Orlicz–hardy Spaces Associated With Operators Satisfying Davies–gaffney Estimates

Abstract

Abstract. Let X be a metric space with doubling measure, L a nonnegative self-adjoint operator in L 2 (X ) satisfying the Davies-Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type p ω ∈ (0, 1] and ρ(t) = t −1 /ω −1 (t −1 ) for all t ∈ (0, ∞). The authors introduce the Orlicz-Hardy space H ω,L (X ) via the Lusin area function associated to the heat semigroup, and the BMO-type space BMO ρ,L (X ). The authors then establish the duality between H ω,L (X ) and BMO ρ,L (X ); as a corollary, the authors obtain the ρ-Carleson measure characterization of the space BMO ρ,L (X ). Characterizations of H ω,L (X ), including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let X = R n and L = −∆ + V be a Schrödinger operator, where V ∈ L 1 loc (R n ) is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L; moreover, if there exist q 1 , q 2 ∈ (0, ∞) such that q 1 < 1 < q 2 and [ω(t q 2 )] q 1 is a convex function on (0, ∞), then several characterizations of the Orlicz-Hardy space H ω,L (R n ), in terms of the Lusin-area functions, the nontangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1).

Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

“…For the second direction (in generalizing the Laplacian to some other operator L) we cite the body of work in [13,12,14,15,16,19,25,26,27]. The starting point here is to replace the semigroup e −t 2 ∆ in (i) and (ii) by some other semigroup e −t 2 L , but one can define an adaptation of (iii) by encoding the cancellation of atoms using L in a certain way (see [25] and also Definition 2.1 below).…”

Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

“…This kind of Musielak-Orlicz-Hardy spaces associated with operators generalizes the (Orlicz-)Hardy space and the (weighted) Hardy space associated with operators, which has attracted great interests in recent years. Such function spaces associated with operators play important roles in the study for the boundedness of singular integrals associated with some differential operators, which may not fall within the scope of the classical Calderón-Zygmund theory (see, for example, [3], [5], [11], [12], [13], [17], [16], [18], [29], [30]). …”

Herz-Type Hardy Spaces Associated with Operators

“…For example, Auscher, Duong, and McIntosh [7], then Duong and Yan [8,9], introduced the Hardy and BMO spaces adapted to the operator which satisfies the Gaussian heat kernel upper bounds. Yang and his cooperators discussed new Orlicz-Hardy spaces associated with operators [10][11][12][13]. For more results, we refer to [14][15][16][17][18][19] and the references therein.…”