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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