Abstract

Abstract. 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 ‖ √ Lu‖p � ‖∇u‖p and the H ∞-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.

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