在球化和展平下保持Besov（分数Sobolev）能量
Preserving Besov (fractional Sobolev) energies under sphericalization and flattening
AI总结：
本文提出一种新的球化映射，适用于广泛情况，包括完全不连通的分形集。通过构造带有加权测度的测度，证明了该映射保持测度的二重性质和Besov（分数Sobolev）能量。同时，展平有界完备度量空间的结果也得以获得，并证明球化与展平的组合空间与原空间是biLipschitz等价的。
摘要：
我们介绍了一种新的球化映射，适用于非常广泛的情况，包括完全不连通的分形类型集。对于一个无界完备度量空间，在足够大的半径下在基准点处是均匀完美的，并配备了二重测度，我们基于测度进行更具体的构造，并赋予其加权测度。该映射随后被证明保持测度的二重性质和Besov（分数Sobolev）能量。同时，也获得了有界完备度量空间的展平结果。最后，证明了球化与展平的组合，或反之，所得到的空间与原空间是biLipschitz等价的，且所得测度与原测度成比例。
英文摘要：
We introduce a new sphericalization mapping for metric spaces that is applicable in very general situations, including totally disconnected fractal type sets. For an unbounded complete metric space which is uniformly perfect at a base point for large radii and equipped with a doubling measure, we make a more specific construction based on the measure and equip it with a weighted measure. This mapping is then shown to preserve the doubling property of the measure and the Besov (fractional Sobolev) energy. The corresponding results for flattening of bounded complete metric spaces are also obtained. Finally, it is shown that for the composition of a sphericalization with a flattening, or vice versa, the obtained space is biLipschitz equivalent with the original space and the resulting measure is comparable to the original measure.
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