Preserving Besov (fractional Sobolev) energies under sphericalization and flattening

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.

💡 Research Summary

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This paper introduces a novel sphericalization mapping for metric spaces that works in very general settings, including totally disconnected fractal-type sets. The authors consider an unbounded complete metric space (Z, d) that is uniformly perfect at a base point b for large radii and carries a doubling measure ν. They define a deformed metric ˆd by a chain‐infimum formula using a non‑increasing density function ρ(t) and, crucially, replace the product ρ(x)ρ(y) used in earlier works with the sum ρ(x)+ρ(y). This modification allows the construction to apply without any path‑connectedness assumptions.

For sphericalization they set m₀ = 1, for flattening m₀ = 0, and choose
 ρ(t) = 1/(t + m₀)·ν(B(b, t + m₀))^{‑1/σ}, σ>0,
and define a weighted measure ˆν by dˆν = ρ^{σ} dν. The parameter σ is later taken to be pθ, where p≥1 and θ>0 are the parameters of the Besov (fractional Sobolev) seminorm