Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces

Denis Brazke; Armin Schikorra; Po Lam Yung

doi:10.1007/s00526-022-02382-6

Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces

Denis Brazke, Armin Schikorra^*, Po Lam Yung

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

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 W^s^,^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 W^s^,^p into Fp,qs for q∈ { p, 2 } with sharp constants with respect to s∈ (0 , 1). As a consequence we obtain an R^N-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.

Original language	English
Article number	41
Journal	Calculus of Variations and Partial Differential Equations
Volume	62
Issue number	2
DOIs	https://doi.org/10.1007/s00526-022-02382-6
Publication status	Published - 24 Dec 2022

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title = "Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces",

abstract = "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.",

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T1 - Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces

AU - Brazke, Denis

AU - Schikorra, Armin

AU - Yung, Po Lam

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 - https://www.scopus.com/pages/publications/85144846287

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 -

Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces

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