On a Poincaré-type Inequality for Energy Forms in L ^p

Abstract.

We consider Dirichlet spaces (${\mathcal{E}}, {\mathcal{F}}$) in L ² and more general energy forms $(\mathcal{E}^{(p)}, {\mathcal{F}}_{1,p})$ in L ^p, $1 < p < +\infty$ . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that ${\mathcal{F}}$ , resp. ${\mathcal{F}}_{1,p}$ , are compactly embedded in L ², resp. L ^p, we prove a Poincaré inequality for transient (Dirichlet) forms. If both $(T_{t})_{t\geq0}$ and its adjoint $(T^{*}_{t} )_{t\geq0}$ are sub-Markovian semigroups, we show that the transience of T _t is independent of $p \in (1, +\infty$) and that it is implied by the transience of the energy form $\mathcal{E}^{(p)}$ of $(T_{t})_{t\geq0}$ and the form ${\mathcal{E}}_{*}^{(q)}$ belonging to $(T^{*}_{t} )_{t\geq0}$ .