A generalized Helmholtz-type decomposition of symmetric tensor fields and applications to ray transforms

Technical Summary: Generalized Helmholtz-type Decompositions of Symmetric Tensor Fields and Applications to Ray Transforms

Problem Statement
The paper addresses the problem of decomposing symmetric $m$-tensor fields in $\mathbb{R}^2$ into solenoidal and potential parts, a generalization of the classical Helmholtz decomposition for vector fields. While Sharafutdinov established such a decomposition for symmetric $m$-tensor fields in $\mathbb{R}^n$, and recent work by the authors in [16] introduced $k$-solenoidal and $k$-potential decompositions, these results were restricted by dimension and order constraints that rendered them inapplicable for the specific case of $n=2$ and $m=2$.

The primary objective is to extend these decomposition results to two dimensions under a mean-zero assumption and to utilize these decompositions to resolve injectivity questions for momentum ray transforms and elastic ray transforms acting on symmetric 2-tensor fields in $\mathbb{R}^2$.

Methodology
The authors employ a combination of Fourier analysis, distribution theory, and integral geometry techniques:

1. Fourier Domain Decomposition: The core methodology involves analyzing tensor fields in the Fourier domain. The authors utilize pointwise decompositions of the Fourier transforms of tensor fields. For symmetric 2-tensors, they adapt Lemma 2 from [16] to the 2D setting, expressing the Fourier transform $\hat{f}$ as a sum of a solenoidal part $\hat{g}$ and a potential part derived from a scalar field.
2. Mean-Zero Condition: A critical methodological step is the imposition of a mean-zero condition ($\int_{\mathbb{R}^2} f(x) dx = 0$). The authors prove that this condition is not merely sufficient but necessary for the existence of the decomposition in $\mathbb{R}^2$. They demonstrate that without this condition, the potential part would contain non-decaying logarithmic terms, violating the required decay properties of the decomposition.
3. Elastic Tensor Analysis: For elastic tensor fields (elements of $E_2(n)$), the authors define specific differential operators $H$ and $K$ (and their adjoints) to characterize solenoidal and potential parts. They extend the decomposition results from [12] (valid for $n \ge 3$) to $n=2$ by explicitly constructing the components of the decomposition using Fourier multipliers and verifying their integrability and decay properties.
4. Injectivity Proofs: The decompositions are applied to integral transforms. The authors show that if the ray transforms vanish, the tensor field must satisfy the mean-zero condition, allowing the application of the decomposition theorem. By analyzing the kernel of the transforms on the solenoidal part, they prove that the solenoidal component must vanish, leaving only the potential part.

Key Contributions and Results

• Generalized Decomposition in $\mathbb{R}^2$: The paper proves that any symmetric 2-tensor field $f \in \mathcal{S}(\mathbb{R}^2; S_2)$ satisfying the mean-zero condition admits a unique decomposition $f = g + d^2v$, where $g$ is a 2-solenoidal field ($\delta^2 g = 0$) and $v$ is a scalar potential. The authors further prove that the mean-zero condition is necessary; if such a decomposition exists with the required decay, the field must have zero mean.
• Elastic Tensor Decomposition: The authors extend the solenoidal-potential decomposition for elastic 2-tensor fields to 2 dimensions. They show that an elastic tensor field $f$ with zero mean can be uniquely decomposed as $f = Hv + Ku + g$, where $g$ is solenoidal (annihilated by $H^*$ and $K^*$) and $Hv + Ku$ represents the potential part.
• Injectivity of Momentum Ray Transforms: The paper establishes that for a symmetric 2-tensor field in $\mathbb{R}^2$, the vanishing of the first two integral moments ($I_0 f = 0$ and $I_1 f = 0$) implies that the tensor field is the Hessian of a scalar function ($f = d^2v$). This result removes the dimensional restrictions present in previous work [16].
• Injectivity of Elastic Ray Transforms: The authors prove that the elastic ray transform $X_2$ on symmetric 2-tensor fields in $\mathbb{R}^2$ is injective modulo the natural obstructions (the potential part $Hv + Ku$). This extends the result of [12] from $n \ge 3$ to the 2D case.
• Connection Between Transforms: A significant theoretical contribution is the proof that the kernel of the momentum ray transform pair $\{I_0, I_1\}$ is identical to the kernel of the elastic ray transform $X_1$ (acting on elastic 1-tensors, which are symmetric 2-tensors). This equivalence allows the characterization of the kernels of these distinct integral transforms using the same structural properties.

Significance and Claims
The paper claims to resolve a gap in the literature regarding the applicability of generalized Helmholtz-type decompositions in two dimensions. By establishing the decomposition under a mean-zero assumption and proving the necessity of this condition, the authors provide a rigorous foundation for studying inverse problems involving tensor fields in $\mathbb{R}^2$.

The significance lies in the application of these decompositions to prove injectivity results for momentum and elastic ray transforms without the dimensional restrictions ($n \ge 3$) that previously limited such analyses. The authors note that these injectivity results are crucial for understanding Calderón-type inverse problems for biharmonic and polyharmonic operators. Furthermore, the identification of the kernel equivalence between momentum and elastic ray transforms provides a unified framework for analyzing the solvability of these integral geometry problems in 2D.

The work does not propose new experimental applications but rather strengthens the theoretical underpinnings of integral geometry, specifically regarding the characterization of kernels and the solenoidal injectivity of ray transforms in low-dimensional settings.