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A characterization is given of those measures n on U, the upper half plane U\ or the unit disk, such that differentiation of order m maps H p boundedly into L q {\i), where 00. We use the absolute value symbol | -1 to denote the Euclidean norm in W and in U n+l (so \z\ 2 = \x\ 2 + y 2 ). When z = (x,y), let z* = (x,-y). The pseudohyperbolic metric p is defined on U by p(z, w) = \z -w\/\z -w*\. Clearly p is invariant under rigid motions in the x-variable and under dilations in IR" +1 . Let D e (w) = {zeU: p{z, w) < e) when weU a n d 0 < e < 1. L e t Y a (t) = {(x, y) e U: \x -t\ < acy) where a>0. In discussions where the actual value of e or oc is irrelevant, they may be omitted from the subscripts.If 0 < r < oo and / is a measurable function on U, define and A(/)(0= supwhere B{x, y) denotes the Euclidean ball in R n centred at x with radius y. We say that E is the 'tent' over E. Define, for 0 < r < oo and / measurable on U, DERIVATIVES OF HARDY SPACES 597

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