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 Hölder constant of its -th derivative.
We apply these inequalities in order to obtain pointwise estimates on the derivative of the -th root of a function of class whose derivative of order is -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.
@article{ASNSP_2013_5_12_4_1001_0, author = {Ghisi, Marina and Gobbino, Massimo}, title = {Higher order {Glaeser} inequalities and optimal regularity of roots of real functions}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {1001--1021}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 12}, number = {4}, year = {2013}, mrnumber = {3184577}, zbl = {1317.26010}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_1001_0/} }
TY - JOUR AU - Ghisi, Marina AU - Gobbino, Massimo TI - Higher order Glaeser inequalities and optimal regularity of roots of real functions JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2013 SP - 1001 EP - 1021 VL - 12 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2013_5_12_4_1001_0/ LA - en ID - ASNSP_2013_5_12_4_1001_0 ER -
%0 Journal Article %A Ghisi, Marina %A Gobbino, Massimo %T Higher order Glaeser inequalities and optimal regularity of roots of real functions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2013 %P 1001-1021 %V 12 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2013_5_12_4_1001_0/ %G en %F ASNSP_2013_5_12_4_1001_0
Ghisi, Marina; Gobbino, Massimo. Higher order Glaeser inequalities and optimal regularity of roots of real functions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 1001-1021. http://www.numdam.org/item/ASNSP_2013_5_12_4_1001_0/
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