Extrapolation to mixed Herz spaces and its applications

Abstract

In this paper, we extend the extrapolation theory to mixed Herz spaces ${\dot{K}}_{\vec{q}}^{\alpha ,p}\left(R^n\right)$ and $K_{\vec{q}}^{\alpha ,p}\left(R^n\right)$. To prove the main result, we first study the dual spaces of mixed Herz spaces, and then give the boundedness of the Hardy–Littlewood maximal operator on mixed Herz spaces. By using the extrapolation theorems, we obtain the boundedness of many integral operators on mixed Herz spaces. We also give a new characterization of $\mathrm{bounded}\mathrm{mean}\mathrm{oscillation}\mathrm{space}\left(\mathrm{BMO}\right)\left(R^n\right)$ via the boundedness of commutators of some operators on mixed Herz spaces.