Comparison of Linear Differential Operators

In this paper, we study the question of the existence of the inequality
$$ \|Q(D)f\|_{L_q}\le\gamma_0\|P(D)f\|_{L_p}, $$
where $P$ and $Q$ are algebraic polynomials, $D=d/dx$, and $\gamma_0$ is independent of the function $f$. We obtain criteria (necessary and simultaneously sufficient conditions) for the existence of such inequalities for functions on the circle, on the whole line, and on the semiaxis. Besides, for the semiaxis, we obtain an inequality for $q=\infty$ and any $p\ge1$ with the smallest constant $\gamma_0$.