Estimates of commutators on Herz-type spaces with variable exponent and applications

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Let [b, T] be the commutator generated by b and T, where $$b\in \mathrm {BMO}({\mathbb {R}}^{n})$$ and T is a Calderon–Zygmund singular integral operator. In this paper, the authors establish some strong type and weak type boundedness estimates including the $$L\log L$$ type inequality for [b, T] on the Herz-type spaces with variable exponent. Meanwhile, the similar results for the commutators $$[b,I_l]$$ of fractional integral operator are also obtained. As applications, we consider the regularity in the Herz-type spaces with variable exponent of strong solutions to nondivergence elliptic equations with $$\mathrm {VMO}$$ coefficients.