Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

Bui, The Anh ; Cao, Jun ; Ky, Luong Dang ; Yang, Dachun ; Yang, Sibei

Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 69-129

Résumé

Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order (qL/l(φ))′, where (qL/l(φ))′ denotes the conjugate exponent of qL/l(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ;L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ,L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ,L(Rn) and the classical Musielak-Orlicz-Hardy space Hv(Rn) is given. Moreover, for the Musielak-Orlicz-Hardy space Hφ,L(Rn) associated with the second order elliptic operator in divergence form on Rn or the Schrödinger operator L := −Δ + V with 0 ≤ V ∊ L1loc(Rn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(Rn) to the Musielak-Orlicz space Lφ(Rn) when i(φ) ∊ (0; 1], from Hφ,L(Rn) to Hφ(Rn) when i(φ) ∊ ( [...] ; 1], and from Hφ,L(Rn) to the weak Musielak-Orlicz-Hardy space WHφ(Rn) when i(φ)= [...] is attainable and φ(·; t) ∊ A1(X), where i(φ) denotes the uniformly critical lower type index of φ

Détail

Mots-clés : Musielak-Orlicz-Hardy space, molecule, atom, maximal function, Lusin area function, Schrödinger operator, elliptic operator, Riesz transform