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 H p boundedly into L p (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 H p ‐functions at the points of an ‘ n ‐lattice’. This may be of independent interest.