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

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

2011-07-13

arxiv.org/abs/1107.2694v2

Mathematics

Classical Analysis and ODEs

20 pages (references updated, restatement of the main result in terms of weak L^p spaces, new proof of the key step)

Scientific paper

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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