Asymptotic behaviour of Besov norms via wavelet type basic expansions

Anna Kamont. "Asymptotic behaviour of Besov norms via wavelet type basic expansions." Annales Polonici Mathematici 116.2 (2016): 101-144. <http://eudml.org/doc/280508>.

@article{AnnaKamont2016,
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 $Ω ⊂ ℝ^d$ 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.},
author = {Anna Kamont},
journal = {Annales Polonici Mathematici},
keywords = {Besov spaces; Sobolev spaces; embeddings; wavelet and wavelet type bases},
language = {eng},
number = {2},
pages = {101-144},
title = {Asymptotic behaviour of Besov norms via wavelet type basic expansions},
url = {http://eudml.org/doc/280508},
volume = {116},
year = {2016},
}

TY - JOUR
AU - Anna Kamont
TI - Asymptotic behaviour of Besov norms via wavelet type basic expansions
JO - Annales Polonici Mathematici
PY - 2016
VL - 116
IS - 2
SP - 101
EP - 144
AB - 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.
LA - eng
KW - Besov spaces; Sobolev spaces; embeddings; wavelet and wavelet type bases
UR - http://eudml.org/doc/280508
ER -