We define a scale of Hardy spaces H p F IO (R n ), p ∈ [1, ∞], that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for p = 1 [34]. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of R n , and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about L p -boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.2010 Mathematics Subject Classification. Primary 42B35. Secondary 42B30, 35S30, 58J40.

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.

Abstract. We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-M c Intosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds, such that A always has a bounded H ∞ -functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces L p (R n ; X) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMDvalued context. Even when X = C, our approach gives refined p-dependent versions of known results.