Luong Dang Ky

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

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 Campanato Spaces

In this chapter, we study the Musielak-Orlicz Campanato space \(\mathcal{L}_{\varphi,q,s}(\mathbb{R}^{n})\) and, as an application, prove that some of them is the dual space of the Musielak-Orlicz Hardy space \(H^{\varphi }(\mathbb{R}^{n})\).

Weak Musielak-Orlicz Hardy Spaces

In this chapter, we introduce the weak Musielak-Orlicz Hardy space \(W\!H^{\varphi }(\mathbb{R}^{n})\) via the grand maximal function and then obtain its vertical or its non-tangential maximal function characterizations. We also establish other real-variable characterizations of \(W\!H^{\varphi }(\mathbb{R}^{n})\), respectively, by means of the atom, the molecule, the Lusin area function, the Littlewood-Paley g-function or the g λ ∗-function. As an application, the boundedness of Calderon-Zygmund operators from \(H^{\varphi }(\mathbb{R}^{n})\) to \(W\!H^{\varphi }(\mathbb{R}^{n})\) in the critical case is presented.

Local Musielak-Orlicz Hardy Spaces

In this chapter, we introduce a local Musielak-Orlicz Hardy space \(h^{\varphi }(\mathbb{R}^{n})\) by the local grand maximal function, and a local BMO-type space \(\mathrm{bmo}^{\varphi }(\mathbb{R}^{n})\) which is further proved to be the dual space of \(h^{\varphi }(\mathbb{R}^{n})\). As an application, we prove that the class of pointwise multipliers for the local BMO-type space \(\mathrm{bmo}^{\phi }(\mathbb{R}^{n})\), characterized by E. Nakai and K. Yabuta, is just the dual of \(L^{1}(\mathbb{R}^{n}) + h^{\Phi _{0}}(\mathbb{R}^{n})\), where ϕ is an increasing function on (0, ∞) satisfying some additional growth conditions and \(\Phi _{0}\) a Musielak-Orlicz function induced by ϕ. Characterizations of \(h^{\varphi }(\mathbb{R}^{n})\), including the atoms, the local vertical or the local non-tangential maximal functions, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of \(h^{\varphi }(\mathbb{R}^{n})\), from which, we further deduce some criterions for the boundedness on \(h^{\varphi }(\mathbb{R}^{n})\) of some sublinear operators. Finally, we show that the local Riesz transforms and some pseudo-differential operators are bounded on \(h^{\varphi }(\mathbb{R}^{n})\).

Musielak-Orlicz Hardy Spaces

In this chapter, we first recall the notion of growth functions, establish some technical lemmas and introduce the Musielak-Orlicz Hardy space \(H^{\varphi }(\mathbb{R}^{n})\) which generalize the Orlicz-Hardy space of Janson and the weighted Hardy space of Garcia-Cuerva, Stromberg and Torchinsky.

Littlewood-Paley Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces

In this chapter, we establish the Littlewood-Paley function and the molecular characterizations of the Musielak-Orlicz Hardy space \(H^{\varphi }(\mathbb{R}^{n})\).

Maximal Function Characterizations of Musielak-Orlicz Hardy Spaces

In this chapter, we establish some real-variable characterizations of \(H^{\varphi }(\mathbb{R}^{n})\) in terms of the vertical or the non-tangential maximal functions, via first establishing a Musielak-Orlicz Fefferman-Stein vector-valued inequality.

Bilinear Decompositions and Commutators of Calderón-Zygmund Operators

As another application of Musielak-Orlicz Hardy space \(H^{\log }(\mathbb{R}^{n})\), we consider the boundedness of commutators in this chapter. It is well known that the linear commutator [b, T], generated by a BMO function b and a Calderon-Zygmund operator T, may not be bounded from \(H^{1}(\mathbb{R}^{n})\) into \(L^{1}(\mathbb{R}^{n})\).