Orlicz-Hardy spaces associated with operators

Abstract

Let L be a linear operator in L 2 (R n ) and generate an analytic semigroup {e −tL } t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞).Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(tWe introduce the Orlicz-Hardy space Hω, L(R n ) and the BMO-type space BMOρ, L(R n ) and establish the John-Nirenberg inequality for BMOρ, L(R n ) functions and the duality relation between Hω, L(R n ) and BMOρ, L * (R n ), where L * denotes the adjoint operator of L in L 2 (R n ).Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMOρ, L * (R n ) and the molecular characterization of Hω, L(R n ); the latter is used to establish the boundedness of theis the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = t p for t ∈ (0, ∞) and p ∈ (n/(n + θ(L)), 1].

Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates

“…It is well-known that the Hardy space H p (R n ) when p ∈ (0, 1] is a good substitute of L p (R n ) when studying the boundedness of operators, and the Hardy space H p (R n ) is essentially related to the Laplacian Δ ≡ − n i=1 ∂ 2 ∂x 2 i . In recent years, the study of the real-variable theory of function spaces associated with operators has attracted great interest; see, for example, [3,4,8,13,14,16,19,20,22,27,28,30,32,33,47].…”

“…Moreover, Orlicz-Hardy spaces are also suitable substitutes of the Orlicz spaces in the study of boundedness of operators; see, for example, [26,28,30,33,46]. Recall that Orlicz-Hardy spaces and their dual spaces were studied by Janson [26] on R n and Viviani [46] on spaces of homogeneous type in the sense of Coifman and Weiss [11].…”

“…Recall that Orlicz-Hardy spaces and their dual spaces were studied by Janson [26] on R n and Viviani [46] on spaces of homogeneous type in the sense of Coifman and Weiss [11]. Recently, Orlicz-Hardy spaces associated with some differential operators and their dual spaces were introduced and studied in [28,30,33]. Now we briefly review the development of Orlicz-Hardy spaces associated with operators as follows.…”

“…We point out that in [31,33], Φ is only assumed to be of critical lower type p − Φ ∈ (n/(n + θ(L)), 1], which seems not to be enough; see Remark 2.2(ii) below for more details. Let…”

“…for all t ∈ (0, ∞), where Φ −1 is the inverse function of Φ. Jiang et al [33] introduced the Orlicz-Hardy space H Φ, L (R n ) and the BMO-type space BMO ρ, L (R n ) and proved that the dual space of H Φ, L (R n ) is the space BMO ρ, L * (R n ). Jiang et al [33] also obtained the molecular characterization of H Φ, L (R n ) in terms of the molecules in [4].…”