Characterization of Higher-Order Sobolev Spaces on the Sphere via Generalized Averaging of Function
We present a new characterization of higher-order Sobolev spaces on the sphere. Building on the approach of Barceló et al. (2020), we refine the square function they introduced for this purpose. In particular, we provide a detailed analysis of the weight function and the averaging range in its definition. Our results demonstrate that averaging can be performed using a weight function from a broad class, and that the averaging domain can be restricted to a local coordinate patch on the sphere.
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