Schauder estimates in generalized H\"older spaces

TL;DR

This paper establishes Schauder estimates within generalized H"older spaces characterized by a modulus of continuity, for solutions to linear integrodifferential equations with non-standard differentiability orders.

Contribution

It extends Schauder estimates to generalized H"older spaces with non-representable moduli and differentiability orders, covering a broader class of integrodifferential operators.

Findings

Proves Schauder estimates in $C^\psi$ spaces for integrodifferential operators.

Shows solutions in $C^{\varphi\psi}$ satisfy a priori estimates.

Handles operators with coefficients continuous in the generalized H"older sense.

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 $C^{\varphi\psi}(\mathbb{R}^d)$.