Asymptotic behaviour of Besov norms via wavelet type basic expansions

Anna Kamont

Anna Kamont

Annales Polonici Mathematici, Tome 116 (2016), p. 101-144 / Harvested from The Polish Digital Mathematics Library

Résumé

J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $Ω \subset ℝ^{d}$ is a smooth bounded domain, 1 ≤ p < ∞ and $f \in W^{1, p} (Ω)$ , then $l i m_{s ↗ 1} (1 - s) \int_{Ω} \int_{Ω} {(| f (x) - f (y) |}^{p} {) / (| | x - y | |}^{d + s p}) d x d y = K \int_{Ω} {| \nabla f (x) |}^{p} d x$ , where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_{p}^{s, p} (Ω)$ . The purpose of this paper is to obtain analogous asymptotic formulae for some other equivalent seminorms, defined using coefficients of the expansion of f with respect to a wavelet or wavelet type basis. We cover both the case of the usual (isotropic) Besov and Sobolev spaces, and the Besov and Sobolev spaces with dominating mixed smoothness.

Publié le : 2016-01-01

Zbl 06574974

EUDML-ID : urn:eudml:doc:280508

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     author = {Anna Kamont},
     title = {Asymptotic behaviour of Besov norms via wavelet type basic expansions},
     journal = {Annales Polonici Mathematici},
     volume = {116},
     year = {2016},
     pages = {101-144},
     zbl = {06574974},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap3540-11-2015}
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Anna Kamont. Asymptotic behaviour of Besov norms via wavelet type basic expansions. Annales Polonici Mathematici, Tome 116 (2016) pp. 101-144. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap3540-11-2015/