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Continuité de la divergence dans les espaces de Sobolev rélatifs à l’espace de Wiener. (French) Zbl 0531.46034
Let P be the Wiener measure for time \(t\in I=[0,1]\). The reproducing Hilbert space of P is the space of functions \(g(t)=\int^{t}_{0}f(s)ds\) on I with \(f\in L^ 2(I)\). Let k and \(\ell\) be integers \(\geq 0\) and \(1<p<\infty\). The derivation D on \(\Omega\), its transpose - DIV and the vectorial Sobolev spaces \(W^{p,k}(\Omega,\ell)\) have been defined by P. Krée [Lect. Notes Math. 474, 16-47 (1975; Zbl 0314.46039)] using the space \(0_{M cyl}(\Omega)\) of cylindrical test functionals on \(\Omega\). Later, this has been introduced in probability theory and denoted \(C^{\infty}_{\uparrow}(\Omega)\). The following two results have been resp. proven only for \(p=2\) by B. Lascar [Sémin. P. Krée, 1re année 1974/75, Equat. Deriv. Part Dim. Infinie, Exposé No.11, 16 p. (1975; Zbl 0331.46023)] and by M. Krée [C. R. Acad. Sci. Paris, Sér. A 278, 753-755 (1974; Zbl 0273.35034)]:
a) For \(\ell>1\), DIV is continuous: \(W^{p,k}(\Omega,\ell)\to W^{p,k- 1}(\Omega,\ell -1).\)
b) Introducing \(N=-DIV.D\), the space \(W^{p,k}(\Omega,0)\) coincides with equivalence of norms with the space of all \(f\in L^ p(\Omega)\) such that \(N^{j/2}f\in L^ p(\Omega)\) for \(j=1,2,...,k.\)
Using P. A. Meyer’s inequality [Lecture Notes Math. 920, 95-132 (1982; Zbl 0481.60041)] some spectral multipliers and some commutation identities the present note extends these two results for arbitrary p with \(1<p<\infty\).

MSC:
46G12 Measures and integration on abstract linear spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46F25 Distributions on infinite-dimensional spaces
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
42B15 Multipliers for harmonic analysis in several variables
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