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The well-known curvature method initiated in works of Melnikov and Verdera isnow commonly used to relate the $L^2(\mu)$-boundedness of certain singularintegral operators to the geometric properties of the support of measure $\mu$,e.g. rectifiability. It can be applied however only if Menger curvature-likepermutations, directly associated with the kernel of the operator, arenon-negative. We give an example of an operator in the plane whosecorresponding permutations change sign but the $L^2(\mu)$-boundedness of theoperator still implies that the support of $\mu$ is rectifiable. To the best ofour knowledge, it is the first example of this type. We also obtain severalrelated results with Ahlfors-David regularity conditions.

Singular integrals unsuitable for the curvature method whose $L^2$-boundedness still implies rectifiability