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

• Marina Ghisi ^[1] ; Massimo Gobbino ^[1]
  1. [1] Università degli Studi di Pisa, Italia
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 12, Nº 4, 2013, págs. 1001-1021
• Idioma: inglés
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• Resumen

  • We prove a higher order generalization of the Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself and the H¨older constant of its k-th derivative.

    We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k + α)-th root of a function of class Ck whose derivative of order k is α-Hölder 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