科研之友
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摘要
Let E subset of C be a Borel set such that 0 < H-1(E) < infinity. David and Leger proved that the Cauchy kernel 1/z (and even its coordinate parts Re z/vertical bar z vertical bar(2) and Im z/vertical bar z vertical bar(2), z is an element of C\{0}) has the following property: the L-2(H-1 [ 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 (Re z)(2n-1)/vertical bar z vertical bar(2n), n is an element of N. In this paper, we prove that the above-mentioned property holds for operators associated with the much wider class of the kernels (Re z)(2N-1)/vertical bar z vertical bar(2N) + t . (Re z)(2n-1)/vertical bar z vertical bar(2n), where n and N are positive integer numbers such that N >= n, and t is an element of R\(t(1), t(2)) with t(1), t(2) depending only on n and N.
- 出版日期2017-10
