Some maximal inequalities on Triebel-Lizorkin spaces for $p=\mathrm{\infty }$

In this work we give some maximal inequalities in Triebel-Lizorkin spaces, which are “${\dot{F}}_{\mathrm{\infty }}^{s,q}$-variants” of Fefferman-Stein vector-valued maximal inequality and Peetre’s maximal inequality. We will give some applications of the new maximal inequalities and discuss sharpness of some results.