Distributions and Fundamental Solutions - Main Theorem

Existence and properties of fundamental solutions are governed by deep results connecting analysis, algebra, and geometry of differential operators.

TheoremExistence of Fundamental Solutions

Every linear differential operator $L$ with constant coefficients on $\mathbb{R}^n$ has a fundamental solution $E \in \mathcal{D}'(\mathbb{R}^n)$ satisfying: $LE = \delta$

Moreover, $E$ can be chosen to be tempered: $E \in \mathcal{S}'(\mathbb{R}^n)$ .

The proof uses Fourier transform: $\hat{E}(\xi) = 1/P(\xi)$ where $P$ is the symbol of $L$ . The key difficulty is showing $1/P$ defines a tempered distribution despite potential zeros of $P$ .

TheoremMalgrange-Ehrenpreis Theorem

Every non-zero linear differential operator with constant coefficients has a fundamental solution. This extends the previous theorem to complex coefficients and establishes fundamental solutions constructively.

Remark

For elliptic operators (like Laplacian), fundamental solutions have singularities only at the origin and decay at infinity. For hyperbolic operators (like wave operator), fundamental solutions have support on characteristic surfaces, reflecting finite propagation speed.

TheoremRegularity of Convolutions

If $T \in \mathcal{D}'(\Omega)$ and $\phi \in C^\infty$ , then $T * \phi \in C^\infty$ .

More generally, if $T \in \mathcal{E}'$ (compact support) and $u$ is a distribution, then $T * u$ is $C^\infty$ and: $\partial^\alpha(T * u) = (\partial^\alpha T) * u = T * (\partial^\alpha u)$

This explains why solutions to $Lu = f$ obtained via $u = E * f$ are often smoother than $f$ itself—the fundamental solution $E$ acts as a smoothing kernel.

TheoremHypoellipticity

An operator $L$ is hypoelliptic if $Lu = f$ with $f \in C^\infty$ implies $u \in C^\infty$ locally.

Examples:

Laplacian $\Delta$ is elliptic (hence hypoelliptic)
Heat operator $\partial_t - \Delta$ is hypoelliptic (but not elliptic)
Wave operator $\partial_t^2 - \Delta$ is NOT hypoelliptic

Hypoellipticity is characterized by properties of the fundamental solution and symbol of $L$ .

These theorems establish when PDEs can be solved via convolution with fundamental solutions and when solutions inherit smoothness from the data.