Asymptotic behaviour of Besov norms via wavelet type basic expansions

• Autorzy: Anna Kamont
• Języki publikacji: EN

Abstrakty

EN

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 $Ω ⊂ ℝ^d$ is a smooth bounded domain, 1 ≤ p < ∞ and $f ∈ W^{1,p}(Ω)$, then
$lim_{s↗1} (1-s)∫_{Ω} ∫_{Ω} (|f(x)-f(y)|^p)/(||x-y||^{d+sp}) dxdy = K∫_{Ω} |∇f(x)|^p dx$,
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.

Kategorie tematyczne

• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2016
• Tom: 116
• Numer: 2
• Strony: 101-144

Daty

wydano

2016

Twórcy

autor

Anna Kamont

• Institute of Mathematics, Polish Academy of Sciences, Branch in Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Identyfikatory

DOI

10.4064/ap3540-11-2015