The boundedness on $ BMO^\beta_{L_\alpha} $ of maximal operators for semigroups related to Laguerre operator

Li Yuan; Jinglan Jia; Ping Li; Zhu Wen; Li Yuan; Jinglan Jia; Ping Li; Zhu Wen

doi:10.3934/era.2025275

Electronic Research Archive

2025, Volume 33, Issue 10: 6219-6240. doi: 10.3934/era.2025275

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Research article

The boundedness on $ BMO^\beta_{L_\alpha} $ of maximal operators for semigroups related to Laguerre operator

Li Yuan ¹,
Jinglan Jia ^{2
,
,},
Ping Li ³,
Zhu Wen ⁴

1.
Department of Mathematics and Computer Science, Hanjiang Normal University, Shiyan 442000, China
2.
School of Information and Mathematics, Yangtze University, Jingzhou 434023, China
3.
College of Science, Wuhan University of Science and Technology, Wuhan 430065, China
4.
School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

Received: 21 July 2025 Revised: 23 September 2025 Accepted: 11 October 2025 Published: 21 October 2025

For $ \alpha > -\frac{1}{2} $, the Laguerre differential operator is defined as
$ L_\alpha = \frac{1}{2}\Big(-\frac{d^2}{dx^2}+x^2+\frac{1}{x^2}\big(\alpha^2-\frac{1}{4}\big)\Big), \,x\in(0,\infty). $
For sufficiently good function $ f $, the maximal functions associated with heat and Poisson semigroups are defined by
$ Tf(x) = \sup\limits_{t>0}|T_tf(x)|,\,x\in(0,\infty), $
where $ \{T_t\}_{t > 0} $ is the heat semigroup $ \{e^{-tL_\alpha}\}_{t > 0} $ or Poisson semigroup $ \{e^{-t\sqrt{L_\alpha}}\}_{t > 0} $ related to the Laguerre differential operator $ L_\alpha $. In this paper, we first established a $ T1 $ criterion for the boundedness of the $ \gamma $-Laguerre-Calderón-Zygmund operator on $ BMO^{\beta}_{L_{\alpha}}((0, \infty)) \; (0\leq\beta\leq1) $ spaces related to the Laguerre differential operator $ L_\alpha $. As applications, using this $ T1 $ criterion, we proved the boundedness on $ BMO^{\beta}_{L_{\alpha}}((0, \infty)) \; (0\leq\beta\leq1) $ of the maximal operators for semigroups related to the Laguerre differential operator $ L_\alpha $.
- Laguerre operators,
- heat semigroup,
- Poisson semigroup,
- T1 theorem,
- Campanato type spaces
Citation: Li Yuan, Jinglan Jia, Ping Li, Zhu Wen. The boundedness on $ BMO^\beta_{L_\alpha} $ of maximal operators for semigroups related to Laguerre operator[J]. Electronic Research Archive, 2025, 33(10): 6219-6240. doi: 10.3934/era.2025275

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Abstract

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

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

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

$ Tf(x) = \sup\limits_{t>0}|T_tf(x)|,\,x\in(0,\infty), $

where $ \{T_t\}_{t > 0} $ is the heat semigroup $ \{e^{-tL_\alpha}\}_{t > 0} $ or Poisson semigroup $ \{e^{-t\sqrt{L_\alpha}}\}_{t > 0} $ related to the Laguerre differential operator $ L_\alpha $. In this paper, we first established a $ T1 $ criterion for the boundedness of the $ \gamma $-Laguerre-Calderón-Zygmund operator on $ BMO^{\beta}_{L_{\alpha}}((0, \infty)) \; (0\leq\beta\leq1) $ spaces related to the Laguerre differential operator $ L_\alpha $. As applications, using this $ T1 $ criterion, we proved the boundedness on $ BMO^{\beta}_{L_{\alpha}}((0, \infty)) \; (0\leq\beta\leq1) $ of the maximal operators for semigroups related to the Laguerre differential operator $ L_\alpha $.

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