Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L 2 (? 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 p o (ω) ? (n/(n+θ(L)),1] and ρ(t) = t t1/ω ?1(t ?1) for t ∈ (0,∞). We introduce the Orlicz-Hardy space H ω, L (? n ) and the BMO-type space BMO ρ, L (? n ) and establish the John-Nirenberg inequality for BMO ρ, L (? n ) functions and the duality relation between H ω, L ((? n ) and BMO ρ, L* (? n ), where L* denotes the adjoint operator of L in L 2 (? n ). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMO ρ, L* (? n ) and the molecular characterization of H ω, L (? n ); the latter is used to establish the boundedness of the generalized fractional operator L ρ ?γ from H ω, L (? n ) to H L 1 (? n ) or L q (? n ) with certain q > 1, where H L (? n ) is 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].

Let L be a linear operator in L2(Rn) 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(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L*(Rn), where L* denotes the adjoint operator of L in L2(Rn). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMOρ, L*(Rn) and the molecular characterization of Hω, L(Rn); the latter is used to establish the boundedness of the generalized fractional operator Lρ- γfrom Hω, L(Rn) to HL1(Rn) or Lq(Rn) with certain q > 1, where HL1(Rn) is the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = tp for t ∈ (0, ∞) and p ∈ (n/(n + θ(L)), 1].