Dependence of the differentiability of functions of several variables on their rate of approximation by rational functions

Let $E$ be a Lebesgue measurable subset of a $k$-dimensional cube ($k\geqslant1$), let $f\in L_p[E]$, where $0<p\leqslant\infty$, and let $R_n[f,p,E]$ be the least deviation of $f$, in the metric of $L_p[E]$, from the rational functions of degre $\leqslant n$. If $R_n[f,p,E]=O(n^{-\lambda})$, then, for $0<\mu<\lambda$, $f$ has a local differential of order $\mu$ in the $L_p$-metric at each point $\xi\in E$, except perhaps points $\xi$ of some set of metric dimension $\leqslant k-1+(p\mu+1)/(p\lambda+1)$ (this inequality is sharp). In addition, $f$ has a global differential of order $\mu$ in the metric of $L_q [E]$ for any $q<p/(p\mu+1)$.
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