Variations on the theme of the inequality \(| f'| ^ 2\leq 2f\,\sup | f''|\)
type: Item
instance of: scholarly article
title: Variations on the theme of the inequality \(| f'| ^ 2\leq 2f\,\sup | f''|\)
zbMATH Open document ID: 0605.26010
DOI: 10.1007/BF01171035
author: Alois Kufner
author: V. Maz'ya
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155159
review text: 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.
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 3977314
zbMATH Keywords: inequality between derivatives
zbMATH Keywords: differential and pseudodifferential operators
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
cites work: Hypoellipticity of Certain Degenerate Elliptic Boundary Value Problems
cites work: Mixed-type boundary conditions for second order elliptic differential equations
cites work: On stability for difference schemes; a sharp form of gårding's inequality
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: Solvability of a first order linear partial differential equation
description: scientific article; zbMATH DE number 3977314