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

Abstract

Consider the space X = ( 0 , ∞ ) X=(0,\infty ) equipped with the Euclidean distance and the measure d μ α ( x ) = x α d x d\mu _\alpha (x)=x^{\alpha }dx where α ∈ ( − 1 , ∞ ) \alpha \in (-1,\infty ) is a fixed constant and d x dx is the Lebesgue measure. Consider the Laguerre operator L = − d 2 d x 2 − α x d d x + x 2 \displaystyle L=-\frac {d^2}{dx^2} -\frac {\alpha }{x}\frac {d}{dx}+x^2 on X 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 L^{-\gamma }, \gamma >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 L p L^p spaces or the weighted Sobolev spaces in Laguerre settings.

Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

“…Similar results was also proved for the Laguerre operator in [52]. The theory of these function spaces was further developed in [17,18] where the authors proved molecular and atomic decompositions theorems and square function characterizations for these spaces. (iii) In [11] the authors introduced the theory of Besov spacesḂ s,L p,q associated to an operator L satisfying Poisson estimates on metric spaces with a measure enjoying a polynomial upper bound on volume growth.…”

Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators

“…Motivated by these two works, Hofmann et al [22] further established the theory of the Hardy spaces H p L (X), 1 ≤ p < ∞, on a metric measure space (X, d, µ) associated to a general nonnegative self-adjoint operator L satisfying Davies-Gaffney estimates. For further developments concerning Hardy spaces associated to operators, we refer to [8,10,11,12,24,28,32,37,38], among many others.…”

“…It is worth pointing out that Besov and Triebel-Lizorkin spaces associated to some particular operators were earlier studied by some authors. For instance, Besov and Triebel-Lizorkin spaces in the context of Hermite were studied by Petrushev and Xu [30] and Bui and Duong [7], while these spaces in the context of Laguerre were studied by Kerkyacharian et al [26] and Bui and Duong [8]. The spaces in [7] and [8] were introduced via continuous Littlewood-Paley functions associated to the heat semigroup (or Poisson semigroup) and, in some restricted cases (e.g., q = 2), their Lusin function characterization was obtained.…”

On the flows associated to selfadjoint operators on metric measure spaces

“…We refer to [6,Lemma 3.1] for the proof of the Gaussian upper bounds (46) and [37,Lemma 3.5] for the proof of the dispersive estimate (48).…”