Square functions and H∞ calculus on subspaces of Lp and on Hardy spaces

Abstract

Let X be a (closed) subspace of L p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H1(ℝ N ) is the usual Hardy space, for an appropriate choice of || || F . For example if N=1, the right choice is the sum for h ∈ H1(ℝ), where H denotes the Hilbert transform.