Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity

Abstract

We study the stability of Triebel-Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel-Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel-Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.

Figures (4)

• Figure 2.1: On the first graphic, $f\in W^{5,p}\cap W^{1,\infty}$, so $\nabla^2 f\in L^{4p}$, $\nabla^3 f\in L^{2p}$ and $\nabla^4 f\in L^{\frac{4p}{3}}$. On the second we depict the case $f\in F^s_{p,q}\cap W^{1,\infty}$, with $4<s=4+\sigma<5$; in that case $\nabla f\in F^{\sigma/M}_{\frac{(s-1)p}{\sigma/M},r}$, $\nabla^2 f\in F^{\sigma/M}_{\frac{(s-1)p}{1+\sigma/M},r}$, $\nabla^3 f\in F^{\sigma/M}_{\frac{(s-1)p}{2+\sigma/M},r}$, and $\nabla^4 f\in F^{\sigma/M}_{\frac{(s-1)p}{3+\sigma/M},q}$ ($q$ can be replaced by $r$ if $M>1$). The circular dots describe the case $M=1$, the squares describe the case $M=2$. See Lemma .
• Figure 2.2: Composition rule for Sobolev and Triebel-Lizorkin scales of a bounded continuous function $g$ and a bi-Lipschitz function $f$ in Lemmata and .
• Figure 3.1: Interior corkscrew domains may have cusps, in opposition to uniform domains.
• Figure 3.2: In uniform domains we can join cubes by admissible chains, which play the role of cigar paths. Shadows are the cubes under the influence of a given cube, and they play the role of Carleson boxes.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Lemma 2.1: Chain rule
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Proposition 2.6: see RunstSickel