Asymptotic behaviour of Besov norms via wavelet type basic expansions
Volume 116 / 2016
Abstract
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.
