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We introduce a new family of refined Sobolev-Malliavin spaces that capturethe integrability in time of the Malliavin derivative. We consider duality inthese spaces and derive a Burkholder type inequality in a dual norm. The theorywe develop allows us to prove weak convergence with essentially optimal ratefor numerical approximations in space and time of semilinear parabolicstochastic evolution equations driven by Gaussian additive noise. Inparticular, we combine a standard Galerkin finite element method with backwardEuler timestepping. The method of proof does not rely on the use of theKolmogorov equation or the It\={o} formula and is therefore non-Markovian innature. Test functions satisfying polynomial growth and mild smoothnessassumptions are allowed, meaning in particular that we prove convergence ofarbitrary moments with essentially optimal rate.

Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE