Monotonicity of certain functionals under rearrangement

Garsia, Adriano; Rodemich, Eugène

Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 67-116.

Nous montrons qu’une large classe d’inégalités intégrales concernant des fonctions sur $[0,1]$ peut être obtenue par des méthodes purement combinatoires. De façon plus précise, nous obtenons des modules de continuité ou d’autres estimations de normes d’ordre élevé pour des fonctions vérifiant des conditions du type

$${\int }_0^1{\int }_0^1\Psi \left(\frac{f\left(x\right)-f\left(y\right)}{p(x-y)}\right)dxdy<\infty$$

où $\Psi \left(u\right)$ et $p\left(u\right)$ sont des fonctions croissantes de $\left|u\right|$. On en déduit différentes applications. On montre en particulier que ces méthodes donnent une nouvelle condition pour la continuité des chemins d’un processus stochastique général.

We show here that a wide class of integral inequalities concerning functions on $[0,1]$ can be obtained by purely combinatorial methods. More precisely, we obtain modulus of continuity or other high order norm estimates for functions satisfying conditions of the type ${\int }_0^1{\int }_0^1\Psi \left(\frac{f\left(x\right)-f\left(y\right)}{p(x-y)}\right)dxdy<\infty$ where $\Psi \left(u\right)$ and $p\left(u\right)$ are monotone increasing functions of $\left|u\right|$.

Several applications are also derived. In particular these methods are shown to yield a new condition for path continuity of general stochastic processes

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