Schauder estimates in generalized H\"older spaces

Abstract

We prove Schauder estimates in generalized H\"older 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 $\varphi$ which, again, is not represented by a real number. Assuming that $\mathcal{L}$ has $\psi$-continuous coefficients, we prove that solutions $u \in C^{\varphi\psi}(\mathbb{R}^d)$ to linear equations $\mathcal{L} u = f \in C^\psi(\mathbb{R}^d)$ satisfy a priori estimates in $C^{\varphi\psi}(\mathbb{R}^d)$.