The best approximation of the differentiation operator in the metric of $L_p$

For Stechkin's problem of the best approximation for the differentiation operator
$$ E_n=\inf_{\substack{L_q\\ ||V||_{L_p}\leqslant n}}\sup_{||f^{(l)}||_{L_r(S)}\leqslant 1}||f^{(k)}-Vf||_{L_q(S)} $$
we indicate the necessary and sufficient conditions that $E_n$ be finite. We study some properties of continuous linear operators $V$ from $L_p$ into $L_q$.