Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L-P (R-n; X) of X-valued functions on R-n. We characterize Kato's square root estimates parallel to root Lu parallel to(p) similar or equal to parallel to Delta u parallel to(p) and the H-infinity-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative L-P space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X = C, we get a new approach to the L-P theory of square roots of elliptic operators, as well as an L-P version of Carleson's inequality. (C) 2007 Elsevier Inc. All rights reserved.