The characterization of weighted local hardy spaces on domains and its application

Wang, Henggeng; Yang, Xiaoming

doi:10.1631/jzus.2004.1148

Cited by 6 publications

(7 citation statements)

References 4 publications

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“…In Sect. 2 Recently, Wang and Yang [19] (see also Bui [5]) have studied weighted local Hardy spaces on domains. Bownik, Li, Yang and Zhou [4] investigated weighted anisotropic Hardy spaces.…”

mentioning

confidence: 99%

Anisotropic Local Hardy Spaces

Damián

2010

J Fourier Anal Appl

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mentioning

confidence: 99%

Anisotropic Local Hardy Spaces

Damián

2010

J Fourier Anal Appl

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“…A variety of similar spaces, including spaces on domains, weighted spaces on domains and Sobolev-type spaces based on the H p norm, have since been studied e.g. in [34,48,35,36,57,6,26].…”

Section: The Trace Of a Weighted Hardy-sobolev Spacementioning

confidence: 99%

Traces of weighted function spaces: Dyadic norms and Whitney extensions

2017

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Abstract. The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well adapted to extending functions using the Whitney extension operator.

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“…Proof of Theorem 1.7. We borrow some ideas from [66,65] and [84]. We prove Theorem 1.7 by using the following strategy: first, we show that (i) and (iv) are equivalent; then we prove the equivalence between (ii) and (iii); finally, we show that (ii) implies (i), which, together with the standard proof of the implication (iv) =⇒ (iii), completes the proof of Theorem 1.7.…”

Section: Proof Of Theorem 17mentioning

confidence: 99%

“…Let Ω be a bounded Lipschitz domain of R n . Wang and Yang [84] introduced the weighted local Hardy spaces on Ω by restricting arbitrary elements of h p ω (R n ) to Ω with ω ∈ A ∞ (R n ). They characterized the space h p ω, r (Ω) in terms of the grand maximal function, the radial maximal function and the atomic decomposition.…”

Section: Introductionmentioning

confidence: 99%

Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems

Cao

Chang

Yang

et al. 2013

Trans. Amer. Math. Soc.

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Let Ω be either R n or a strongly Lipschitz domain of R n , and ω ∈ A ∞ (R n) (the class of Muckenhoupt weights). Let L be a second-order divergence form elliptic operator on L 2 (Ω) with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by L has the Gaussian property (G 1) with the regularity of their kernels measured by μ ∈ (0, 1]. Let Φ be a continuous, strictly increasing, subadditive, positive and concave function on (0, ∞) of critical lower type index p − Φ ∈ (0, 1]. In this paper, the authors first introduce the "geometrical" weighted local Orlicz-Hardy spaces h Φ ω, r (Ω) and h Φ ω, z (Ω) via the weighted local Orlicz-Hardy spaces h Φ ω (R n), and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by L when p − Φ ∈ (n/(n + μ), 1]. Second, the authors furthermore establish three equivalent characterizations of h Φ ω, r (Ω) in terms of the grand maximal function, the radial maximal function and the atomic decomposition when the complement of Ω is unbounded and p − Φ ∈ (0, 1]. Third, as applications, the authors prove that the operators ∇ 2 G D are bounded from h Φ ω, r (Ω) to the weighted Orlicz space L Φ ω (Ω), and from h Φ ω, r (Ω) to itself when Ω is a bounded semiconvex domain in R n and p − Φ ∈ (n n+1 , 1], and the operators ∇ 2 G N are bounded from h Φ ω, z (Ω) to L Φ ω (Ω), and from h Φ ω, z (Ω) to h Φ ω, r (Ω) when Ω is a bounded convex domain in R n and p − Φ ∈ (n n+1 , 1], where G D and G N denote, respectively, the Dirichlet Green operator and the Neumann Green operator.

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The characterization of weighted local hardy spaces on domains and its application

Abstract: In this paper, we give the four equivalent characterizations for the weighted local hardy spaces on Lipschitz domains. Also, we give their application for the harmonic function defined in bounded Lipschitz domains.

Cited by 6 publications

References 4 publications

Anisotropic Local Hardy Spaces

Anisotropic Local Hardy Spaces

Traces of weighted function spaces: Dyadic norms and Whitney extensions

Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems

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