Higher order Glaeser inequalities and optimal regularity of roots of real functions

Marina Ghisi; Massimo Gobbino

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arXiv · 1107.2694

Higher order Glaeser inequalities and optimal regularity of roots of real functions

Marina Ghisi·Massimo Gobbino

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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.

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BibTeX RIS

Marina Ghisi, Massimo Gobbino. 2011-07-13. Higher order Glaeser inequalities and optimal regularity of roots of real functions. https://arxiv.org/abs/1107.2694

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