Hardy spaces meet harmonic weights

Abstract

We investigate the Hardy space H L 1 H^1_L associated with a self-adjoint operator L L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L L -harmonic non-negative function h h such that the semigroup exp ⁡ ( − t L ) \exp (-tL) , after applying the Doob transform related to h h , satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H L 1 H^1_L in terms of a simple atomic decomposition associated with the L L -harmonic function h h . Our approach also yields a natural characterisation of the B M O BMO -type space corresponding to the operator L L and dual to H L 1 H^1_L in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in R n {\mathbb {R}^n} , Schrödinger operators with certain potentials, and Bessel operators.

Sparse bounds for pseudo-multipliers associated to Grushin operators, II

“…Grushin operators is well developed and we recall it according to our requirement. For κ = 1, the Hardy space for the Grushin operator G was studied in [DJ16] and subsequently more general results were established in [PSY19].…”

“…In practice, for 1 ≤ q < ∞, in the spirit of [DY05,PSY19], we consider the atomic Hardy space H 1 at,q as well as BMO spaces on the homogeneous space (R n 1 +n 2 , d, | • |) as follows.…”

“…where the infimum is taken over all such representations of f . Since the heat semigroup {e −tGκ } t>0 is conservative, that is, R n 1 +n 2 e −tGκ (x, y) dy = 1 for all t > 0 and x ∈ R n 1 +n 2 , invoking Theorem A in [PSY19] we have that the spaces H 1…”

“…Here we have used the fact that the heat semigroup {e −tGκ } t>0 is conservative, that is, e −tG (1) = R n 1 +n 2 e −tGκ (x, y) dy = 1, for all t > 0, and therefore one can do the analysis of [PSY19] with h = 1 itself.…”