Variations on the theme of the inequality \(| f'| ^ 2\leq 2f\,\sup | f''|\)
The well-known inequality in the title is useful in the theory of differential and pseudodifferential operators. The authors present simple proofs of several inequalities related to that one, some of which being generalizations or refinements of inequalities proved by V. G. Maz'ja and other authors. They start with estimates juxtaposing the values of f(x), f'(x) and of the modulus of continuity of f' at x for non-negative functions f. This implies an inequality which differs from the title one by the Hölder constant of f' standing on the place of \(\sup f''.\) Further, a weighted \(L^ p\)-estimate of the first derivative of \(f^{1/2}\) is derived as well as a non-weighted multidimensional analogue for powers of functions. Some results are extended from non- negative functions to functions without simple zeros.
differential and pseudodifferential operators
scientific article; zbMATH DE number 3977314