On the equivalence of Sobolev norms in Malliavin spaces

We investigate the problem of equivalence Lq-Sobolev norms in Malliavin spaces for q∈[1,∞), focusing on graph norm k-th derivative operator and full Sobolev involving all derivatives up to order k, where k is any positive integer. The case q=1 infinite-dimensional setting challenging, since at such extreme standard approach Meyer's inequalities fails. In this direction, we are able establish mentioned k=2 relying a vector-valued Poincaré inequality that prove independently turns out be new level generality, while k>2 issue remains open, even if obtain some functional estimates independent interest. With our argument (that also resorts Wiener chaos) recover q∈(1,∞) setting; latter known eighties, however proof more direct than those existing literature, allows give explicit bounds multiplying constants involved inequalities. Finally, deal with finite-dimensional q∈[1,∞) (where equivalence, without constants, follows from compactness arguments): much simpler, Gaussian integration-by-parts formulas an adaptation Euclidean spaces, it provides again quantitative which blow when dimension diverges ∞ (whence need different setting).