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We investigate the weak order of convergence for space-time discreteapproximations of semilinear parabolic stochastic evolution equations driven byadditive square-integrable L\'evy noise. To this end, the Malliavin regularityof the solution is analyzed and recent results on refined Malliavin-Sobolevspaces from the Gaussian setting are extended to a Poissonian setting. For aclass of path-dependent test functions, we obtain that the weak rate ofconvergence is twice the strong rate.

Malliavin regularity and weak approximation of semilinear SPDE with L\'evy noise