The Huovinen transform and rectifiability of measures

Abstract

For a set E of positive and finite length, we prove that if the Huovinen transform (the convolution operator with kernel z k /|z| k+1 for an odd number k) associated to E exists in principal value, then E is rectifiable.1 building upon a number of important results including [Me, MMV, Dav3, DM, MM, NTV2]2 More precisely, these techniques play an important role in Theorem A below.3 We say that a Borel measure µ is s-rectifiable if there exist Lipschitz maps

Rectifiability; a survey

“…Jaye and Merchán [JM20a] strengthened this by replacing zero density by the condition that modifications of Tolsa's αs (recall Section 7.4) tend to zero. See also [JM20b] for related results and recall the discussion in 10.5 on [JM21]. 11.4.…”