Higher order Glaeser inequalities and optimal regularity of roots of real functions
Classical Analysis and ODEs
2012-01-27 v2
Abstract
We prove a higher order generalization of Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself, and the Holder constant of its k-th derivative. We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k+alpha)-th root of a function of class C^{k} whose derivative of order k is alpha-Holder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent. Some examples show that our results are optimal.
Keywords
Cite
@article{arxiv.1107.2694,
title = {Higher order Glaeser inequalities and optimal regularity of roots of real functions},
author = {Marina Ghisi and Massimo Gobbino},
journal= {arXiv preprint arXiv:1107.2694},
year = {2012}
}
Comments
20 pages (references updated, restatement of the main result in terms of weak L^p spaces, new proof of the key step)