Best Constants for the Riesz Projection

We prove the following inequality with a sharp constant,‖P+f‖L p(T)⩽csc πp ‖f‖Lp(T), f∈Lp(T),where 1<p<∞, and P+: Lp(T)→Hp(T) is the Riesz projection onto the Hardy space Hp(T) on the unit circle T. (In other words, the “angle” between the analytic and co-analytic subspaces of Lp(T) equals π/p* where p*=max (p, pp−1).) This was conjectured in 1968 by I. Gohberg and N. Krupnik. We also prove an analogous inequality in the nonperiodic case where P+f=F−1 (χR+Ff) is the half-line Fourier multiplier on R. Similar weighted inequalities with sharp constants for Lp(R, |x|α), −1<α<p−1, are obtained. In the multidimensional case, our results give the norm of the half-space Fourier multiplier on Rn