Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

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642 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 Remark 7. We shall firstly prove the approximation of xkn in the above sense for the one vector 1 ∈ L2 (Xm ,μm B ). Secondly, the approximation also holds for some dense set D of analytic vectors in the space L2 (Xm ,μm B ) D = X α = 1km, k
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 643 det C = exp − (2π) m 1 exp(tEpq) 2 ∗ C exp(tEpq)x, x dx = dμBpq(t)(x), where (Bpq(t)) −1 = Cpq(t) = exp(tEpq) ∗ C exp(tEpq) (we note that det C = det Cpq(t)). Hence, using (2) we get We shall prove that H μ LI+tEpq B ,μB = det Cpq(t) + C 2 det Cpq(t) det C det 2 Cpq(t)+C 2 1/4 = det C det Cpq(t)+C 2 1/2 . (3) = det C + t2 4 cppA q q(C), (4) where A p q (C), 1 p,q m, denote the cofactors of the matrix C corresponding to the row p and the column q. Wehave hence det Cpq(t)+C 2 det C and finally, using (3) we get where H μ mLI+tEpq m B ,μB = B (n) = 1r,sm = det C + t2 det C det Cpq(t)+C 2 ∞ n=q+1 4 cppA q q(C) = 1 + det C t2 4 cppbqq, 1/2 = 1 + t2 4 cppbqq −1/2 H μ LI+tEpq B (n) ,μB (n) = ∞ n=q+1 b (n) rs Ers and C (n) := B (n) −1 = 1 + t2 4 c(n) 1r,sm pp b(n) qq c (n) rs Ers. −1/2 , So using the properties of the Hellinger integral for two Gaussian measures we conclude that m LI+tEpq m μB ⊥ μB ∀t ∈ R \{0} ⇔ ∞ n=q+1 1 + t2 4 c(n) ⇔ S L m pq μB =∞. pp b(n) qq −1/2 = 0
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The assumption tells us that the fo
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3.4. The zeroes of ( √ z, η) D.
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4.1. Persistence of surface waves D
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This polynomial satisfies D. Serre
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we now have (cf. (7.8)) that H. Sch
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lim n→∞ Ω lim sup n→∞
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