Integral inequalities for the Hilbert transform applied to a nonlocal transport equation
نویسندگان
چکیده
We prove several weighted inequalities involving the Hilbert transform of a function f (x) and its derivative. One of those inequalities, − ∫ fx(x)[Hf (x)−Hf (0)] |x|α dx Cα ∫ (f (x)− f (0))2 |x|1+α dx, is used to show finite time blow-up for a transport equation with nonlocal velocity. © 2006 Elsevier Masson SAS. All rights reserved. Résumé Dans cet article nous présentons la démonstration de plusieurs inegalités satisfaites par la transformation de Hilbert d’une fonction f (x) et sa derivée f ′(x). Nous avons l’estimation suivante : − ∫ fx(x)[Hf (x)−Hf (0)] |x|α dx Cα ∫ (f (x)− f (0))2 |x|1+α dx, où la constante Cα est strictement positive. Nous avons aussi utilisé cette inegalité pour démontrer l’explosion en un temps fini des solutions d’une équation de transport avec une vitesse nonlocale. © 2006 Elsevier Masson SAS. All rights reserved. MSC: 35Q35; 76B03; 42A38; 35L67
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