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

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680 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 Using (76) we have fk ˆλkA k k (Ck) A k k Ck(ˆλ) −1 = ˆλkA kk+1...m kk+1...m (Cm) A kk+1...m kk+1...m Cm(ˆλ) −1 hence I k m = fkA k k (Cm(ˆλ)) − ˆλkA k k (Cm(λ [k] )) I k m (ˆλ), where the function I k m (ˆλ) is defined by kk+1...m Akk+1...m Cm(ˆλ) −1 kk+1...m A ∅⊆δ⊆{k+1,k+2,...,m} Cm(ˆλ) − ˆλkA k k Cm ˆλ [k] I k m (ˆλ) := ˆλk kk+1...m (Cm)A k k = ˆλk kk+1...m Akk+1...m Cm(ˆλ) −1 A kk+1...m kk+1...m (Cm) Ak k (Cm(ˆλ [k] )) A kk+1...m kk+1...m (Cm(ˆλ)) Ak k (Cm(ˆλ)) (80), (81) = ˆλk kk+1...m Akk+1...m Cm(ˆλ) −1 A × ˆλδ kk+1...m kk+1...m (Cm) Ak∪δ k∪δ (Cm) A kk+1...m kk+1...m (Cm(ˆλ)) Ak∪δ k∪δ (Cm(ˆλ {k} )) . Using (26) or (27) we conclude for λ = (0,λ2,...,λm) ∈ C m Finally we obtain I k m (ˆλ) = ˆλk A kk+1...m kk+1...m Cm(λ) = A k∪δ {k} k∪δ Cm λ = × kk+1...m Akk+1...m Cm(ˆλ) −1 ∅⊆γ ⊆{2,3,...,k−1} ∅⊆γ ⊆{2,3,...,k−1} ∅⊆γ ⊆{2,3,...,k−1} ˆλγ γ ∪{k,k+1,...m} λγ Aγ ∪{k,k+1,...m} (Cm), ∅⊆δ⊆{k+1,k+2,...,m} A kk+1...m kk+1...m (Cm) A A k∪δ k∪δ (Cm) A γ ∪{k}∪δ λγ Aγ ∪{k}∪δ (Cm). ˆλδ γ ∪{k,k+1,...m} γ ∪{k,k+1,...m} (Cm) γ ∪{k}∪δ γ ∪{k}∪δ (Cm) 0 due to the Hadamard–Fisher’s inequality (Lemma A.6), for α ={k,k + 1,...,m} and β = γ ∪ {k}∪δ. This completes the proof of Lemma 16. ✷ References [1] S. Albeverio, R. Höegh-Krohn, The energy representation of Sobolev–Lie group, Compos. Math. 36 (1978) 37–52. [2] S. Albeverio, A. Kosyak, Quasiregular representations of the infinite-dimensional Borel group, J. Funct. Anal. 218 (2) (2005) 445–474. [3] S. Albeverio, A. Kosyak, Group action, quasi-invariant measures and quasiregular representations of the infinitedimensional nilpotent group, Contemp. Math. 385 (2005) 259–280. [4] S. Albeverio, R. Höegh-Krohn, D. Testard, Irreducibility and reducibility for the energy representation of a group of mapping of a Riemannian manifold into a compact Lie group, J. Funct. Anal. 41 (1981) 378–396. [5] S. Albeverio, R. Höegh-Krohn, D. Testard, A. Vershik, Factorial representations of path groups, J. Funct. Anal. 51 (1983) 115–131. [6] S. Albeverio, R. Höegh-Krohn, J. Marion, D. Testard, B. Torrésani, Noncommutative Distributions, Unitary Representation of Gauge Groups and Algebras, Monogr. Textbooks Pure Appl. Math., vol. 175, Dekker, New York, 1993.
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 681 [7] E.F. Beckenbach, R. Bellmann, Inequalities, Springer, Berlin, 1961. [8]J.Dixmier,LesC∗-algèbres et leur représentation, Gautier–Villars, Paris, 1969. [9] J. Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien, second ed., Gauthier–Villars, Paris, 1969. [10] R.F. Gantmacher, Matrizenrechnung, Teil 1, VEB, Berlin, 1958. [11] I.M. Gel’fand, A.M. Vershik, M.I. Graev, Representations of SL2(R),whereRis a ring of functions, Uspekhi Mat. Nauk 28 (1973) 83–128. [12] R.A. Horn, C.R. Jonson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1989. [13] R.A. Horn, C.R. Jonson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991. [14] R.S. Ismagilov, Representations of the group of smooth mappings in a compact Lie group, Funkts. Anal. i Priložhen. 15 (2) (1981) 73–74 (in Russian). [15] S. Kakutani, On equivalence of infinite product measures, Ann. of Math. 4 (1948) 214–224. [16] A.A. Kirillov, Elements of the Theory of Representations, Grundlehren Math. Wiss., vol. 220, Springer, Berlin, 1976 (translated from the Russian). [17] A.A. Kirillov, Introduction to the theory of representations and noncommutative harmonic analysis, in: Representation Theory and Noncommutative Harmonic Analysis, I, in: Encyclopaedia Math. Sci., vol. 22, Springer, Berlin, 1994, pp. 1–156. [18] A.V. Kosyak, Irreducibility criterion for regular Gaussian representations of group of finite upper triangular matrices, Funkts. Anal. i Priložhen. 24 (3) (1990) 82–83 (in Russian); transl. in: Funct. Anal. Appl. 24 (3) (1990) 243–245. [19] A.V. Kosyak, Criteria for irreducibility and equivalence of regular Gaussian representations of group of finite upper triangular matrices of infinite order, Selecta Math. Soviet. 11 (1992) 241–291. [20] A.V. Kosyak, Irreducible regular Gaussian representations of the group of the interval and the circle diffeomorphisms, J. Funct. Anal. 125 (1994) 493–547. [21] A.V. Kosyak, Regular representations of the group of finite upper-triangular matrices, corresponding to product measures, and criteria for their irreducibility, Methods Funct. Anal. Topology 6 (2000) 43–65. [22] A.V. Kosyak, The generalized Ismagilov conjecture for the group BN 0 . I, Methods Funct. Anal. Topology 8 (2) (2002) 33–49. [23] A.V. Kosyak, The generalized Ismagilov conjecture for the group BN 0 . II, Methods Funct. Anal. Topology 8 (3) (2002) 27–45. [24] A.V. Kosyak, A criterion for irreducibility of quasiregular representations of the group of finite upper-triangular matrices, Funkts. Anal. i Priložhen. 37 (1) (2003) 78–81 (in Russian); transl. in: Funct. Anal. Appl. 37 (1) (2003) 65–68. [25] A.V. Kosyak, Quasi-invariant measures and irreducible representations of the inductive limit of the special linear groups, Funkts. Anal. i Priložhen. 38 (1) (2004) 82–84 (in Russian); transl. in: Funct. Anal. Appl. 38 (1) (2004) 67–68. [26] H.H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math., vol. 463, Springer, Berlin, 1975. [27] S.M. Malamud, A converse to the Jensen inequality, its matrix extensions and inequalities for minors and eigenvalues, Linear Algebra Appl. 322 (1–3) (2001) 19–41. [28] M.P. Malliavin, Naturality of quasi-invariance of some measures, in: A.B. Cruzeiro, J.-C. Zambrini (Eds.), Stochastic Analysis and Applications, Lisbon, 1989, in: Progress in Probability, vol. 26, Birkhäuser Boston, Boston, MA, 1991, pp. 144–154. [29] M.P. Malliavin, Probability and geometry, in: Taniguchi Conference on Mathematics, Nara’98, in: Adv. Stud. Pure Math., vol. 31, Math. Soc. Japan, Tokyo, 2001, pp. 179–209. [30] M.P. Malliavin, P. Malliavin, Measures quasi invariantes sur certain groupes de dimension infinie, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 765–768. [31] M.P. Malliavin, P. Malliavin, Integration on loop groups, I, J. Funct. Anal. 93 (1990) 207–237. [32] M.P. Malliavin, P. Malliavin, Integration on loop groups, III: Asymptotic Peter–Weyl orthogonality, J. Funct. Anal. 108 (1992) 13–46. [33] N.I. Nessonov, Examples of factor-representations of the group GL(∞), in: Mathematical Physics, Functional Analysis, Naukova Dumka, Kiev, 1986, pp. 48–52 (in Russian). [34] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. I, Academic Press, New York, 1972. [35] E.T. Shavgulidze, Distributions on infinite-dimensional spaces and second quantization in string theories, II, in: V International Vilnius Conference on Probability Theory and Math. Statistics, Abstracts of Comm., Vilnius, June 26–July 1, 1989, pp. 359–360. [36] G.E. Shilov, Fan Dik Tun’, Integral, Measure, and Derivative on Linear Spaces, Nauka, Moscow, 1967 (in Russian). [37] D.P. Zhelobenko, A.I. Shtern, Representations of Lie Groups, Nauka, Moscow, 1983.
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The assumption tells us that the fo
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When G = η ⊗ r, we obtain Becaus
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This polynomial satisfies D. Serre
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