Best constants for the Riesz projection

We prove the following inequality with a sharp constant, \\P+ f\\(Lp(T)) le ss than or equal to csc pi/p \\f\\(Lp(T)), f is an element of L-p(T), where 1 < p < infinity, and P+:L-p(T) --> H-p(T) is the Ricsz projection onto th e Hardy spate H-p(T) on the unit circle T. (In other words, the "angle" bet ween the analytic and to-analytic subspaces of L-p(T) equals pi/p* where p* = max (p, p/p-1).) This was conjectured in 1968 by I. Gohberg and N. Krupn ik. We also prove an analogous inequality in the nonperiodic case where P(+)f = F-1 (chi(R+) Ff) is the half-line Fourier multiplier on R. Similar weighte d inequalities with sharp constants for L-p(R, \x\(alpha)), -1 < alpha < p - 1, are obtained. In the multidimensional case, our results give the norm of the half-space Fourier multiplier on R-n. (C) 2000 Academic Press.