On the equivalence of Sobolev norms in Malliavin spaces (Q2155288)

Consider a separable Hilbert space \(\mathcal{H}\) and a centred Gaussian process \((W(h);\,h\in\mathcal{H})\) defined on a probability space \(\Omega\), with covariance \(\mathbb{E}[W(g)W(h)]=\langle g,h\rangle_{\mathcal{H}}\). Let \(\mathcal{V}\) be another Hilbert space and consider a functional \(F=f(W):\Omega\to\mathcal{V}\). For smooth functionals \(F=f(W(h_1),\ldots,W(h_n))\) we can define the iterated Malliavin derivatives \(D^kF\) taking values in spaces \(L^q(\Omega,\mathcal{H}^{\otimes k}\otimes\mathcal{V})\) with norms \(\bigl\|D^kF\bigr\|_q\). After the introduction of all these notions, the equivalence of the norms \[ \bigl\|F\bigr\|_q+\sum_{\ell=1}^k\bigl\|D^\ell F\bigr\|_q\text{ and }\bigl\|F\bigr\|_q+\bigl\|D^kF\bigr\|_q \] is studied. This equivalence is first proved in the finite dimensional case for any \(q\ge1\). In the infinite-dimensional case, the equivalence is proved when \(q>1\) (this result was already known), and when \(q=1\) and \(k=2\) (this result is new). The case \(q=1\) and \(k\ge3\) is open.