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.

Abstract. Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.

No abstract

No abstract

No abstract

No abstract

Theorem A. Assume u ǫ ∈ H 1 0 (Ω), ∇u ǫ 2 = 1 and u ǫ ⇀ u 0 weakly in H 1 0 (Ω). Then for any q < 1/(1 − ∇u 0 2 2), we have lim sup ǫ→0 Ω

No abstract