Sobolev–Orlicz inequalities, ultracontractivity and spectra of time changed Dirichlet forms

Abstract

Let \(\mathcal{E}\) be a regular Dirichlet form on L 2(X,m), μ a positive Radon measure charging no sets of zero capacity and Φ an N-function. We prove that the Sobolev-Orlicz inequality(SOI) \(\|f^2\|_{L^{\Phi}(X,\mu)}\leq C\mathcal{E}_1[f]\) for every \(f\in D(\mathcal{E})\) is equivalent to a capacitary-type inequality. Further we show that if \(D(\mathcal{E})\) is continuously embedded into L 2(X,μ), the latter one implies some integrability condition, which is nothing else but the classical uniform integrability condition if μ is finite. We also prove that a SOI for \(\mathcal{E}\) yields a Nash-type inequality and if further μ = m and Φ is admissible, it yields the ultracontractivity of the corresponding semigroup. After, in the spirit of SOIs, we derive criteria for \(D(\mathcal{E})\) to be compactly embedded into L 2(μ), provided μ is finite. As an illustration of the theory, we shall relate the compactness of the latter embedding to the discreteness of the spectrum of the time changed Dirichlet form and shall derive lower bounds for its eigenvalues in term of Φ.