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We show uniqueness in law for the critical SPDE  $$  dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\;  X_0 =x \in H,  $$  where $A$ $ : dom(A) \subset H \to H$ is a negative definite self-adjointoperator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and$W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can becontinuous with at most linear growth (some functions $F$ which grow more thanlinearly can also be considered). This leads to new uniqueness results forgeneralized stochastic Burgers' equations and for three-dimensional stochasticCahn-Hilliard type equations which have interesting applications. To get weakuniqueness we also establish a new optimal regularity result for the Kolmogorovequation $ \lambda u - Lu = f$ on $H$, where $\lambda >0$, $ f: H \to {\mathbbR}$ is Borel and bounded and $L$ is the Ornstein-Uhlenbeck operator related tothe SPDE when $F=0$. In particular we show that the first derivative $Du : H\to H$ verifies $Du(x) \in \text{dom}((-A)^{1/2})$, for any $x \in H,$ andmoreover $$ \sup_{x \in H} |(-A)^{1/2}Du (x)|_H = \| (-A)^{1/2}Du \|_{0} \le C\, \| f\|_{0}. $$

An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs