BOUNDEDNESS OF THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION IN VARIABLE EXPONENT SPACES

In this paper, we show that the commutator of the intrinsic square function with 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 $M{\dot{K}}^{{\alpha}({\cdot}),{\lambda}}_{q,p({\cdot})}({\mathbb{R}}^n)$ and Herz type Hardy spaces $H{\dot{K}}^{{\alpha}({\cdot}),q}_{p({\cdot})}({\mathbb{R}}^n)$, 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.