Schauder theorems for a class of (pseudo-)differential operators on finite and infinite dimensional state spaces

Abstract

We prove maximal regularity results in Hölder and Zygmund spaces for linear stationary and evolution equations driven by a large class of differential and pseudo-differential operators L, both in finite and in infinite dimension. The assumptions are given in terms of the semigroup generated by L. We cover the cases of fractional Laplacians and Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and several types of local and nonlocal Ornstein-Uhlenbeck operators, as well as the Gross Laplacian and its negative powers, in infinite dimension.

Ornstein-Uhlenbeck semigroups in infinite dimension

“…If e tA maps H = Q 1/2 (X) into itself, and S H (t) = e tA |H : H → H is a strongly continuous semigroup in H, Schauder theorems similar to the ones stated in Sect. 2 were proved in [33]: for every α ∈ (0, 1), λ > 0 and…”

“…See [31,Sect. 2], and [33] for representation formulae and estimates for any order H-derivatives of T (t)f when f ∈ C b (X).…”

“…In both cases, we gain "1/θ degrees" of regularity. Both for α = 0 and for α > 0, in the critical cases α + 1/θ = k ∈ N we cannot expect that u ∈ C k (X); in [33] it is proved that u belongs to a suitable Zygmund space, which is continuously embedded in all spaces C k−ε b (X) with ε ∈ (0, 1). This difficulty arises even in finite dimension, for instance if…”

“…Schauder type regularity results are available also for evolution equations with bounded and continuous data, see [33].…”

“…As a consequence, T (t) is smoothing along H. See [31, Sect. 2], and [33] for representation formulae and estimates for any order H-derivatives of T (t)f when f ∈ C b (X). Again, as in the Hilbert case, Schauder type theorems were proved in [33], that generalize the one stated in Sect.…”