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BV Capacity and Sobolev Capacity for the Laguerre Operator

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Abstract

This paper considers some topics of geometric measure theory related to the Laguerre operator. Firstly, this paper is devoted to introducing and investigating the so-called Laguerre bounded variation capacity, thereby discovering some Poincaré type inequalities and BV-isocapacity inequalities. Secondly, we define the Laguerre p-capacity and discuss properties of capacity for the space \(W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)\). Moreover, we study the relation between the \(\mathcal {L}^{\alpha }\)-BV capacity and the Laguerre 1-capacity. Finally, we prove the Laguerre p-capacitary-strong-type inequality and the trace inequality for \(W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)\).

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Acknowledgements

Y. Liu was supported by the National Natural Science Foundation of China (No. 11671031, No. 12271042) and Beijing Natural Science Foundation of China (No. 1232023).

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  1. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    He Wang & Yu Liu

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  1. He Wang
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  2. Yu Liu
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Correspondence to Yu Liu.

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Communicated by Rosihan M. Ali.

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Wang, H., Liu, Y. BV Capacity and Sobolev Capacity for the Laguerre Operator. Bull. Malays. Math. Sci. Soc. 46, 104 (2023). https://doi.org/10.1007/s40840-023-01500-7

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