On a Sobolev type inequality and its applications

Witold Bednorz¹

¹Department of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 176 (2006) no. 2, pp. 113-137

Résumé

Assume $\|\cdot\|$ is a norm on $\mathbb R^n$ and $\|\cdot\|_{\ast}$ its dual. Consider the closed ball $T:=B_{\|\cdot\|}(0,r)$, $r>0$. Suppose $\varphi$ is an Orlicz function and $\psi$ its conjugate. We prove that for arbitrary $A,B>0$ and for each Lipschitz function $f$ on $T$, $$\eqalign{ \sup_{s,t\in T}|f(s)-f(t)|\leq{} 6AB\bigg(\int^r_{0}\psi\bigg(\frac{1}{A\varepsilon^{n-1}}\bigg) \varepsilon^{n-1}\,d\varepsilon\cr {}+\frac{1}{n|B_{\|\cdot\|}(0,1)|}\int_{T}\varphi\bigg(\frac{1}{B}\,\|\nabla f(u)\|_{\ast}\bigg)\,du\bigg), \cr}$$ where $|\cdot|$ is the Lebesgue measure on $\mathbb R^n$. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function $\eta:\mathbb R_{+} \rightarrow \mathbb R$ with $\eta(0) = 0$, a necessary and sufficient condition on $\varphi$ so that each separable process $X(t)$, $t\in T$, which satisfies $$ \|X(s)-X(t)\|_{\varphi}\leq \eta(\|s-t\|)\quad\ \hbox{for}\ s,t\in T $$ is a.s. sample bounded.

Détail

DOI : 10.4064/sm176-2-2

Keywords: assume cdot norm mathbb cdot ast its dual consider closed ball cdot suppose varphi orlicz function psi its conjugate prove arbitrary each lipschitz function eqalign sup f leq bigg int psi bigg frac varepsilon n bigg varepsilon n varepsilon frac cdot int varphi bigg frac nabla ast bigg bigg where cdot lebesgue measure mathbb strengthening sobolev inequality obtained nbsp talagrand inequality state given concave strictly increasing function eta mathbb rightarrow mathbb eta necessary sufficient condition varphi each separable process which satisfies x varphi leq eta s t quad hbox sample bounded