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

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652 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 Γ(g1,g2,...,gm) = det Ĉ = det(Ĉ − cppEqq + cppEqq) So we have proved the following lemma. = det(Ĉ − cppEqq) + cppA q q(Ĉ − cppEqq) = Γ g1,g2,...,g p q ,...,gm + cppΓ(g1,g2,...,gq−1,gq+1,...,gm). Lemma 15. Let 1 r p
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 653 (see (40)) and theirs generalizations (see (42)). We cannot calculate explicitly n = max pq Mξn (t) 2 , Ξ pq t∈R p but we are able by Lemmas B.1 and B.2 to obtain the estimation Ξ pq n >Ψ pq n , Ψ pq n := (M 12...p−1p 12...p−1q (C(n) p,q)) 2 exp(−1) M 12...p−1 12...p−1 (C(n) p )M 12...p 12...p (C(n) p ) + p k=2 ˆλk(A p k (C(n) p )) 2 (see (47) and (48)). The crucial for proving (17) is Lemma 16 dealing with some inequalities involving the generalized characteristic polynomials. This lemma is proved in Appendix C. We use the notations of Lemma 8 (see Remark 9): Let S L m pq μB = ∞ n=q+1 c (n) pp b(n) qq = ∞ n=q+1 c (n) pp A q q(C (n) m ) det C (n) m = ∞ n=q+1 cppA q q(Cm) det Cm λ = (λk) m k=1 ∈ Rm , ˆλ = (ˆλk) m k=1 , k−1 ˆλ1 = 0, ˆλk = crr, 2 k m, (19) fq = e Ĉm = 1rp
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H.-Q. Li / Journal of Functional An
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x(s) = H.-Q. Li / Journal of Functi
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et et Donc, H.-Q. Li / Journal of F
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et Posons H.-Q. Li / Journal of Fun
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⎛ ⎜ ⎝ − H.-Q. Li / Journal
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Donc, H.-Q. Li / Journal of Functio
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5. Quelques problèmes ouverts H.-Q
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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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When G = η ⊗ r, we obtain Becaus
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This polynomial satisfies D. Serre
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A. Wi´snicki / Journal of Function
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Therefore, A. Wi´snicki / Journal
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we now have (cf. (7.8)) that H. Sch
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2.2. Definition J. Van Schaftingen
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Since ϕ = ϕ0 + ϕ1 ∧ ωN, one h
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3. The Poisson transform K. Koufany
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Its limit when t → 0is K. Koufany
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1.2. Spectral and growth bounds L.
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lim n→∞ Ω lim sup n→∞
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