## Functional Inequalities for Uniformly Integrable Semigroups and Applications (0)

Venue: | Forum Math |

Citations: | 8 - 2 self |

### BibTeX

@ARTICLE{Gong_functionalinequalities,

author = {Fu-Zhou Gong and Feng-Yu Wang},

title = {Functional Inequalities for Uniformly Integrable Semigroups and Applications},

journal = {Forum Math},

year = {},

volume = {14},

pages = {293--313}

}

### OpenURL

### Abstract

Let (E; F ; ) be a probability space, (E ; D(E)) a (not necessarily symmetric) Dirichlet form on L 2 (), and P t the associated sub-Markov semigroup. The equivalence of the following eight properties is studied: (i) the L 2 -uniform integrability of the unit ball in the Sobolev space; (ii) the super-Poincar'e inequality (1.2); (iii) the F-Sobolev inequality (1.3); (iv) the L 2 -uniform integrability of P t ; (v) the L 2 -uniform integrability of the associated resolvents; (vi) the compactness of P t ; (vii) the compactness of the associated resolvents; (viii) empty essential spectrum of the associated generator. The main results can be summarized as follows. In general, (i), (ii) and (iii) are equivalent to each other, and they imply (iv) which is equivalent to (v). If P t has transition density and F is -separable, then the first seven properties from (i) to (vii) are equivalent. If in addition (E ; D(E)) is symmetric, then all the above eight properties are equivalent. Moreo...

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Citation Context ... [0; r \Gamma1 0 )"oe ess (\GammaL): Let r 1 ? r 0 be such that r 1s! 1 and take " = min n 1 \Gamma r 1s2 p 2 r 1 ; e \Gamma2t (1 \Gamma r 1 ) 8tfi(r 1 ) o which is positive. By Weyl's crite=-=rion (see [15], The-=-orem VII.12 and comments on page 264), there exists a sequence ff n g ae D(L) such that kf n k 2 = 1; (hf n ; f m i H ) = 0 for n 6= m, and k( + L)f n k 2s"; ns1: (4.9) For any m; ns1; let h(s) =... |

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Citation Context ...(2.11) (2) The statement (iv) is equivalent to (v). Proof. a) For a linear operator (T; D(T )) on L 2 (), we denote by (T C ; D(T C )) its complexification on the complex L 2 space L 2 C (), see e.g. =-=[13]-=- and [14] for details. Since (E ; D(E)) is a coercive closed form, by the proof of Corollary I-2.21 in [14], L 0 := (1 \Gamma L) C satisfies the strong sector condition (see pages 15 and 16 in [14]). ... |

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Citation Context ...i in (1.2) and F in (1.3) are also available in [20]. The first aim of this paper is to extend the above results to general framework. Our another motivation comes from the famous work [10] (see also =-=[11]-=-, [2] and [6]), which proved the equivalence of the log-Sobolev inequalities and the hypercontractivity of corresponding semigroups. It is then very natural for us to study semigroup properties for th... |

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Citation Context ...and the essential spectrum of L The aim of this section is to complete the proof of Theorem 1.2. We first present the following lemma which is an extension of a result due to Wu (see Proposition 3 in =-=[21]-=- and Theorem 2.3 in [22]). This lemma enables one to prove the compactness of semigroups on a Hilbert space-valued L p -space. Lemma 3.1. Assume that F is -separable and let p 2 [1; 1) be fixed. Let H... |

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Citation Context ...egrable, we may choose K ? 0 such that kPf n;K \Gamma Pf m;K k ps" 2 ; n 6= m; (3.2) where f n;K = f n 1 fjfn j HKg : Since F is -separable, L p H () is separable. By Bourbaki theorem (see page 1=-=1 in [3]-=-), the set ff : kfk1sKg ae L 1 H () is weakly compact and metrisable w.r.t. the weak topology oe(L 1 H (); L 1 H ()): Therefore, there exist 10 f 2 L 1 H () and a subsequence ff n i ;K g such that f n... |

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Citation Context ...cally compact Hausdorff topological space, F the Borel oe-field, andsa probability measure on (E; F ). Let (E ; D(E)) be a strongly local, regular, irreducible symmetric Dirichlet form on L 2 () (see =-=[9]-=- for detailed definitions). Assume that the metric induced by E is equivalent to the original topology (see [9] for the definition of the metric). Then inf oe ess (\GammaL) = sup A is compact inffE(f;... |

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Citation Context ...n H ffi is L 1 -uniformly integrable and that the L 1 -norm and the L 2 -norm are equivalent on H ffi . By Lemma 3.1, P 0 t j H ffi is L 1 -compact and hence also L 2 -compact. By Theorems 1 and 2 in =-=[12], the-=- essential spectrum of L 0 j H ffi is empty. By (4.6) and noting that (L 0 Q ffi )j H ffi = L 0 j H ffi since Q ffi is a projection, we have ; = oe ess (L 0 j H ffi ) = oe ess (L 0 ) " D ffi : 13... |

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(Show Context)
Citation Context ...nack inequality. We shall only consider the above L on a manifold, but the Harnack inequalities are also available for other cases, see e.g. [1] for diffusion semigroups on abstract Wiener spaces and =-=[16]-=- for generalized Mehler semigroups. 17 Let L = \Delta +Z for some C 1 -vector field Z. Assume that Ric\Gammahr \Delta Z; \Deltai is bounded from below. By the proof of Lemma 2.1 in [18], for any t ? 0... |