Embedding Derivatives of Hardy Spaces into Lebesgue Spaces

A characterization is given of those measures μ on U, the upper half plane R+2 or the unit disk, such that differentiation of order m maps Hp boundedly into Lp (μ), where 0<p<∞ and 0<q<∞. The cases where 0<p = q <2 and 0<q <p are the only two not previously known. The solution is presented in the n real variable setting R+n+1 of Fefferman and Stein [7] with an arbitrary differential monomial of order m replacing complex differentiation. Defining the function k(x, y) as the μ-measure of a hyperbolic ball of fixed radius centred at (x, y), we may describe the characterization here briefly, if opaquely, in terms of membership of k in a ‘weighted tent space’ or an L∞-analogue of one (depending on the size of q). In the course of the proof there is developed a theory of ‘tent spaces’ with respect to arbitrary measures on R+n+1. A consequence of the theory is an interpolation theorem for the values of derivatives of Hp-functions at the points of an ‘n-lattice’. This may be of independent interest.