Let [b, T] be the commutator generated by b and T, where \(b\in \mathrm {BMO}({\mathbb {R}}^{n})\) and T is a Calderón–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.

令[ b，  T ]为b和T生成的换向器，其中\（b \ in \ mathrm {BMO}（{\ mathbb {R}} ^ {n}）\），T为Calderón–Zygmund奇异积分操作员。在本文中，作者建立了具有可变指数的Herz型空间上[ b，  T ]的\（L \ log L \）型不等式的一些强类型和弱类型有界估计。同时，换向器\（[b，I_1] \）的结果相似还获得分数积分算子的。作为应用，我们考虑具有\（\ mathrm {VMO} \）系数的非散度椭圆型方程的强解的可变指数的Herz型空间中的正则性。