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were found independently by Pichorides, [6] and Cole, [4] in the prototypical case where T is the conjugation operator Tu = u∗ on harmonic functions in the unit disk. Cole’s work on the conjugation operator in uniform algebras was unpublished but later appeared in the monograph [4]. In the case that T is the Riesz transform, Banuelos and Wang, in [1], found the sharp constants in (1) by probabilistic methods following upon ideas of Burkholder (see for example [2]), and Gundy and Varopolous [5]. The inequalities in (1) do not hold for p = 1 or for p = ∞, but when T is the conjugation operator Zygmund proved

Sharp estimates in some inequalities of Zygmund type for Riesz transforms