Exact constants in inequalities between norms of derivatives of functions

V. N. Gabushin

Matematičeskie zametki, Tome 4 (1968) no. 2, pp. 221-232.

Résumé

In this article we shall concern ourselves with determining exact (least possible) constants in the inequalities of the form $\|f^{(k)}\|_{L_q}\le K\|f\|_{L_p}^{\frac{l-k-r^{-1}+q^{-1}}{l-r^{-1}+p^{-1}}}\|f^{(l)}\|_{L_r}^{\frac{k-q^{-1}+p^{-1}}{l-r^{-1}+p^{-1}}}$ for functions defined on the entire $(-\infty,\infty)$, absolutely continuous on any interval together with their $(l-1)$-th derivatives, and having finite $$ l=2,\quad k=0,\quad k=1,\quad q=r=\infty,\quad 1\leqslant p\infty $$ is considered.