Let \((E,H,\mu)\) be an abstract Wiener space, \(D\) the Malliavin derivative, \(V\) a closed, densely defined operator from \(H\) to some other Hilbert space \(\underline H\), and \(D_V:= VD\). Given a bounded operator \(B\) on \(\underline H\) coercive on the range \(\overline{R(V)}\), consider \(A:= V^*BV\), \(A^*:= VV^*B\), and the realisations of \(L:= D_{V^*}BD_V\) in \(L^p(E,\mu)\) and of \(L:= D_V D_{V^*}B\) in \(L^p(E,\mu;\underline H)\), \(1< p<\infty\).
As a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus, the equivalence of the following four assertions is proved:
(1) \(D(\sqrt{L})= D(D_V)\) with \(\|\sqrt{L}f\|_p\overline\sim\| D_Vf\|_p\) for \(f\in D(\sqrt{L})\);
(2) \(L\) admits a bounded \(H^\infty\)-functional calculus on \(\overline{R(D_V)}\);
(3) \(D(\sqrt{A})= D(V)\) with \(\|\sqrt{A}h\|\overline\sim\| Vh\|\) for \(h\in D(\sqrt{A})\);
(4) \(A\) admits a bounded \(H^\infty\)-functional calculus on \(\overline{R(V)}\).
If (1)–(4) hold, then \(D(L)= D({D_V}^2)\cap D(D_A)\). A one-sided version of (1)–(4) leads to \(L^p\)-boundedness of the Riesz transform \(D_V/\sqrt{L}\) in terms of a square function estimate. If \(-A\) generates an analytic \(C_0\)-contraction semigroup on a Hilbert space and \(-L\) is the \(L^p\)-realisation of the generator of its second quantisation, the results imply that two-sided bounds of the Riesz transform of \(L\) are equivalent with the Kato square root property for \(A\). The boundedness of the Riesz transform is used to obtain an \(L^p\)-domain characterisation for \(L\).