Schauder estimates on smooth and singular spaces

Abstract

In this paper, we present a proof of Schauder estimate on Euclidean space and use it to generalize Donaldson's Schauder estimate on space with conical singularities in the following two directions. The first is that we allow the total cone angle to be larger than 2π and the second is that we discuss higher order estimates. Contents 1. Introduction 1 2. Hölder space on R n 5 3. A proof of the Schauder estimates on R n 9 4. Preliminaries about X β 12 4.1. Notations 12 4.2. Basics on the Poisson equation 13 5. Bounded harmonic functions on X β 13 5.1. The expansion 14 5.2. The approximating X β -polynomial is still harmonic 16 6. Generalized Hölder space on X β 17 6.1. Formal discussion 17 6.2. The definition. 18 6.3. Comparison with Donaldson's C 2,α β space 21 7. Schauder estimates for X β 24 7.1.

Schauder estimates on products of cones

“…There, Donaldson proved the Schauder estimate for the Laplacian of such metrics, establishing an important step toward the eventual solution of the Yau-Tian-Donaldson conjecture relating the existence of Kähler-Einstein metrics on Fano manifolds to K-stability [CDS15a,CDS15b,CDS15c]. The Schauder estimate for metrics with cone singularities along a smooth hypersurface was later reproved by Guo-Song [GS16] and Gui-Yin [GY18].…”

“…× C(S 1 2πβn ) × R m−2n as considered in this paper. Yet another approach for the Schauder estimates on C(S 1 2πβ ) × R m−2 was given by Gui-Yin [GY18], considering also β > 1 and higher order estimates. A main ingredient in Gui-Yin's work is an expansion formula for bounded harmonic functions, [GY18, Proposition 5.2], and their methods are closer to ours.…”

Higher order neck analysis of harmonic maps and its applications

“…The basic idea is to compare the solution of PDE with polynomials (see [Caf89], [CC95]). Here we adapt an approach in [GY18]. There is a key lemma in the argument (see Lemma 3.1) that improves the regularity by solving Poisson equations.…”