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.

中文翻译：

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变指数局部哈代空间上双线性伪微分算子的换向器

本文的目的是建立由双线性伪微分算子$$T_{\sigma }$生成的交换子$$[b_{1}, b_{2}, T_{\sigma }]$$的有界性$ 与平滑符号和 $$b_{1},\ b_{2}\in \mathrm {BMO}({\mathbb {R}}^{n})$$ 在局部哈代空间与可变指数的乘积上。通过应用精细的原子分解结果，作者证明了双线性伪微分算子 $$T_{\sigma }$$ 是有界于勒贝格空间 $$L^{p}({\mathbb {R}}^{ n})$$ 变成 $$h^{p_{1}(\cdot )}({\mathbb {R}}^{n})\times h^{p_{2}(\cdot )}({\ mathbb {R}}^{n})$$。此外，还得到了换向子$$[b_{1}, b_{2}, T_{\sigma }]$$对局部具有变指数的Hardy空间的乘积的有界性。