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

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 478 wydano: 2011
Zawartość
Warianty tytułu
Abstrakty
EN
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.
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Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 478
Liczba stron
78
Liczba rozdzia³ów
Opis fizyczny
Daty
wydano
2011
Twórcy
autor
• School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
autor
• School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
Bibliografia
Języki publikacji
 EN
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