We prove an interior Schauder estimate for the Laplacian on metric products of two dimensional cones with a Euclidean factor, generalizing the work of Donaldson and reproving the Schauder estimate of Guo–Song. We characterize the space of homogeneous subquadratic harmonic functions on products of cones, and identify scales at which geodesic balls can be well approximated by balls centered at the apex of an appropriate model cone. We then locally approximate solutions by subquadratic harmonic functions at these scales to measure the Hölder continuity of second derivatives.

中文翻译：

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肖特对圆锥产品的估计

我们用欧几里得因数证明了二维圆锥的公制乘积上拉普拉斯算子的内部Schauder估计，推广了唐纳森的工作并推翻了Guo-Song的Schauder估计。我们在圆锥的乘积上表征了均次次谐波函数的空间，并确定了以适当的模型圆锥的顶点为中心的球可以很好地逼近测地线的尺度。然后，我们在这些尺度上通过次二次谐波函数对局部解进行近似，以测量二阶导数的Hölder连续性。