SPECTRAL ESTIMATES OF CAUCHY TRANSFORM IN L-2(OMEGA)

We prove that the singular numbers of the Cauchy transform (Cf)(z) = integral D f(xi)/z-xi dA(xi) on L2(D) are asymptotically s(n)(C) almost-equal-to 1/square-root n, while s(n)(C\L(alpha)2(D)) almost-equal-to 1/n (where L(alpha)2(D) is the subspace of analytic functions in L2(D)). Also, the singular numbers of the logarithmic potential (Lf)(z) = integral D f(xi) log (1/\z-xi\) dA(xi) on L2(D) are asympotically s(n)(L) almost-equal-to 1/n, while s(n)(L\L(alpha)2(D)) almost-equal-to 1/n2. Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform from L(alpha)2(mu) into L2(nu) where mu and nu are rotation-invariant measures on DBAR.