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 ‖ L u ‖ 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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