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

Autor(en): Ben Amor, Ali
Stichwörter: BOUNDS; CAPACITARY INEQUALITIES; Mathematics; measure; OPERATORS; Sobolev-Orlicz inequality; spectrum; TRACE; ultracontractivity
Erscheinungsdatum: 2007
Volumen: 255
Ausgabe: 3
Startseite: 627
Seitenende: 647
Let epsilon be a regular Dirichlet form on L-2 (X,m), mu a positive Radon measure charging no sets of zero capacity and phi an N-function. We prove that the Sobolev-Orlicz inequality(SOI) parallel to f(2)parallel to(L)phi((X,mu)) <= C epsilon(1) [f] for every f epsilon D(epsilon) is equivalent to a capacitary-type inequality. Further we show that if D(epsilon) is continuously embedded into L-2(X,mu), the latter one implies some integrability condition, which is nothing else but the classical uniform integrability condition if mu is finite. We also prove that a SOI for epsilon yields a Nash-type inequality and if further mu = m and phi is admissible, it yields the ultracontractivity of the corresponding semigroup. After, in the spirit of SOIs, we derive criteria for D(epsilon) to be compactly embedded into L-2(mu), provided mu 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 phi.
ISSN: 00255874
DOI: 10.1007/s00209-006-0037-8

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