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 language | English (US) |
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Pages (from-to) | 1294-1365 |
Number of pages | 72 |
Journal | Communications on Pure and Applied Mathematics |
Volume | 56 |
Issue number | 9 |
DOIs | |
State | Published - Sep 2003 |