[Paper Review] On homogeneous Sobolev and Besov spaces on the whole and the half space

Anatole Gaudin|arXiv (Cornell University)|Nov 14, 2022

Advanced Harmonic Analysis Research29 references4 citations

TL;DR

This paper presents a novel, distribution-based construction of homogeneous Sobolev and Besov spaces on $\mathbb{R}^n$ and the upper half-space $\mathbb{R}^n_+$, avoiding quotient spaces by polynomials to ensure well-defined product laws and trace operators. Using real and complex interpolation on the space $\mathcal{S}'_h(\mathbb{R}^n)$, it establishes trace theorems and applies the framework to the Dirichlet and Neumann Laplacians with sharp estimates.

ABSTRACT

In this paper, we propose an elementary construction of homogeneous Sobolev spaces of fractional order on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. This construction completes the construction of homogeneous Besov spaces on $\mathcal{S}'_h(\mathbb{R}^n)$ started by Bahouri, Chemin and Danchin on $\mathbb{R}^n$. We will also extend the treatment done by Danchin and Mucha on $\mathbb{R}^n_+$, and the construction of homogeneous Sobolev spaces of integer orders started by Danchin, Hieber, Mucha and Tolksdorf on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Properties of real and complex interpolation, duality, and density are discussed. Trace results are also reviewed. Our approach relies mostly on interpolation theory and yields simpler proofs of some already known results in the case of Besov spaces. The lack of completeness on the whole scale will lead to consideration of intersection spaces with decoupled estimates to circumvent this issue. As standard and simple applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous functions spaces.

Motivation & Objective

• To provide a consistent, distribution-based realization of homogeneous Sobolev and Besov spaces on $\mathbb{R}^n$ and $\mathbb{R}^n_+$, avoiding the ambiguity of quotient spaces modulo polynomials.
• To resolve the issue of non-uniqueness in product laws and trace operators that arises when working with equivalence classes of tempered distributions modulo polynomials.
• To establish trace theorems for homogeneous spaces on the half-space using interpolation theory and decay conditions on low-frequency projections.
• To apply the framework to the Dirichlet and Neumann Laplacians on $\mathbb{R}^n_+$, proving boundedness and regularity results in homogeneous function spaces.
• To extend previous constructions of homogeneous spaces (e.g., by Bahouri-Chemin-Danchin, Danchin-Mucha) to fractional and non-integer orders via interpolation on $\mathcal{S}'_h$.

Proposed method

• Define homogeneous function spaces as subspaces of $\mathcal{S}'_h(\mathbb{R}^n)$, the space of tempered distributions with vanishing low-frequency norms as $\lambda \to \infty$, ensuring decay of low-frequency projections.
• Use real and complex interpolation theory to construct homogeneous Besov and Sobolev spaces, leveraging the structure of $\mathcal{S}'_h$ to preserve distributional character and avoid completeness issues.
• Establish trace operators via extension operators $T$ mapping data on $\mathbb{R}^{n-1}$ to functions in $\mathbb{R}^n_+$, using the Poisson kernel and Fourier multiplier techniques.
• Apply dilation arguments to derive sharp trace estimates: for $f \in \dot{B}^{s - 1/p}_{p,p}(\mathbb{R}^{n-1})$, $\|Tf\|_{\dot{H}^{s,p}(\mathbb{R}^n_+)} \lesssim \|f\|_{\dot{B}^{s - 1/p}_{p,p}(\mathbb{R}^{n-1})}$.
• Use real interpolation with $L^p$-based norms and weighted estimates to extend results to non-endpoint $q$-norms in Besov spaces.
• Prove boundedness of the Poisson semigroup and use it to derive estimates for the extension operator $T$ via frequency localization and scaling arguments.

Experimental results

Research questions

• RQ1 How can homogeneous Sobolev and Besov spaces on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ be consistently defined as subspaces of tempered distributions without quotienting by polynomials?
• RQ2 What role does the space $\mathcal{S}'_h(\mathbb{R}^n)$ play in enabling well-defined product laws and trace operators in homogeneous function spaces?
• RQ3 Can interpolation theory be used to extend constructions of homogeneous spaces from integer to fractional orders on the half-space?
• RQ4 What are the sharp trace estimates for functions in homogeneous Besov and Sobolev spaces on $\mathbb{R}^n_+$, and how do they depend on the regularity and integrability parameters?
• RQ5 How can the framework be applied to the Dirichlet and Neumann Laplacians to establish boundedness and regularity in homogeneous function spaces?

Key findings

• The space $\mathcal{S}'_h(\mathbb{R}^n)$, defined by uniform vanishing of low-frequency projections, provides a consistent domain for homogeneous function spaces, avoiding the ambiguity of polynomial equivalence classes.
• For $s \geq 0$, $p \in (1, \infty)$, and $f \in \dot{B}^{s - 1/p}_{p,p}(\mathbb{R}^{n-1})$, the extension operator $T$ satisfies $\|Tf\|_{\dot{H}^{s,p}(\mathbb{R}^n_+)} \lesssim \|f\|_{\dot{B}^{s - 1/p}_{p,p}(\mathbb{R}^{n-1})}$, establishing a sharp trace estimate.
• For $s > 0$, $p \in (1, \infty)$, $q \in [1, \infty)$, and $f \in \dot{B}^{s - 1/p}_{p,q}(\mathbb{R}^{n-1})$, the estimate $\|Tf\|_{\dot{B}^{s}_{p,q}(\mathbb{R}^n_+)} \lesssim \|f\|_{\dot{B}^{s - 1/p}_{p,q}(\mathbb{R}^{n-1})}$ holds via real interpolation and dilation arguments.
• The construction allows for well-defined (para-)product laws and trace operators, resolving issues arising from non-uniqueness in quotient spaces of distributions modulo polynomials.
• The framework successfully extends previous results on homogeneous spaces to fractional orders and provides a unified treatment of Dirichlet and Neumann Laplacians on the half-space.
• The use of intersection spaces and decoupled estimates circumvents the lack of completeness in homogeneous spaces, enabling consistent analysis of non-linear PDEs.