Abstract. Let X be a metric space with doubling measure, and L be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a non-negative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, we define Hardy spaces H p L (X) for p > 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L (X) spaces by the complex method.

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We prove that given any k ∈ N, for each open set Ω ⊆ R n and any closed subset D of Ω such that Ω is locally an (ε, δ)-domain near ∂Ω \ D there exists a linear and bounded extension operatoris defined as the completion in the classical Sobolev space W k,p (O) of (restrictions to O of) functions from C ∞ c (R n ) whose supports are disjoint from D. In turn, this result is used to develop a functional analytic theory for the class W k,p D (Ω) (including intrinsic characterizations, boundary traces and extensions results, interpolation theorems, among other things) which is then employed in the treatment of mixed boundary value problems formulated in locally (ε, δ)-domains.

We explore the extent to which basic differential operators (such as Laplace-Beltrami, Lamé, Navier-Stokes, etc.) and boundary value problems on a hypersurface S in R n can be expressed globally, in terms of the standard spatial coordinates in R n . The approach we develop also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations.Boundary value problems (BVP's) for partial differential equations (PDE's) on surfaces arise in a variety of situations and have many practical applications. See, for example, [15, §72] for the heat conduction by surfaces, [3, §10] for the equations of surface flow, [7], [8] and [13] for shell problems in elasticity, [2] for the vacuum Einstein equations describing gravitational fields, [29] for the Navier-Stokes equations on spherical domains, as well as the references therein. Furthermore, while studying the asymptotic behavior of solutions to elliptic boundary value problems in the neighborhood of a conical point one is led to considering a one-parameter family of boundary value problems in a subdomain S of S n−1 , the unit sphere in R n , naturally associated (via the Mellin transform) with the original elliptic problem. A classical reference in this regard is [16]. Finally, PDE's on surfaces also turn up naturally in the limit case, as the thickness goes to zero, of equations in thin layers or shells. Cf. [7, §3] for the case of elasticity, and [29] and [30] for the case of Navier-Stokes equations.A hypersurface S in R n has the natural structure of a (n − 1)-dimensional Riemannian manifold and the aforementioned PDE's are not the immediate analogues of the ones corresponding to the flat, Euclidean case, since they have to take into consideration geometric characteristics of S such as curvature. Inherently, these PDE's are originally written in local coordinates, intrinsic to the manifold structure of S.The main aim of this paper is to explore the extent to which the most basic partial differential operators (PDO's), as well as their associated boundary value problems, on a hypersurface S in R n , can be expressed globally, in terms of the standard spatial coordinates in R n . It turns out that a convenient way to carry out this program is by employing the so-called Günter derivatives (cf. [11], [14] and [17]): D := (D 1 , D 2 , . . . , D n ) . (0.1)Here, for each 1 ≤ j ≤ n, the first-order differential operator D j is the directional derivative along πe j , where π : R n → T S is the orthogonal projection onto the tangent plane to S and, as usual, e j = (δ jk ) 1≤k≤n ∈ R n , with δ jk denoting the Kronecker symbol. The operator D is globally defined on (as well as tangential to) S, and *

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space.The main goal is to develop the corresponding theory for L p -integrable bounday data for optimal values of p's. We also discuss a number of relevant applications in electromagnetic scattering. Statement of the Problems and Introductory RemarksLet us consider the electromagnetic wave propagation in a homogeneous, isotropic medium that occupies the exterior of a bounded domain in R 3 and has electric conductivity σ ≥ 0, electric permittivity > 0, and magnetic permeability µ. If we denote by E, H the electric and the magnetic fields, respectively, and if J stands for the current density, then the Maxwell equations readAlso, in an isotropic conductor, the electric field satisfies Ohm's law σ E = J . An excellent exposition of this material can be found in [24, Vol. I]; cf. also [15].We assume time-harmonic dependency for E and H, that is, that for some time-independent vector fields E, H the following separation of variables holds:H(X, t) = µ − 1 2 H (X) e −iωt , P P