Quantifying Curvelike Structures of Measures by Using L2 Jones Quantities

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Abstract

We study the curvelike structure of special measures on ℝn in a multiscale fashion. More precisely, we consider the existence and construction of a sufficiently short curve with a sufficiently large measure. Our main tool is an L2 variant of Jones' β numbers, which measure the scaled deviations of the given measure from a best approximating line at different scales and locations. The Jones function is formed by adding the squares of the L2 Jones numbers at different scales and the same location. Using a special L2 Jones function, we construct a sufficiently short curve with a sufficiently large measure. The length and measure estimates of the underlying curve are expressed in terms of the size of this Jones function.

Original languageEnglish (US)
Pages (from-to)1294-1365
Number of pages72
JournalCommunications on Pure and Applied Mathematics
Volume56
Issue number9
DOIs
StatePublished - Sep 1 2003

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