We consider the elliptic equation λ φ – Lφ = f where λ > 0, f is θ-Hölder continuous and L is an Ornstein−Uhlenbeck operator in a Hilbert space H. We show that the mapping D_2_φ (with values in the space of Hilbert−Schmidt operators on H) is θ-Hölder continuous.
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Giuseppe Da Prato, Optimal regularity results in spaces of Hölder continuous functions for some infinite dimensional Ornstein−Uhlenbeck semigroup. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 21 (2010), no. 1, pp. 15–31