Bounded variation of measure convolutions
In the paper ``On the radial variation of bounded analytic functions'', Duke Math. J. 69, No. 3, 671-682 (1993; Zbl 0787.30020)] the author proved that if \(F\) is a bounded analytic function on the unit disc, then the radial variation at \(t \in [0,2\pi)\) \[ V(F,t) = \int^ 1_ 0 | F'(\text{re}^{it}) | dr \] is bounded on a set of Hausdorff dimension 1. By adapting the argument from the above paper the author proves the following result. Let \(f\) be a function on \(\mathbb{R}\) satisfying \(\int | x f(x)|^ 2 dx < \infty\) and \[ \int [(1 + | x |) f'(x) | + (1 + x^ 2) | f''(x)| ] dx < \infty. \] Define \(f_ h = {1\over h} f\left( {x\over h}\right)\), \(h>0\) and let \(\mu\) be a positive measure. Then \((\mu * f_ h)_{h > 0}\) has bounded variation on a set of Hausdorff dimension 1 in any open interval.
scientific article; zbMATH DE number 687518