The paper deals with generalizations of the classical Landau inequality on the real line \({\mathbb R}\): \[ \mid u^{\prime}(x)\mid^2 \leq 2\| u\| _{L_{\infty}({\mathbb R})}\| u^{\prime \prime}\| _{L_{\infty}({\mathbb R})},\tag{1} \] in the way, when \(|u^{\prime}(x)|^2\) and the \(L_{\infty}\)-norms in (1) are replaced by values at the point \(x\) of certain operators acting on \(u\). The pointwise interpolation inequalites generalizing (1) and involving gradients of the first and any natural orders, the Hardy-Littlewood maximal operator, operator in the theory of the fractional Sobolev spaces, the Riesz and Bessel potential operators of fractional order are given.
The results presented in the article were basically published in the papers by the authors [Math. Bohem. 124, No. 2–3, 131–148 (1999; Zbl 0936.26008); in: Analytical and computational methods in scattering and applied mathematics. CRC Res. Notes Math. 417, 217–229 (2000; Zbl 0968.47007) and Funct. Anal. Appl. 36, No. 1, 30–48 (2002); translation from Funkts. Anal. Prilozh. 36, No. 1, 36–58 (2002; Zbl 1034.42018)].
Remark. There are two misprints on p. 217: 1) on line 16 up, in the right-hand side of the formula “\(\partial^{\alpha -1}_{x_1}\)” must be replaced by “\(\partial^{\alpha_1}_{x_1}\)”; 2) on line 12 up the citation “[MS1]” must be replaced by the citation “[15]” in the References.
For the entire collection see [Zbl 1023.00025].