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

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636 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 Conjecture 2. (Ismagilov, 1985) The right regular representation T R,μ : G → U(L 2 (G, μ)) is irreducible if and only if : (1) μ Lt ⊥ μ ∀t ∈ G \{e}, (2) the measure μ is G-ergodic. Remark 3. In the case of the right regular representation, the group α(G) ′ = R(G) ′ ⊂ Aut(G) obviously contains the group L(G), the image of the group G with respect to the left action. The work [11] initiated the study of representations of current groups, i.e. groups C(X,U) of continuous mappings X ↦→ U, where X is a finite-dimensional Riemannian manifold and U is a finite-dimensional Lie group. The regular representation of infinite-dimensional groups, in the case of current groups, was studied firstly in [1,4,5,14] (see also the book [6]). An analog of the regular representation for an arbitrary infinite-dimensional group G, usingaG-quasi-invariant measure on some completion G of such a group, is defined in [18,20]. For X = S1 , U a compact or non-compact connected Lie group, Wiener measures on the loop groups G = C(X,U) were constructed and their quasi-invariance were proved in [1,4–6,28–32]. and the measure μ Conjecture 2 was formulated by R.S. Ismagilov for the group G = B N 0 being the product of arbitrary one-dimensional centered Gaussian measures on the group G = B N and was proved for this case in [18,19]. The first result in this direction was proved in [33]. For the complex infinite-dimensional Borel group Bor c,N 0 and the standard Gaussian measure on its completion Bor c,N the irreducibility of the corresponding regular representation was proved there. Here Bor c,N 0 (respectively Bor c,N )is the group of matrices of the form x = exp t + s where t is a diagonal matrix with a finite number of nonzero real elements (respectively arbitrary real elements) and s is a finite (respectively arbitrary) complex strictly upper-triangular matrix. For the product of arbitrary one-dimensional measures on the group B N Conjecture 2 was proved in [21] under some technical assumptions on the measure. In [20] Conjecture 2 was proved for the groups of the interval and circle diffeomorphisms. For the group of the interval diffeomorphisms the Shavgulidze measure [35] was used, the image of the classical Wiener measure with respect to some bijection. For the group of circle diffeomorphisms the Malliavin measure [30] was used. Whether Conjecture 2 holds in the general case is an open problem. In [25] it was shown that Conjecture 1 holds for the inductive limit G = SL0(2∞, R) = lim −→n SL(2n − 1, R), of the special linear groups (simple groups) acting on a strip of length m ∈ N in the space of real matrices which are infinite in both directions, the measure μ being a product Gaussian measure. Let us consider the special case of a G-space, namely the homogeneous space X = H \ G, where H is a subgroup of the group G and μ is some quasi-invariant measure on X (if it exists) with respect to the right action R of the group G on the homogeneous space H \ G. In this case we call the corresponding representation π R,μ,H\G an analog of the quasiregular or geometric representation of the group G (see [22]). In [2] Conjecture 1 was proved for the solvable infinite-dimensional real Borel group G = Bor N 0 acting on G-spaces Xm , m ∈ N, where X m is the set of left cosets Gm \ Bor N , and Gm is some subgroups of the group Bor N of all upper-triangular matrices of infinite order with non-
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 637 zero elements on the diagonal. The measure μ on Xm is the product of infinitely many onedimensional Gaussian measures on R. In [23,24] Conjecture 1 was proved for the nilpotent group G = BN 0 and some G-spaces Xm , m ∈ N, being the set of left cosets Gm \B N , where Gm are some subgroups of the group BN .Here the measure μ on Xm is the infinite product of arbitrary one-dimensional Gaussian measures on R. In this case the variables xpq,1 p
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x(s) = H.-Q. Li / Journal of Functi
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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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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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lim n→∞ Ω lim sup n→∞
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