We solve the Kato square root problem for divergence form operators on complete Riemannian manifolds that are embedded in Euclidean space with a bounded second fundamental form. We do this by proving local quadratic estimates for perturbations of certain first-order differential operators that act on the trivial bundle over a complete Riemannian manifold with at most exponential volume growth and on which a local Poincaré inequality holds. This is based on the framework for Dirac-type operators that was developed by Axelsson, Keith and McIntosh.x denotes the value of a function or section at x in M . The components A 00 and A 10 act by multiplication, as in (A 10 ) x ((u 0 ) x ) := (A 10 ) x × u 0 (x). The notation for the components of A is chosen to reflect that T 0,0 M := C, T 1,0 M = T M, T 0,1 M = T * M and T 1,1 M ∼ = End(T M). The bilinear formGiven A as above and a in L ∞ (M ), suppose that there exist constants κ 1 , κ 2 > 0 such that the following accretivity conditions are satisfied:( 1.2)The divergence form operator L A,a :We solve the Kato square root problem for the operator L A,a as in the following theorem.Theorem 1.1. Let n ∈ N and suppose that M is a complete Riemannian manifold that is embedded in R n with a bounded second fundamental form. If a and A satisfy the accretivity conditions in (1.2), then the divergence form operator L A,a defined by (1.3) has a square root L A,a with domain D( L A,a ) = W 1,2 (M ) and L A,a u L 2 (M ) u W 1,2 (M ) for all u ∈ W 1,2 (M ).

We consider layer potentials associated to elliptic operators Lu=−div(A∇u) acting in the upper half‐space R+n+1 for n⩾2, or more generally, in a Lipschitz graph domain, where the coefficient matrix A is L∞‐ and t‐independent, and solutions of Lu=0 satisfy interior estimates of De Giorgi/Nash/Moser type. A ‘Calderón–Zygmund’ theory is developed for the boundedness of layer potentials, whereby sharp Lp and endpoint space bounds are deduced from L2‐bounds. Appropriate versions of the classical ‘jump relation’ formulae are also derived. The method of layer potentials is then used to establish well‐posedness of boundary value problems for L with data in Lp and endpoint spaces.

We define local Hardy spaces of differential forms h p D (∧T * M ) for all p ∈ [1, ∞] that are adapted to a class of first order differential operators D on a complete Riemannian manifold M with at most exponential volume growth. In particular, if D is the Hodge-Dirac operator on M and ∆ = D 2 is the Hodge-Laplacian, then the local geometric Riesz transform D(∆ + aI) −1/2 has a bounded extension to h p D for all p ∈ [1, ∞], provided that a > 0 is large enough compared to the exponential growth of M . A characterisation of h 1 D in terms of local molecules is also obtained. These results can be viewed as the localisation of those for the Hardy spaces of differential forms H p D (∧T * M ) introduced by Auscher, McIntosh and Russ.

We establish new Calderón reproducing formulas for self-adjoint operators D that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with D through holomorphic functional calculus whilst the synthesising function interacts with D through functional calculus based on the Fourier transform. We apply these to prove the embeddingfor the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where D = d + d * is the Hodge-Dirac operator on a complete Riemannian manifold M that has polynomial volume growth. This fills a gap in that work. The new reproducing formulas also allow us to obtain an atomic characterisation of H 1 D (∧T * M ). The embedding H p L ⊆ L p , 1 ≤ p ≤ 2, where L is either a divergence form elliptic operator on R n , or a nonnegative self-adjoint operator that satisfies Davies-Gaffney estimates on a doubling metric measure space, is also established in the case when the semigroup generated by the adjoint −L * is ultracontractive.