Sharp $L{\log }^{\alpha }L$ inequalities for conjugate functions

Essén, Matts; Shea, Daniel F.; Stanton, Charles S.

Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 623-659.

Nous donnons une méthode pour la construction des fonctions $\phi$ et $\psi$ telles que $H(x,y)=\phi \left(x\right)-\psi \left(y\right)$ aî t une minorante sousharmonique spécifiée. D’après un théorème de B. Cole, ce procédé établit des inégalités d’intégrales pour les fonctions conjuguées. Nous appliquons cette méthode pour déduire des inégalités optimales pour les conjuguées des fonctions de la classe $L{log}^{\alpha }L$, pour $-1\le \alpha <\infty$. En particulier, le cas $\alpha =1$ procure une amélioration de la version de Pichorides de l’inégalité classique de Zygmund pour les conjuguées des fonctions de $LlogL$. Nous appliquons aussi cette méthode pour obtenir une nouvelle preuve de l’inégalité de M. Riesz pour les fonctions de $L^p$ ($1<p<2$), avec meilleure constante. Toutes ces inégalités sont des cas spéciaux d’une inégalité générale et optimale pour les fonctions conjuguées (cf. Théorème 6).

We give a method for constructing functions $\phi$ and $\psi$ for which $H(x,y)=\phi \left(x\right)-\psi \left(y\right)$ has a specified subharmonic minorant $h(x,y)$. By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class $L{log}^{\alpha }L$, for $-1\le \alpha <\infty$. In particular, the case $\alpha =1$ yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in $LlogL$. We also apply the method to produce a new proof of the M. Riesz’s inequality for functions in $L^p$, $(1<p<2)$, also with sharp constant. All these inequalities are special cases of a general sharp inequality for conjugate functions (cf. Theorem 6).

Classification : 42A50, 30D55, 31A15
Keywords: conjugate functions, norm estimates, minimal thinness
Mot clés : fonctions conjuguées, estimation des normes, effilement minimal

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