By Michal A.D., Botsford J.L.
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Additional resources for An Extension of the New Einstein Geometry
20) can be derived from the results in [Dor]. 20). 24) ∞ α(x, 2 −j r) q 1 q . j=0 The “right” choice of q is 2, but to get this one should modify the definition of α(x, t) (in most dimensions) so that the measurement of approximation of f by an affine function uses a suitable Lp norm, rather than the supremum. (That the choice of q = 2 is the “right” one reflects some underlying orthogonality, and is a basic point of Littlewood–Paley theory. ) In short, harmonic analysis provides a fairly thorough understanding of the sizes of the α(x, t)’s and related quantities, and with quantitative estimates.
Uniform rectifiability is a notion of controlled geometry that trades topology for estimates. It tolerates some amount of singularities, like holes and crossings, and avoids some common difficulties with homeomorphisms, such as manifold factors. The precise definition is slightly technical, and relies on measure theory in a crucial way. In many respects it is analogous to the notion of BMO from Section 2. The following is a preliminary concept that helps to set the stage. 1 (Ahlfors regularity) Fix n and d, with n a positive integer and 0 < d ≤ n.
The bad behavior cannot always be confined to a single scale — one might have sums of functions like the gρ ’s, but with very different choices of ρ — but, on average, the bad behavior is limited to a bounded number of scales. Let us be more precise, and define a family of functions which try to count the number of “bad” scales associated to a given point x. Fix a radius r > 0, and also a small number , which will provide our threshold for what is considered “small”. We assume that a lipschitz function f on Rd has been fixed, as before.
An Extension of the New Einstein Geometry by Michal A.D., Botsford J.L.