Article

Science in China Series A: Mathematics

, Volume 52, Issue 5, pp 1042-1080

Orlicz-Hardy spaces associated with operators

  • RenJin JiangAffiliated withLaboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University
  • , DaChun YangAffiliated withLaboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University Email author 
  • , Yuan ZhouAffiliated withLaboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University

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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 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 (ℝ n ) and the BMO-type space BMO ρ, L (ℝ n ) and establish the John-Nirenberg inequality for BMO ρ, L (ℝ n ) functions and the duality relation between H ω, L ((ℝ n ) and BMO ρ, 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 BMO ρ, L* (ℝ n ) and the molecular characterization of H ω, L (ℝ n ); the latter is used to establish the boundedness of the generalized fractional operator L ρ γ from H ω, L (ℝ n ) to H L 1 (ℝ n ) or L q (ℝ n ) with certain q > 1, where H L (ℝ 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].

Keywords

Orlicz function Orlicz-Hardy space BMO duality molecule fractional integral

MSC(2000)

42B30 42B35 42B20 42B25