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.