Comparison of Linear Differential Operators

R. M. Trigub

Matematičeskie zametki, Tome 82 (2007) no. 3, pp. 426-440

Résumé

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 $\gamma_0$.

Détail

Keywords: linear differential operator, Kolmogorov multiplicative inequality, Fourier series, Schoenberg spline, Hölder's inequality, Minkowski's inequality.
Mots-clés : algebraic polynomial