We study connections between the Boyd indices in Orlicz spaces and the growth conditions frequently met in various applications, for instance, in the regularity theory of variational integrals with non-standard growth. We develop a truncation method for computation of the indices and we also give characterizations of them in terms of the growth exponents and of the Jensen means. Applications concern variational integrals and extrapolation of integral operators.

Some examples of the close interaction between inequalities and interpolation are presented and discussed. An interpolation technique to prove generalized Clarkson inequalities is pointed out. We also discuss and apply to the theory of interpolation the recently found facts that the Gustavsson-Peetre class P+- can be described by one Carlson type inequality and that the wider class P0 can be characterized by another Carlson type inequality with blocks.

On démontre dans cet article des versions probabilistes des injections de Sobolev sur une variété riemannienne compacte, $(M,g)$. Plus précisément on démontre que pour des mesures de probabilité naturelles sur l’espace $L^2\left(M\right)$, presque toute fonction appartient à tous les espaces $L^p\left(M\right)$, $p\<+\infty$. On donne ensuite des applications à l’étude des harmoniques sphériques sur la sphère ${𝕊}^d$ : on démontre (encore pour des mesures de probabilité naturelles) que presque toute base hilbertienne de $L^2\left({𝕊}^d\right)$ formée d’harmoniques sphériques...

In the setting of spaces of homogeneous-type, we define the Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi$, where $\phi$ is a function of positive lower type and upper type less than $1$, and show that $I_{\phi }$ and $D_{\phi }$ are bounded from Lipschitz spaces ${\Lambda }^{\xi }$ to ${\Lambda }^{\xi \phi }$ and ${\Lambda }^{\xi /\phi }$ respectively, with suitable restrictions on the quasi-increasing function $\xi$ in each case. We also prove that $I_{\phi }$ and $D_{\phi }$ are bounded from the generalized Besov ${\dot{B}}_p^{\psi ,q}$, with $1\le p,q<\infty$, and Triebel-Lizorkin spaces ${\dot{F}}_p^{\psi ,q}$, with $1<p,q<\infty$, of order $\psi$ to those of order $\phi \psi$ and $\psi /\phi$ respectively,...

We study the integral representation properties of limits of sequences of integral functionals like $\int f(x,Du)\mathrm{d}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that$${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$Under weak assumptions on the continuous function $p\left(x\right)$, we prove $\Gamma$-convergence to integral functionals of the same type. We also analyse the case of integrands $f(x,u,Du)$ depending explicitly on $u$; finally we weaken the assumption allowing $p\left(x\right)$ to be discontinuous on nice sets.

We study the integral representation properties of limits of sequences of integral functionals like  $\int f(x,Du)\mathrm{d}x$  under nonstandard growth conditions of (p,q)-type: namely, we assume that $${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$ Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets.

Nous précisons, dans le contexte microlocal Sobolev, les résultats de propagations de singularités obtenus par N. Hanges dans le contexte microlocal $C^{\infty }$ pour les opérateurs pseudo-differentiels à symbole principal réel et dont la variété caractéristique est la réunion de deux hypersurfaces lisses d’intersection non involutive. Nous obtenons également un résultat de propagation dans un cas non linéaire. Nos démonstrations consistent essentiellement à étudier l’action des paramétrices constantes par...