On the theory of homogeneous Lipschitz spaces and Campanato spaces

Greenwald, Harvey Charles

doi:10.2140/pjm.1983.106.87

Cited by 11 publications

(4 citation statements)

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“…Let α ∈ [0, ∞), q ∈ [1, ∞] and s be an integer that is no less than nα , where and in what follows, s denotes the maximal integer no more than s. It was proved by Taibleson and Weiss [17] that the classical Morrey-Campanato spaces L(α, q, s; R n ) are dual spaces of Hardy spaces on R n . It was also pointed out by Janson, Taibleson and Weiss in [10] that for α = 0, the spaces L(α, q, s; R n ) are variants of BMO (R n ) (see [11]) and for α ∈ (0, ∞), they are variants of homogenous Besov-Lipschitz spaces with smoothness of order nα (see [9]). Throughout the whole paper, we only consider the case when α ∈ (0, ∞).…”

Section: Introductionmentioning

confidence: 96%

New Characterizations of Weighted Morrey-campanato Spaces

Yang¹,

Yang²

2011

Taiwanese J. Math.

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Let α ∈ (0, ∞), q ∈ [1, ∞], s be a nonnegative integer, ω ∈ A 1 (R n ) (the class of Muckenhoupt's weights). In this paper, the authors introduce the weighted Morrey-Campanato space L(α, q, s, ω; R n ) and obtain its equivalence on different q ∈ [1, ∞] and integers s ≥ nα (the integer part of nα). The authors then introduce the weighted Lipschitz space ∧(α, ω; R n ) and prove that ∧(α, ω;Using this, the authors further establish a new characterization of L(α, q, s, ω; R n ) by using the convolution ϕ tB * f to replace the minimizing polynomial P s B f on any ball B of a function f in its norm whenwhere ϕ is an appropriate Schwartz function, t B denotes the radius of the ball B and ϕ tB (•) ≡ t −n B ϕ(t −1 B •).

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Section: Introductionmentioning

confidence: 96%

New Characterizations of Weighted Morrey-campanato Spaces

Yang¹,

Yang²

2011

Taiwanese J. Math.

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“…REMARK. The idea of considering the expression J A k h f(x,s)K(h,t)dt is taken from [7], where a similar but isotropic case is treated.…”

Section: A K H F(x)k(ht)dh\\mentioning

confidence: 99%

On non-isotropic homogeneous Lipschitz spaces

Meda

1989

J Aust Math Soc A

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We prove that in a non-isotropic Euclidean space, homogeneous Lipschitz spaces of distributions, defined in terms of (generalized) Weierstrass integrals, can be characterized by means of higher order difference operators. The main purpose of this paper is to prove that, in the homogeneous case, these two points of view give rise to equivalent spaces of non-isotropic R".To do this we need non-isotropic higher order difference operators and a substitute for the Weierstrass kernel. Non-isotropic difference operators were introduced by Soardi in [15]; he also defined the homogeneous Lipschitz spaces A^°° (in our notation) and studied their connection with some Morrey-Campanato spaces (see also [12] for related results).

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“…Let α ∈ [0, ∞), p ∈ [1, ∞], and s be an integer no less than nα , where and in what follows, s denotes the maximal integer no more than s. It was proved by Taibleson and Weiss [19] that the classical Morrey-Campanato spaces L(α, p, s; R n ) are dual spaces of Hardy spaces on R n . It is also well known that when α = 0, the spaces L(α, p, s; R n ) are variants of BMO (R n ) (see [13]), and when α ∈ (0, ∞), they are variants of homogenous Besov-Lipschitz spaces with smoothness of order nα (see [11]). Throughout the whole paper, we only consider the case when α ∈ (0, ∞).…”

Section: §1 Introductionmentioning

confidence: 99%

Elementary characterizations of generalized weighted Morrey-Campanato spaces

Yang

2010

Appl. Math. J. Chin. Univ.

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Let α ∈ (0, ∞), p, q ∈ [1, ∞), s be a nonnegative integer, and ω ∈ A1(R n ) (the class of Muckenhoupt's weights). In this paper, we introduce the generalized weighted MorreyCampanato space L(α, p, q, s, ω; R n ) and obtain its equivalence on different p ∈ [1, β) andWe then introduce the generalized weighted Lipschitz space ∧(α, q, ω; R n ) and prove that L (α, p, q, s, ωIt is well known that the classical Morrey-Campanato spaces play an important role in the study of partial differential equations and harmonic analysis; see, for example, [1][2][3][4][5]7,8,12,13,[15][16][17]. Let α ∈ [0, ∞), p ∈ [1, ∞], and s be an integer no less than nα , where and in what follows, s denotes the maximal integer no more than s. It was proved by Taibleson and Weiss[19] that the classical Morrey-Campanato spaces L(α, p, s; R n ) are dual spaces of Hardy spaces on R n . It is also well known that when α = 0, the spaces L(α, p, s; R n ) are variants of BMO (R n ) (see [13]), and when α ∈ (0, ∞), they are variants of homogenous Besov-Lipschitz spaces with smoothness of order nα (see [11]). Throughout the whole paper, we only consider the case when α ∈ (0, ∞).For α ∈ (0, ∞), p ∈ [1, ∞], a nonnegative integer s and ω ∈ A 1 (R n ), the weighted MorreyCampanato spaces L(α, p, s, ω; R n ) were recently introduced in [22] and their certain equivalent characterizations were also given therein. We point out that when s = 0, Tang in [20] first introduced the weighted Morrey-Campanato spaces L(α, p, s, ω; R n ) and established some of their equivalent characterizations. Here and in what follows, A 1 (R n ) denotes the class of Muckenhoupt's weights. In this paper, we continue this research by introducing the generalized weighted Morrey-Campanato spaces L(α, p, q, s, ω; R n ) for q ∈ [1, ∞]. When q = ∞,

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On the theory of homogeneous Lipschitz spaces and Campanato spaces

Abstract: In this paper the equivalence between the Campanato spaces and homogeneous Lipschitz spaces is shown through the use of elementary and constructive means. These Lipschitz spaces can be defined in terms of derivatives as well as differences.

Cited by 11 publications

References 8 publications

New Characterizations of Weighted Morrey-campanato Spaces

New Characterizations of Weighted Morrey-campanato Spaces

On non-isotropic homogeneous Lipschitz spaces

Elementary characterizations of generalized weighted Morrey-Campanato spaces

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