Orlicz type inequalities with powers (Q2788779)

The paper establishes an Orlicz type inequality for the Dirichlet operator with powers. Let \(P\) be a suitable non-negative definite self-adjoint operator on \(L^2\) and let \(P_t:=e^{-tP}\) be the contractive \(C_0\) Markov semigroup on \(L^2\). For example, one could take the diffusion operator \(P\) to be a second-order elliptic differential operator. Let \(\mathcal{E}(f,f):=(f,Pf)_{L^2}\) be the corresponding symmetric quadratic form. Assume \(\ker(P)=\{0\}\) and let \(P^p\) be the \(p^{\text{th}}\) power and let \(\mathcal{E}_p\) be defined by \(L^p\). The authors improve the super-Poincaré inequality to get an Orlicz type inequality for \(\mathcal{E}_p\) based on an Orlicz type inequality for the original Dirichlet form \(\mathcal{E}\). Examples are provided showing the optimality of the results.