  We study the stability of Triebel-Lizorkin regularity of bounded functions
and Lipschitz functions under bi-Lipschitz changes of variables and the
regularity of the inverse function of a Triebel-Lizorkin bi-Lipschitz map in
Lipschitz domains. To obtain the results we provide an equivalent norm for the
Triebel-Lizorkin spaces with fractional smoothness in uniform domains in terms
of the first-order difference of the last weak derivative available averaged on
balls.
