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On weighted compactness of commutators of square function and semi-group maximal function associated to Schrödinger operators

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Abstract

Let \(\Delta\) be the Laplacian operator on \({\mathbb{R}}^n\) and V be a nonnegative potential satisfying an appropriate reverse Hölder inequality. The Littlewood–Paley square function g associated with the Schrödinger operator \(L=-\Delta +V\) is defined by:

$$\begin{aligned} g(f)(x)=\Big (\int _{0}^{\infty }\Big |\frac{d}{dt}e^{-tL}(f)(x)\Big |^2tdt\Big )^{1/2}. \end{aligned}$$

In this paper, we show that the commutators of g are compact operators on \(L^p(w)\) for \(1<p<\infty\) if \(b\in {\rm{CMO}}_\uptheta (\uprho )\) and \(w\in A_p^{\uprho ,\uptheta }\) , where \({\rm{CMO}}_\uptheta (\uprho ) ({\mathbb{R}}^n)\) denotes the closure of \(\mathcal{C}_c^\infty ({\mathbb{R}}^n)\) in the \(\mathrm{BMO}_\uptheta (\uprho )\) topology and \(A_p^{\uprho ,\uptheta }\) is a weighted class which is more larger than Muckenhoupt \(A_p\) weight class. An extra weight condition in a previous weighted compactness result is removed for the commutators of the semi-group maximal function defined by \(\mathcal{T}^*(f)(x)=\sup _{t>0}|e^{-tL}f(x)|.\)

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Acknowledgements

The authors want to express their sincere thanks to the referees for his/her valuable remarks and suggestions, which clearly improved the exposition of this paper. The authors were partly supported by the National Key R &D Program of China (No. 2020YFA0712900) and NNSF of China (No. 12271041).

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  1. Shifen Wang

    Present address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Shifen Wang & Chunmei Zhang

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Qingying Xue

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  1. Shifen Wang
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Wang, S., Xue, Q. & Zhang, C. On weighted compactness of commutators of square function and semi-group maximal function associated to Schrödinger operators. Collect. Math. 75, 129–148 (2024). https://doi.org/10.1007/s13348-022-00381-6

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