A New Family of Singular Integral Operators Whose L² -Boundedness Implies Rectifiability

© 2017, Mathematica Josephina, Inc. Let E⊂ C be a Borel set such that 0 < H1(E) < ∞. David and Léger proved that the Cauchy kernel 1 / z (and even its coordinate parts Rez/|z|2 and Imz/|z|2,z∈C\{0}) has the following property: the L2(H1⌊ E) -boundedness of the corresponding singular integral operator implies that E is rectifiable. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form (Rez)2n-1/|z|2n,n∈N. In this paper, we prove that the above-mentioned property holds for operators associated with the much wider class of the kernels (Rez)2N-1/|z|2N+t·(Rez)2n-1/|z|2n, where n and N are positive integer numbers such that N⩾ n, and t∈ R\ (t1, t2) with t1, t2 depending only on n and N.