Journal of the
Korean Mathematical Society JKMS

In this paper, we show that the commutator of the intrinsic square function with {\rm BMO} symbols is bounded on the variable exponent Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$ applying a generalization of the classical Rubio de Francia extrapolation. As a consequence we further establish its boundedness on the variable exponent Morrey spaces $\mathcal{M}_{p(\cdot), u}$, Morrey-Herz spaces $M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot), \lambda}({\mathbb { R}}^n)$ and Herz type Hardy spaces $H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\mathbb { R}}^n)$, where the exponents $\alpha(\cdot)$ and $p(\cdot)$ are variable. Observe that, even when $\alpha(\cdot)\equiv \alpha$ is constant, the corresponding main results are completely new.

Keywords: the intrinsic square function, commutator, variable exponents, Morrey spaces, Herz type spaces

MSC numbers: Primary 46E30; Secondary 42B35