Abstract
We prove that the singular numbers of the Cauchy transform\((Cf)(z) = \int D \tfrac{{f(\xi )}}{{z - \xi }}dA(\xi )\) onL 2(D) are asymptotically\(s_n (C) \approx \tfrac{1}{{\sqrt n }}\), whiles n (C | L 2 a (D))≈1/n (whereL 2 a (D) is the subspace of analytic functions inL 2(D)). Also, the singular numbers of the logarithmic potential\((Lf)(z) = \int D f(\xi )\log \left( {\tfrac{1}{{\left| {z - \xi } \right|}}} \right)dA(\xi )\) onL 2(D) are asympoticallys n (L)≈1/n, whiles n(L |L 2 a (D))≈1/n 2. Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform fromL 2 L (μ) intoL 2(ν) where μ and ν are rotation-invariant measures on\(\bar D\).
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The author was partly supported by a grant from the national Science Foundation.