted Hardy spaces, the weighted Lp spaces or the weighted Sobolev spaces in Laguerre settings.

AB - Consider the space X = (0,∞) equipped with the Euclidean distance and the measure dμα(x) = xαdx where α ∈ (−1,∞) is a fixed constant and dx is the Lebesgue measure. Consider the Laguerre operator (Formula Presented) on X. The aim of this article is threefold. Firstly, we establish a Calderón reproducing formula using a suitable distribution of the Laguerre operator. Secondly, we study certain properties of the Laguerre operator such as a Harnack type inequality on the solutions and subsolutions of Laplace equations associated to Laguerre operators. Thirdly, we establish the theory of the weighted homogeneous Besov and Triebel-Lizorkin spaces associated to the Laguerre operator. We define the weighted homogeneous Besov and Triebel-Lizorkin spaces by the square functions of the Laguerre operator, then show that these spaces have an atomic decomposition. We then study the fractional powers L−γ, γ > 0, and show that these operators map boundedly from one weighted Besov space (or one weighted Triebel-Lizorkin space) to another suitable weighted Besov space (or weighted Triebel-Lizorkin space). We also show that in particular cases of the indices, our new weighted Besov and Triebel-Lizorkin spaces coincide with the (expected) weighted Hardy spaces, the weighted Lp spaces or the weighted Sobolev spaces in Laguerre settings.

UR - http://www.scopus.com/inward/record.url?scp=85006170214&partnerID=8YFLogxK

U2 - 10.1090/tran/6745

DO - 10.1090/tran/6745

M3 - Article

VL - 369

SP - 2109

EP - 2150

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -