TY - JOUR
T1 - Sobolev spaces revisited
AU - Brezis, Haïm
AU - Seeger, Andreas
AU - van Schaftingen, Jean
AU - Yung, Po Lam
N1 - Publisher Copyright:
© 2022 Accademia Nazionale dei Lincei.
PY - 2022/8/31
Y1 - 2022/8/31
N2 - We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on Rn, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the Lp norm of functions in Lp(Rn).
AB - We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on Rn, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the Lp norm of functions in Lp(Rn).
KW - BBM formula
KW - Gagliardo seminorms
KW - Gagliardo–Nirenberg interpolation
KW - Sobolev spaces
KW - bounded variation
UR - http://www.scopus.com/inward/record.url?scp=85140097434&partnerID=8YFLogxK
U2 - 10.4171/RLM/976
DO - 10.4171/RLM/976
M3 - Article
SN - 1120-6330
VL - 33
SP - 413
EP - 437
JO - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
IS - 2
ER -