Weighted local Orlicz–Hardy spaces with applications to pseudo-differential operators

Abstract

Let Φ be a concave function on (0, ∞) of strictly lower type pΦ ∈ (0, 1] and ω ∈ A loc ∞ (R n ) (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy. We also introduce the BMO-type space bmoρ, ω (R n ) and establish the duality between h Φ ω (R n ) and bmoρ, ω (R n ). Characterizations of h Φ ω (R n ), including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h Φ ω (R n ), from which, we further deduce that for a given admissible triplet (ρ, q, s)ω and a β-quasi-Banach space B β with β ∈ (0, 1], if T is a B β -sublinear operator, and maps all (ρ, q, s)ω-atoms and (ρ, q)ω-single-atoms with q < ∞ (or all continuous (ρ, q, s)ω-atoms with q = ∞) into uniformly bounded elements of B β , then T uniquely extends to a bounded B β -sublinear operator from h Φ ω (R n ) to B β . As applications, we show that the local Riesz transforms are bounded on h Φ ω (R n ), the local fractional integrals are bounded from 2010 Mathematics Subject Classification: Primary 46E30; Secondary 42B35, 42B30, 42B25, 42B20, 35S05, 47G30, 47B06.

Hausdorff type operators on the power weighted Hardy spaces H|·|αp(Rn)

“…Remark If , we have the following well known facts (see e.g. , , ): (i)The dual space of is the space and they have an pair inequality for any and . (ii) and …”

Local Hardy spaces of Musielak-Orlicz type and their applications

“…The weighted local Orlicz-Hardy space h Φ ω (R n ), with ω ∈ A loc ∞ (R n ) and Φ being an Orlicz function of critical lower type index p − Φ ∈ (0, 1] (see Section 2 below for the notions of Orlicz functions and p − Φ ), via the local grand maximal function was introduced in [64]. Let ρ(t)including the atoms, the local vertical and the local nontangential maximal functions, were presented in [64]. Some criterions for the boundedness of B β -sublinear operators on h Φ ω (R n ) were also given in [64].…”

“…Some criterions for the boundedness of B β -sublinear operators on h Φ ω (R n ) were also given in [64]. As applications, it was showed, in [64], that the local Riesz transforms and some pseudo-differential operators are bounded on h Φ ω (R n ). Moreover, Orlicz-Hardy spaces associated with some differential operators and their dual spaces were introduced and studied in [30,32,38].Let φ be a positive increasing function on (0, ∞), and BMO φ (R n ) and bmo φ (R n ) two classes of BMOtype spaces introduced by Nakai and Yabuta [45] (see also Definition 7.7 below).…”

“…Let ϕ : R n × [0, ∞) → [0, ∞) be a growth function, namely, ϕ(x, ·) is an Orlicz function of upper type 1 and lower type p ∈ (0, 1], and ϕ(·, t) belongs to A loc ∞ (R n ) (see Definition 2.5 below). Motivated by [35,51,64], in this paper, we introduce a new local Musielak-Orlicz Hardy space, h ϕ (R n ), via the local grand maximal function and the local BMO-type space, bmo ϕ (R n ), which is further proved to be the dual space of h ϕ (R n ). As an interesting application, we show that the class of pointwise multipliers for bmo φ (R n ), characterized by Nakai and Yabuta in [45], is the dual of L 1 (R n ) + h Φ0 (R n ), where φ is an increasing function on (0, ∞) satisfying some additional growth conditions (see Theorem 7.9 and Remark 7.10 below), bmo φ (R n ) denotes the local BMO-type space introduced by Nakai and Yabuta in [45], and Φ 0 is a Musielak-Orlicz function induced by φ. Characterizations of h ϕ (R n ), including the atoms, the local vertical and the local nontangential maximal functions, are presented.…”

“…including the atoms, the local vertical and the local nontangential maximal functions, were presented in [64]. Some criterions for the boundedness of B β -sublinear operators on h Φ ω (R n ) were also given in [64].…”