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

Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity