Quantifying Curvelike Structures of Measures by Using L₂ Jones Quantities

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.