[Paper Review] Riesz transforms outside a convex obstacle

Rowan Killip, Monica Vişan+1 more|arXiv (Cornell University)|May 25, 2012

Advanced Mathematical Physics Problems35 references4 citations

TL;DR

This paper establishes sharp $L^p$ boundedness of Riesz transforms and equivalence of homogeneous Sobolev norms for the Dirichlet Laplacian in exterior domains of smooth convex obstacles in dimensions $d \geq 3$. Using heat kernel estimates, spectral theory, and counterexamples, it proves that Riesz transforms are bounded on $L^p$ for $1 < p < d$ and unbounded for $p > d$, with the endpoint $p = d$ remaining unresolved. The results enable transfer of classical Euclidean harmonic analysis tools to the exterior domain setting, with applications to energy-critical nonlinear Schrödinger equations.

ABSTRACT

The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood-Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent $p$ and regularity $s$. Counterexamples are included to show that these results are indeed sharp. In particular, we precisely settle the question of boundedness of Riesz transforms on $L^p$, including the endpoint. The utility of such results in the study of nonlinear PDE is that they allow us to deduce important results, such as the fractional product and chain rules for the Dirichlet Laplacian, directly from the classical Euclidean setting. As an application, we discuss the local well-posedness and stability problems for energy-critical NLS. All the results of this paper play an essential role in the authors' proof of large-data global well-posedness and scattering for the energy-critical NLS in three dimensional exterior domains; see arXiv:1208:4904.

Motivation & Objective

• To develop harmonic analysis tools—Mikhlin multiplier theorem, Littlewood–Paley theory, Hardy inequalities—for the Dirichlet Laplacian in exterior domains of smooth convex obstacles.
• To establish sharp $L^p$ boundedness of Riesz transforms, including the endpoint, for the Dirichlet Laplacian on exterior domains.
• To prove equivalence of homogeneous Sobolev norms defined via the Dirichlet Laplacian and the whole-space Laplacian for sharp ranges of $p$ and $s$.
• To provide counterexamples showing the sharpness of the $L^p$ boundedness results for Riesz transforms.
• To enable transfer of classical Euclidean harmonic analysis results to the exterior domain setting, facilitating analysis of nonlinear PDEs such as energy-critical NLS.

Proposed method

• Utilizes spectral theory and the functional calculus of the Dirichlet Laplacian $-\Delta_\Omega$ to define homogeneous Sobolev norms via $\|(-\Delta_\Omega)^{s/2}f\|_{L^p}$.
• Applies heat kernel estimates and the factorization $\sqrt{t}\nabla e^{t\Delta_\Omega} = [\nabla(-\Delta_{\mathbb{R}^d})^{-1/2}][(-\Delta_{\mathbb{R}^d})^{1/2}(-\Delta_\Omega)^{-1/2}][(-t\Delta_\Omega)^{1/2}e^{t\Delta_\Omega}]$ to relate exterior and Euclidean heat flows.
• Employs a contradiction argument with explicit test functions $u(t,x) = (1 - |x|^{2-d})g(t,x)$, where $g$ is a Gaussian heat solution, to show failure of $L^p$ bounds for $p > d$.
• Uses cancellation properties of a compactly supported $\psi \in C_c^\infty(\mathbb{R}^d)$ with $\int \psi = 0$, $\int x\psi = 0$ to construct a sequence $f_R$ such that $\|(-\Delta_{\mathbb{R}^d})^{s/2}f_R\|_{L^p} \gtrsim 1$ while $\|(-\Delta_\Omega)^{s/2}f_R\|_{L^p} \to 0$, proving sharpness of Sobolev norm equivalence.
• Applies the Hardy inequality and trace theorems to ensure boundary vanishing and regularity of solutions to the heat equation in $H^1_D(\Omega)$.
• Leverages the spectral theorem to define $(-\Delta_\Omega)^{s/2}$ as a self-adjoint operator on $L^2(\Omega)$, with domain $H^1_D(\Omega)$.

Experimental results

Research questions

• RQ1 For which $p \in (1, \infty)$ are the Riesz transforms bounded on $L^p(\Omega)$ for the Dirichlet Laplacian outside a smooth convex obstacle in $d \geq 3$?
• RQ2 Are the homogeneous Sobolev norms $\|(-\Delta_\Omega)^{s/2}f\|_{L^p}$ and $\|(-\Delta_{\mathbb{R}^d})^{s/2}f\|_{L^p}$ equivalent for $f \in C_c^\infty(\Omega)$, and for which ranges of $p$ and $s$?
• RQ3 What is the sharp range of $p$ for which the uniform estimate $\|\sqrt{t}\nabla e^{t\Delta_\Omega}f\|_{L^p} \lesssim \|f\|_{L^p}$ holds uniformly in $t > 0$?
• RQ4 Can counterexamples be constructed to show that the $L^p$ boundedness of Riesz transforms fails for $p > d$?
• RQ5 To what extent do classical harmonic analysis tools—like the Mikhlin multiplier theorem and Littlewood–Paley theory—extend to the Dirichlet Laplacian in exterior domains?

Key findings

• The Riesz transforms associated with the Dirichlet Laplacian on the exterior of a smooth convex obstacle in $d \geq 3$ are bounded on $L^p$ for $1 < p < d$, and unbounded for $p > d$, with the endpoint $p = d$ left unresolved.
• The homogeneous Sobolev norms $\|(-\Delta_\Omega)^{s/2}f\|_{L^p}$ and $\|(-\Delta_{\mathbb{R}^d})^{s/2}f\|_{L^p}$ are equivalent for $f \in C_c^\infty(\Omega)$ precisely when $1 < p < d$ and $s \in (0, d/p)$, and this range is sharp.
• The uniform estimate $\|\sqrt{t}\nabla e^{t\Delta_\Omega}f\|_{L^p} \lesssim \|f\|_{L^p}$ holds for all $t > 0$ if and only if $p < d$, and fails for $p > d$, as shown by a contradiction argument using a perturbed Gaussian heat solution.
• Counterexamples demonstrate that the $L^p$ boundedness of Riesz transforms fails for $p > d$, and the sharpness of the Sobolev norm equivalence is confirmed via a sequence $f_R$ with $\|(-\Delta_{\mathbb{R}^d})^{s/2}f_R\|_{L^p} \gtrsim 1$ but $\|(-\Delta_\Omega)^{s/2}f_R\|_{L^p} \to 0$ as $R \to \infty$.
• The results allow direct transfer of classical Euclidean harmonic analysis tools—such as fractional product and chain rules—for the Dirichlet Laplacian, enabling new proofs of local well-posedness and stability for energy-critical nonlinear Schrödinger equations in exterior domains.
• The paper establishes that the definitions of $\dot{H}_D^{s,p}(\Omega)$ via spectral theory and via restriction of $\dot{H}^{s,p}(\mathbb{R}^d)$ are equivalent for $s < 3/2$, and this extends to $s \leq 1$ via the $s=1$ case.