TY - JOUR
T1 - Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces
AU - Brazke, Denis
AU - Schikorra, Armin
AU - Yung, Po Lam
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022/12/24
Y1 - 2022/12/24
N2 - We study a convergence result of Bourgain–Brezis–Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces Ws,p=Fp,ps, and H1,p=Fp,21. When s→ 1 , the Fp,ps norm becomes the Fp,p1 norm but BBM showed that the Ws,p norm becomes the H1,p=Fp,21 norm. Naively, for p≠ 2 this seems like a contradiction, but we resolve this by providing embeddings of Ws,p into Fp,qs for q∈ { p, 2 } with sharp constants with respect to s∈ (0 , 1). As a consequence we obtain an RN-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.
AB - We study a convergence result of Bourgain–Brezis–Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces Ws,p=Fp,ps, and H1,p=Fp,21. When s→ 1 , the Fp,ps norm becomes the Fp,p1 norm but BBM showed that the Ws,p norm becomes the H1,p=Fp,21 norm. Naively, for p≠ 2 this seems like a contradiction, but we resolve this by providing embeddings of Ws,p into Fp,qs for q∈ { p, 2 } with sharp constants with respect to s∈ (0 , 1). As a consequence we obtain an RN-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.
UR - http://www.scopus.com/inward/record.url?scp=85144846287&partnerID=8YFLogxK
U2 - 10.1007/s00526-022-02382-6
DO - 10.1007/s00526-022-02382-6
M3 - Article
SN - 0944-2669
VL - 62
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 2
M1 - 41
ER -