Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates-猫眼课题宝

DOI：

10.2478/agms-2012-0006

发表时间：

2013-02

期刊：

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其他

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作者：

T. A. Bui;Jun Cao;Luong Dang Ky;Dachun Yang;Sibei Yang

设X是具有加倍测度的度量空间，L是ω型一对一算子，在L2（X）中有界H∞ -泛函演算满足球上的加强（pL; qL）非对角估计，其中pL <$[1; 2]，qL <$（2;∞）。令φ：X × [0; ∞）→ [0; ∞）是一个函数，使得φ（x;·）是Orlicz函数，φ（·;t）∈ A∞（X）（一致Muckenhoupt权类），它的一致临界上型指数l（φ）∈（0;1]，且φ（·; t）满足（qL/l（φ））′阶的一致逆Hölder不等式，其中（qL/l（φ））′表示qL/l（φ）的共轭指数.通过与L相联系的Lusin面积函数，引入了Musielak-Orlicz-哈代空间Hφ;L（X），并建立了它的分子特征.特别地，当L是非负自伴的且满足Davies-Gaffney估计时，还得到了Hφ，L（X）的原子特征.进一步给出了Hφ，L（n）与经典Musielak-Orlicz-哈代空间H_v（n）等价的一个充分条件.此外，对于Musielak-Orlicz-哈代空间Hφ，L（n），当i（φ）≥ 0时，作者进一步得到了它的几个等价的非切向和径向极大函数的刻画;最后，作者证明了Riesz变换<$L−1/2从Hφ，L（n）到Musielak-Orlicz空间Lφ（n）是有界的（0分;当i（φ）=可达且φ（·; t）<$A1（X）时，从H φ，L（n）到弱Musielak-Orlicz-哈代空间WHφ（n），其中i（φ）表示φ的一致临界下型指标

Abstract 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

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

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