Analysis of and on Uniformly Rectifiable Sets

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Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

“…The key idea is that using [Jon88] one gets that Ahlfors regular curves contain what is called 'big pieces of chord-arc curves' (see [DS93] for a definition). For chord-arc curves we have (using a modification of [Oki92]) desired estimates, which can be used with machinery from [DS93] to extend to Ahlfors regular curves.…”

“…The key idea is that using [Jon88] one gets that Ahlfors regular curves contain what is called 'big pieces of chord-arc curves' (see [DS93] for a definition). For chord-arc curves we have (using a modification of [Oki92]) desired estimates, which can be used with machinery from [DS93] to extend to Ahlfors regular curves. All this requires an inspection of some proofs given in the above references, which results in the observation that, even though they are not stated as such, they are dimension independent for the relevant cases (or can be made so with very minor modifications; for example [Oki92] can be made dimension independent in the case of chord-arc curves).…”

“…Note that we have defined a quantity which is scale independent. This quantity is usually referred to as the Jones β ∞ number in order to differentiate it from its L p variants (extensively developed by David and Semmes in [DS93] and generalized in [Ler03]). We omit the ∞ subscript as we will always use β ∞ .…”

“…In many cases one adds for each Q a weight in the above sum. See [BJ90,DS93,Jon90,Ler03,Oki92] for explicit and implicit appearances. One can view the remainder of this essay as an explanation of the right way of generalizing this notion to a larger category of sets.…”

“…These results formed the basis of a theory now called 'quantitative rectifiability' which was extended by many authors, of which we should give special mention to Guy David and Stephen Semmes whose work on Uniform Rectifiability inspired part of this essay. For example see [DS93,BJ90,Ler03]. Also see [Paj02] for a more complete survey and bibliography.…”