Two examples related to conical energies

Abstract

In a recent article we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters, the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

arXiv comment23 pages, 2 figures; reformulated Lemma 2.3, other minor improvements, results unchanged; to appear in Ann. Acad. Sci. Fenn. Math

References in corpus (5)

• Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality
• Characterising the big pieces of Lipschitz graphs property using projections
• Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ${\mathbb{R}}^d$
• Plenty of big projections imply big pieces of Lipschitz graphs
• Cones, rectifiability, and singular integral operators

Cited by in corpus (2)

• Radon measures and Lipschitz graphs
• Cones, rectifiability, and singular integral operators