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In this work we give some maximal inequalities in Triebel-Lizorkin spaces,which are "$\dot{F}_{\infty}^{s,q}$-variants" of Fefferman-Stein vector-valuedmaximal inequality and Peetre's maximal inequality. We will give someapplications of the new maximal inequalities and discuss sharpness of someresults.

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