Hardy spaces for the Lamé equation

TL;DR

This work extends Hardy space theory to the elastic Lamé equation by introducing the vector-valued Hardy spaces $\bm{h}_e^p(\mathbb{B})$ and linking them to boundary data via the elastic Poisson transform $\bm{P}_e$. It establishes Fatou-type boundary behavior and shows $\bm{h}_e^p(\mathbb{B})$ is isomorphic to $\bm{L}^p(\mathbb{S})$ for $1<p\le\infty$ (and to $\bm{M}(\mathbb{S})$ for $p=1$). The main contribution is a precise decomposition of $\bm{h}_e^p(\mathbb{B})$ into three subspaces $\bm{h}_+^p$, $\bm{h}_-^p$, and $\bm{h}_0^p$, with explicit boundary-value descriptions in terms of gradients and cross products of Poisson extensions: $\bm{h}_-^p$ corresponds to Riesz fields ($\text{div}=0$, $\nabla\times=0$), $\bm{h}_0^p$ to $\bm{x}\times\nabla$, and $\bm{h}_+^p$ to more general elastic-harmonic components. These descriptions rely on the vector spherical-harmonic decomposition of $\bm{L}^p(\mathbb{S})$ and its $L^p$ projections, providing a detailed, Lamé-parameter–independent structure for Riesz-field components and a robust framework for boundary-value analysis in elasticity.

Abstract

We study, for $1 \leq p \leq \infty$, the Hardy space $\bm{h}_e^p(\B)$, the elastic analogue of the classical Hardy spaces of harmonic functions in the unit ball of $\mathbb{R}^3$. The space consists of vector-field solutions of the Lamé system satisfying the standard integrability condition on concentric spheres centered at the origin. Using the elastic Poisson kernel, we establish a Fatou-type theorem and show that $\bm{h}_e^p(\B)$ is isomorphic to the $\mathbb{R}^3$-valued Lebesgue space $L^p$ on the unit sphere for $1 < p \leq \infty$, while $\bm{h}_e^1(\B)$ corresponds to the space of $\mathbb{R}^3$-valued Borel measures on the unit sphere. For $1 < p < \infty$, we prove that $\bm{h}_e^p(\B)$ decomposes as the direct sum of three subspaces. The main contribution of this paper is to describe each of these subspaces along with the corresponding spaces of boundary values. In particular, two of these spaces consist of solutions of the Lamé equation for all eligible choices of the Lamé constants: one of them is the space of Riesz fields (solutions of the generalized Cauchy--Riemann equations) in $\bm{h}_e^p(\B)$; the second is the space of fields given by the cross product of $x$ with such Riesz fields. The results rely on the classical decomposition of $L^2$ vector fields on the sphere into the direct sum of three spaces of vector spherical harmonics, which we extend to $L^p$.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Proposition 5
• proof
• Theorem 6
• proof