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Hardy spaces H L p (ℝ$^{ n }$) associated with operators satisfying k-Davies-Gaffney estimates

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Hardy spaces H L p (ℝ$^{ n }$) associated with operators satisfying k-Davies-Gaffney estimates

Content Provider	Springer Nature Link
Author	Cao, Jun Yang, DaChun
Copyright Year	2012
Abstract	Let L be a one-to-one operator of type ω having a bounded H $_{∞}$ functional calculus and satisfying the k-Davies-Gaffney estimates with k ∈ ℕ. In this paper, the authors introduce the Hardy space H L p (ℝ$^{ n }$) with p ∈ (0, 1] associated with L in terms of square functions defined via $$\left\{ {e^{ - t^{2k} L} } \right\}_{t > 0}$$ and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L $_{1}$ with complex bounded measurable coefficients and the 2k-order Schrödinger type operator L $^{2}$:= (−Δ)$^{ k }$ + V $^{ k }$, where Δ is the Laplacian and 0 ⩽ V ∈ L loc k (ℝ$^{ n }$). Moreover, as an application, for i ∈ {1, 2}, the authors prove that the associated Riesz transform ▿ $^{ k }$(L i −1/2 ) is bounded from $$H_{L_i }^p \left( {\mathbb{R}^n } \right)$$ to H $^{ p }$(ℝ$^{ n }$) for p ∈ (n/(n + k), 1] and establish the Riesz transform characterizations of $$H_{L_1 }^p \left( {\mathbb{R}^n } \right)$$ for p ∈ (rn/(n + kr), 1] if $$\left\{ {e^{ - tL_1 } } \right\}_{t > 0}$$ satisfies the L $^{ r }$ − L $^{2}$ k-off-diagonal estimates with r ∈ (1, 2]. These results when k:= 1 and L:= L $_{1}$ are known.
Starting Page	1403
Ending Page	1440
Page Count	38
File Format	PDF
ISSN	16747283
Journal	Science in China Series A: Mathematics
Volume Number	55
Issue Number	7
e-ISSN	18691862
Language	English
Publisher	SP Science China Press
Publisher Date	2012-04-01
Publisher Place	Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Hardy space Hardy-Sobolev space k-Davies-Gaffney estimate Schrödinger type operator higher order elliptic operator semigroup square function higher order Riesz transform molecule Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Mathematics

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