On a Sobolev type inequality and its applications
Tom 176 / 2006
Streszczenie
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.
