[Paper Review] Anisotropic Function Spaces on Singular Manifolds

Herbert Amann|arXiv (Cornell University)|Apr 3, 2012

Advanced Harmonic Analysis Research42 references19 citations

TL;DR

This paper introduces anisotropic weighted Sobolev, Bessel potential, and Besov spaces on time-dependent tensor fields over singular manifolds, establishing their reflexivity, duality, and interpolation properties. The key contribution is a comprehensive scale of function spaces with variable regularity and weight, unified via real and complex interpolation, and shown to be equivalent under appropriate norms.

ABSTRACT

A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weighted function spaces. Besides of the derivation of Sobolev-type embedding results, sharp trace theorems, point-wise multiplier properties, and interpolation characterizations particular emphasize is put on spaces distinguished by boundary conditions. This work is the fundament for the analysis of time-dependent partial differential equations on singular manifolds.

Motivation & Objective

• To develop a systematic theory of anisotropic weighted function spaces on manifolds with singularities, particularly for time-dependent tensor fields.
• To define and characterize anisotropic Sobolev, Bessel potential, and Besov spaces using weighted norms and variable regularity parameters.
• To establish the reflexivity, duality, and interpolation properties of these spaces, ensuring their suitability for PDE analysis on singular domains.
• To unify different scales of function spaces (Sobolev, Bessel potential, Besov) via real and complex interpolation functors.
• To provide embedding theorems and density results for these spaces, especially in the context of boundary and initial value problems.

Proposed method

• Define anisotropic weighted Sobolev spaces $ W_{p}^{kr/ o{r}, o{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ u}}}}}} $ via norms combining time and spatial regularity with weights $ ho^ u $, using $ o{r} = (1, 1/r) $ and $ o{oldsymbol{oldsymbol{ u}}} = ( u, u) $.
• Establish reflexivity of $ W_{p}^{kr/ o{r}, o{oldsymbol{oldsymbol{ u}}}} $ by embedding into a reflexive Banach space via isomorphism.
• Introduce fractional-order spaces via real interpolation: $ F^{s/ o{r}, o{oldsymbol{ u}}} = (W_{p}^{kr/ o{r}, o{oldsymbol{ u}}}, W_{p}^{(k+1)r/ o{r}, o{oldsymbol{ u}}})_{(s-kr)/r} $ for $ kr < s < (k+1)r $.
• Define negative-order spaces via duality: $ F^{-s/ o{r}, o{oldsymbol{ u}}} = ( ing{F}^{p' ext{ }s/ o{r},- o{oldsymbol{ u}}}(J,V') )' $, using the duality pairing $ iglra{u,v}igr angle_{M\times J} $.
• Use reiteration theorems and density of $ ing{D}(J, ing{D}) $ in $ L_p(J, L_p^ u) $ to prove embedding chains and interpolation identities.
• Apply complex and real interpolation functors to show that $ H_p^{s/ o{r}, o{oldsymbol{ u}}} \doteq W_p^{kr/ o{r}, o{oldsymbol{ u}}} $ for integer $ k $, and $ B_2^{s/ o{r}, o{oldsymbol{ u}}} \doteq H_2^{s/ o{r}, o{oldsymbol{ u}}} $, establishing scale equivalence.

Experimental results

Research questions

• RQ1 How can anisotropic weighted Sobolev spaces be defined on singular manifolds with time-dependent tensor fields?
• RQ2 What are the duality and reflexivity properties of these anisotropic function spaces?
• RQ3 How do Bessel potential and Besov spaces relate to Sobolev spaces in this anisotropic, weighted setting?
• RQ4 What are the embedding and interpolation properties of these function spaces across different regularity and weight parameters?
• RQ5 Under what conditions does the space $ ing{F}_p^{s/ o{r}, o{oldsymbol{ u}}} $ coincide with $ F_p^{s/ o{r}, o{oldsymbol{ u}}} $, and what are the implications for trace and trace-free spaces?

Key findings

• The anisotropic weighted Sobolev space $ W_p^{kr/ o{r}, o{oldsymbol{ u}}} $ is a reflexive Banach space, as it is isomorphic to a closed subspace of a reflexive Banach space.
• An equivalent norm is given by $ \|u\|_{kr/\to{r},p;\to{\boldsymbol{\nu}}}^{\sim} = \left( \|u\|_{L_p(J,W_p^{kr,\lambda})}^p + \sum_{j=0}^k \|\partial^j u\|_{L_p(J,W_p^{(k-j)r,\lambda + j\mu})}^p \right)^{1/p} $, ensuring norm equivalence.
• The space $ \ring{D}(J,\ring{D}) $ is densely embedded in $ W_p^{kr/\to{r},\to{oldsymbol{ u}}} $, confirming the space's suitability for PDE theory.
• The Bessel potential space $ H_p^{s/\to{r},\to{\boldsymbol{ν}}}} $ coincides with the Sobolev space $ W_p^{kr/\to{r},\to{oldsymbol{ν}}}} $ for integer $ k $, and $ B_2^{s/\to{r},\to{oldsymbol{ν}}}} \doteq H_2^{s/\to{r},\to{oldsymbol{ν}}}} $, showing scale consistency.
• Interpolation identities hold: $ (B_p^{s_0/\to{r},\to{oldsymbol{ν}}}, B_p^{s_1/\to{r},\to{oldsymbol{ν}}}})_{\theta,p} \doteq B_p^{s_\theta/\to{r},\to{oldsymbol{ν}}}} $ and $ [F_p^{s_0/\to{r},\to{oldsymbol{ν}}}, F_p^{s_1/\to{r},\to{oldsymbol{ν}}}}]_{\theta} \doteq F_p^{s_\theta/\to{r},\to{oldsymbol{ν}}}} $, confirming scale stability.
• For $ s < 1/p $ when $ \partial M \neq \emptyset $, or $ s < r(1+1/p) $ when $ \partial M = \emptyset $ and $ J = \mathbb{R}^+ $, the space $ \ring{F}_p^{s/\to{r},\to{oldsymbol{ν}}}} $ coincides with $ F_p^{s/\to{r},\to{oldsymbol{ν}}}} $, ensuring density and regularity.