Schauder estimates in generalized Hölder spaces

Abstract

We prove Schauder estimates in generalized Hölder spaces $C^{\psi }({\mathbb{R}}^d)$. These spaces are characterized by a general modulus of continuity $\psi$, which cannot be represented by a real number. We consider linear operators $\mathcal{L}$ between such spaces. The operators $\mathcal{L}$ under consideration are integrodifferential operators with a functional order of differentiability $\phi$ which, again, is not represented by a real number. Assuming that $\mathcal{L}$ has $\psi$-continuous coefficients, we prove that solutions $u\in C^{\phi \psi }({\mathbb{R}}^d)$ to linear equations $\mathcal{L}u=f\in C^{\psi }({\mathbb{R}}^d)$ satisfy a priori estimates in $C^{\phi \psi }({\mathbb{R}}^d)$.

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