Abstract
The aim of this paper is to establish the boundedness of the commutator \([b_{1}, b_{2}, T_{\sigma }]\) generated by the bilinear pseudo-differential operator \(T_{\sigma }\) with smooth symbols and \(b_{1},\ b_{2}\in \mathrm {BMO}({\mathbb {R}}^{n})\) on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator \(T_{\sigma }\) is bounded from the Lebesgue space \(L^{p}({\mathbb {R}}^{n})\) into \(h^{p_{1}(\cdot )}({\mathbb {R}}^{n})\times h^{p_{2}(\cdot )}({\mathbb {R}}^{n})\). Moreover, the boundedness of the commutator \([b_{1}, b_{2}, T_{\sigma }]\) on product of local Hardy spaces with variable exponents is also obtained.
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The research is supported by the Doctoral Scientific Research Foundation of Northwest Normal University (No. 0002020203).
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Lu, G. Commutators of Bilinear Pseudo-differential Operators on Local Hardy Spaces with Variable Exponents. Bull Braz Math Soc, New Series 51, 975–1000 (2020). https://doi.org/10.1007/s00574-019-00184-7
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DOI: https://doi.org/10.1007/s00574-019-00184-7
Keywords
- Bilinear pseudo-differential operator
- Commutator
- BMO function
- Local Hardy spaces with variable exponent