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

100%

Yang D., Yang S.

Instytut Matematyczny Polskiej Akademii Nauk

Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{Φ} ∈ (0,1]$ and $ω ∈ A^{loc}_{∞}(ℝ ⁿ)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h^{Φ}_{ω}(ℝ ⁿ)$ via the local grand maximal function. Let $ρ(t)≡ t^{-1}/Φ^{-1}(t^{-1})$ for all t ∈ (0,∞). We also introduce the BMO-type space $bmo_{ρ,ω}(ℝ ⁿ)$ and establish the duality between $h^{Φ}_{ω}(ℝ ⁿ)$ and $bmo_{ρ,ω}(ℝ ⁿ)$. 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 $h^{p}_{ω^{p}}(ℝ ⁿ)$ to $L^{q}_{ω^{q}}(ℝ ⁿ)$ when q > 1 and from $h^{p}_{ω^{p}}(ℝ ⁿ)$ to $h^{q}_{ω^{q}}(ℝ ⁿ)$ when q ≤ 1, and some pseudo-differential operators are also bounded on both $h^{Φ}_{ω}(ℝ ⁿ)$. All results for any general Φ even when ω ≡ 1 are new.

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

67%

Bui T. A., Cao J., Ky L. D., Yang D., Yang S.

Analysis and Geometry in Metric Spaces

2013

tom 1

69-129

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

Ograniczanie wyników

1 Analysis and Geometry in Metric Spaces

2 Yang D.

2 Yang S.

1 Bui T. A.

1 Cao J.

1 Ky L. D.

1 2013

1 2011

Wyniki wyszukiwania

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

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