Asymptotic behaviour of Besov norms via wavelet type basic expansions

Anna Kamont¹

¹ Institute of Mathematics Polish Academy of Sciences Branch in Gdańsk Wita Stwosza 57 80-952 Gdańsk, Poland

Annales Polonici Mathematici, Tome 116 (2016) no. 2, pp. 101-144.

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 $ \varOmega \subset\mathbb{R}^d$ is a smooth bounded domain, $1\le p \lt \infty$ and $f\in W^{1,p}(\varOmega)$, then $$ \lim_{s \nearrow 1}\, (1 -s) \int_{\varOmega} \int_{\varOmega} { |f(x) - f(y) |^p \over \|x-y\|^{d+sp}}\, dx \,dy = K \int_{\varOmega} | \nabla f (x) |^p\, dx, $$ 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 $B_p^{s,p}(\varOmega)$. 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.

Détail

DOI : 10.4064/ap3540-11-2015

Keywords: bourgain brezis mironescu menaldi eds optimal control partial differential equations ios press amsterdam proved following asymptotic formula varomega subset mathbb smooth bounded domain infty varomega lim nearrow s int varomega int varomega x y int varomega nabla where constant depending only double integral left hand side above formula equivalent seminorm besov space varomega purpose paper obtain analogous asymptotic formulae other equivalent seminorms defined using coefficients expansion respect wavelet wavelet type basis cover the usual isotropic besov sobolev spaces besov sobolev spaces dominating mixed smoothness