To the question of smoothness of isometries

Combining the results of a number of authors, one may claim that the general theorem on smoothness of isometries should take the next form: If $M$ and $N$ are two isometric Riemannian manifolds of class $C^{n,\alpha}$, $n\ge0$, $0\ge\alpha\ge1$, $n+\alpha>0$, then any isometrie $f\colon M\to N$ has smoothness of class $C^{n+1,\alpha }$. Such a theorem was lacking in proof for the case $n+\alpha=1$, i.e., foT manifolds of class $C^{0,1}$ and of class $C^{1,0}$. In the article we prove that in the above cases the smoothness of an isometry $f$ is in fact the same as in the general situation: $f$ is of class $C^{1,1}$ or of class $C^{2,0}$ respectively.