NOTE ON THE OPERATOR bP ON Lp(∂D)

Abstract

Let ∂D be the boundary of the open unit disk D in the complex plane and Lp(∂D) the class of all complex, Lebesgue R π −π |f (θ)|pdθ}1/p < ∞. Let measurable function f for which { 1 2π P be the orthogonal projection from Lp(∂D) onto ∩n<0 ker an. For R π f ∈ L1(∂D), ˆf (z) = 1 −π Pr(t − θ)f (θ)dθ is the harmonic ex2π tension of f . Let bP be the composition of P with the harmonic In this paper, we will show that if 1 < p < ∞, then extension. bP : Lp(∂D) → H p(D) is bounded. In particular, we will show that bP is unbounded on L∞(∂D).

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