L∞ to Lp constants for Riesz projections

The norm of the Riesz projection from L∞(Tn) to Lp(Tn) is considered. It is shown that for n=1, the norm equals 1 if and only if p⩽4 and that the norm behaves asymptotically as p/(πe) when p→∞. The critical exponent pn is the supremum of those p for which the norm equals 1. It is proved that 2+2/(n2−1)⩽pn<4 for n>1; it is unknown whether the critical exponent for n=∞ exceeds 2.