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

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660 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 T R,μm B exp( p r=1 tr = exp Ern) = exp i p r=1 tr Arn p p k=1 To obtain ξ pp (t) we generalize the function r=k p = exp i xkrtr T R,μm B exp( p r=1 tr Ern) ykn r=1 . tr r k=1 xkrykn in the following way. We replace in the latter identity the vectors (tr,...,tr) ∈ R p−k+1 by (trk) p r=k ∈ Rp−k+1 and denote the result by ξpp(t): ⎛ ⎜ ξpp(t) = ξpp ⎝ t11 t21 t22 t31 t32 ... tp1 tp2 ... tpp ⎞ p p ⎟ ⎠ := exp i k=1 r=k+1 xkrtrk + tkk To obtain ξ pq (t) we consider the function ξpq(t; tqq) = ξpp(t) exp(itqqyqn). Wehave Finally we have ξ pp (t) = ∂ξpp(t) ∂tpp ⎛ ⎜ ξpq(t; tqq) = ξpq ⎝ := exp i tkr=tk, 1rkp t11 t21 t22 t31 t32 ... tp1 tp2 ... tpp; tqq p p k=1 r=k+1 xkrtrk + tkk ⎞ ⎟ ⎠ ykn + tqqyqn and ξ pq (t) = ∂ξpq(t; tqq) ∂tqq . ykn . (42) tqq=0,tkr=tk, 1rkp Let us define φp(t) = ξpp(t)dμ(x,y), φpq(t) = ξpq(t)dμ(x,y), where μ(x,y) = μI (x) ⊗ ( ∞ n=m+1 μ C (n)(y)) and μI (x) is the standard Gaussian measure in R × R 2 ×···×R m . Using definition (39) and the previous equalities we have finally Ξ pp = max ∂φp(t) t∈R p ∂tpp Ξ pq = max ∂φpq(t) t∈R p ∂tqq 2 2 tkr=tk, 1rkp tqq=0,tkr=tk, 1rkp , . (43) Our aim is to estimate Ξ pq . We shall use the notation Ck := C{1,2,...,k} for Mat(m, C) and 1 k m (see notation Cα for ∅⊆α ⊆{1,...,m} in Lemma B of Section 3). .
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 661 Lemma B.1. For 1 p q m and φpq(t) = ξpq(t)dμ(x,y) we have φpq = = ⎛ ⎜ ⎝ t11 t21 t22 t31 t32 t33 ... tp1 tp2 tp3 ... tpp; tqq R (p−1)(p−2) 2 +p exp i 1 √ det C1(t) exp p p k=1 r=k ⎞ ⎟ ⎠ xkrtrk ykn + tqqyqn dμ(x,y) − 1 (CT , T ) − C1(t) 2 −1 d,d , (44) where we set T = (t11,t22,t33,...,tpp; tqq) ∈ R p+1 , C ∈ Mat(p + 1, C) is defined by ⎛ c11 c12 c13 ... c1p c1q ⎜ c12 c22 c23 ... c2p c2q ⎜ c13 c23 c33 ... c3p c3q C := Cp,q := C{1,2,...,p,q} := ⎜ . ⎜ .. ⎝ c1p c2p c3p ... cpp cpq c1q c2q c3q ... cpq cqq d = d21(t), d31(t), . . . , dp1(t); d32(t), d42(t),...,dp2(t); ...; dpp−1(t) ∈ R (p−1)(p−2) 2 , drs(t) = trses(t), 1 s
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⎛ ⎜ ⎝ − H.-Q. Li / Journal
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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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When G = η ⊗ r, we obtain Becaus
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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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2.2. Definition J. Van Schaftingen
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