The boundedness on $B M O_{L_{\alpha}}^{\beta}$ of maximal operators for semigroups related to Laguerre operator

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Abstract

For $\alpha>-\frac{1}{2}$, the Laguerre differential operator is defined as

$L_{\alpha}=\frac{1}{2}\left(-\frac{d^{2}}{d x^{2}}+x^{2}+\frac{1}{x^{2}}\left(\alpha^{2}-\frac{1}{4}\right)\right), x \in(0, \infty) .$

For sufficiently good function f, the maximal functions associated with heat and Poisson semigroups are defined by

$T f(x)=\sup _{t>0}\left|T_{t} f(x)\right|, x \in(0, \infty),$

where $\left\{T_{t}\right\}_{t>0}$ is the heat semigroup {e^{−tL_α}}_{t>0} or Poisson semigroup $\left\{e^{-t L_{\alpha}}\right\}_{t>0}$ related to the Laguerre differential operator L_α. In this paper, we first established a T1 criterion for the boundedness of the γ-Laguerre-Calderón-Zygmund operator on $B M O_{L_{\alpha}}^{\beta}((0, \infty))(0 \leq \beta \leq 1)$ spaces related to the Laguerre differential operator L_α. As applications, using this T1 criterion, we proved the boundedness on $B M O_{L_{\alpha}}^{\beta}((0, \infty))(0 \leq \beta \leq 1)$ of the maximal operators for semigroups related to the Laguerre differential operator L_α.

Keywords

Laguerre operators / heat semigroup / Poisson semigroup / T1 theorem / Campanato type spaces

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