Abstract. We prove quadratic estimates for complex perturbations of Diractype operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge-Dirac operator on compact manifolds depend analytically on L ∞ changes in the metric. We also recover a unified proof of many results in the Calderón program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces.

Let p > 1 and let (X, d, μ) be a complete metric measure space with μ Borel and doubling that admits a (1, p)-Poincaré inequality. Then there exists ε > 0 such that (X, d, μ) admits a (1, q)-Poincaré inequality for every q > p−ε, quantitatively.

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The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces. Moreover, several competing definitions for the Poincaré inequality are shown to coincide, if the underlying measure is doubling. One such characterization considers only continuous functions and their continuous upper gradients, and extends work of Heinonen and Koskela. Applications include showing that the p-Poincaré inequality (with a doubling measure), for p ≥ 1, persists through to the limit of a sequence of converging pointed metric measure spaces -this extends results of Cheeger. A further application is the construction of new doubling measures in Euclidean space which admit a 1-Poincaré inequality.

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Let α ≥ 1 and let (X, d, µ) be an α-homogeneous metric measure space with conformal Assouad dimension equal to α. Then there exists a weak tangent of (X, d, µ) with uniformly big 1-modulus.

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Abstract. We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in R n . The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smoothability of a Lipschitz manifold in terms of a Sobolev regularity for frames in a cotangent structure. In the proofs, we exploit the duality between flat chains and flat forms, and recently established differential analysis on metric measure spaces. When specialized to R n , our result gives a kind of asymptotic and Lipschitz version of the measurable Riemann mapping theorem as suggested by Sullivan.