Norm estimates of the Cauchy transform and related operators

Abstract

Suppose f∈Lp(D), where p≥1 and D is the unit disk. Let J0 be the integral operator defined as follows: J0[f](z)=∫Dz1−w¯zf(w)dA(w), where z, w∈D and dA(w)=1πdudv, w=u+iv, is the normalized area measure on D. Suppose J0⁎ is the adjoint operator of J0. Then J0⁎=BC, where B and C are the operators induced by the Bergman projection and Cauchy transform, respectively. In this paper, we obtain the L1, L2 and L∞ norm of the operator J0⁎. Moreover, we obtain the Lp(D)→L∞(D) norm of the operators C and J0⁎, provided that p>2. This study is a continuation of the investigations carried out in [4] and [9].