A Marcinkiewicz integral type characterization of the Sobolev space

Hajłasz, Piotr; Liu, Zhuomin

doi:10.5565/publmat_61117_03

Cited by 10 publications

(5 citation statements)

References 24 publications

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“…The case p > 1 follows from the L p boundedness of the Riesz transforms that yields the equivalence between the quantities ∇u L p (R N ) and (−∆) 1/2 u L p (R N ) ; see e.g. [19,Lemma 3.6]. The case p = 1 requires a different argument that can be found in [27,28], based on the approximation of u by smooth functions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On formulae decoupling the total variation of BV functions

Ponce

Spector

2017

Nonlinear Analysis: Theory, Methods & Applications

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To Nicola Fusco, a master of BV, on the occasion of his sixtieth birthday ABSTRACT. In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special case shows that rescalings of the fractional Laplacian of a function u ∈ SBV converge strictly to the singular portion of Du.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On formulae decoupling the total variation of BV functions

Ponce

Spector

2017

Nonlinear Analysis: Theory, Methods & Applications

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“…We refer to [22,23,25,29] for relevant results. See [11] for characterization of the weighted Sobolev space W 1,p w using square functions. Also, we consider discrete parameter versions of T α and U α :…”

Section: Applications To the Theory Of Sobolev Spacesmentioning

confidence: 99%

Littlewood–Paley Equivalence and Homogeneous Fourier Multipliers

Sato

2016

Integr. Equ. Oper. Theory

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Abstract. We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for Hörmander's theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given. IntroductionWe consider the Littlewood-Paley function on R n defined by, where ψ t (x) = t −n ψ(t −1 x). The following result of Benedek, Calderón and Panzone [2] on the L p boundedness, 1 < p < ∞, of g ψ is well-known.Theorem A. We assume (1.1) for ψ andwhereBy the Plancherel theorem, it follows that g ψ is bounded on L 2 (R n ) if and only if m ∈ L ∞ (R n ), where m(ξ) = ∞ 0 |ψ(tξ)| 2 dt/t, which is a homogeneous function of degree 0. Here the Fourier transform is defined aŝ ψ(ξ) = R n ψ(x)e −2πi x,ξ dx, x, ξ = n k=1x k ξ k .

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“…Theorem A was generalized to the weighted Sobolev spaces in [10]. Also, Theorems A and B were extended to the weighted Sobolev spaces in [19] by applying a theorem of [17] for the boundedness of Littlewood-Paley functions g ψ in (1.5) on the weighted L p spaces, which is partly a special case of Theorem 2.1.…”

Section: Let δ *mentioning

confidence: 99%

“…are also applied to characterize Sobolev spaces. See also [10] and [21] for applications of the square function…”

Section: Let δ *mentioning

confidence: 99%