Malliavin's absolute continuity lemma

Template:Short description In mathematics — specifically, in measure theory — Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (smoothness) theorems of the Malliavin calculus. Malliavin's lemma gives a sufficient condition for a finite Borel measure to be absolutely continuous with respect to Lebesgue measure.

Statement of the lemma

Let μ be a finite Borel measure on n-dimensional Euclidean space Rⁿ. Suppose that, for every x ∈ Rⁿ, there exists a constant C = C(x) such that

$|\underset{{𝐑}^n}{\int }\mathrm{D}\phi (y)(x)\mathrm{d}\mu (y)|\le C(x)\Vert \phi {\Vert }_{\mathrm{\infty }}$

for every C^∞ function φ : Rⁿ → R with compact support. Then μ is absolutely continuous with respect to n-dimensional Lebesgue measure λⁿ on Rⁿ. In the above, Dφ(y) denotes the Fréchet derivative of φ at y and ||φ||_∞ denotes the supremum norm of φ.

References

• Template:Cite book Template:MathSciNet (See section 1.3)
• Template:Cite book Template:MathSciNet