Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Abstract

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.

On Hardy and BMO Spaces for Grushin Operator

“…More information about the classical real H p spaces with their characterizations and historical remarks can be also found in [22]. In [13] the authors provide a very general approach to the theory of H 1 spaces for semigroups of linear operators satisfying Davies-Gaffney estimates and in particular Gaussian bounds. Let us point out, that in the context of semigroups, the classical Hardy spaces can be thought as those associated with the Laplace operator on R n .…”

Maximal function characterizations for Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type

“…For the theory of Hardy spaces associated to operators, it has attracted a lot of attention in the last decades, and has been a very active research topic in harmonic analysis -see for example, [1,2,3,7,10,11,12,13,15,16,17,18,21,23,24].…”

“…We now recall the notion of a (p, q, M)-atom associated to an operator L ( [2,11,15]). The atomic Hardy space H p L,at,q,M (X) is defined as follows.…”

“…It can be verified (see [15,11,18]) that for all q > p with 1 ≤ q ≤ ∞ and every number M > n 2 ( 1 p −1), any (p, q, M)-atom a is in H p L,max (X) and so the following continuous inclusion holds:…”

“…holds for certain operators including Schrödinger operators with nonnegative potentials and second order divergence form elliptic operators via particular PDE technique (see for example, [13,14,15,16]). Very recently, the authors of this article have made a reformulation and modification of of a technique due to A. Calderón [5] to obtain an atomic decomposition directly from H p L,max (R n ).…”