Fundamental solution of a linear differential operator

A fundamental solution of a linear differential operator $P(D)$ is a distribution $E$ satisfying $P(D)E={\delta }_0$.

Every nonzero constant-coefficient linear partial differential operator has a distributional fundamental solution.

Let $P({\xi }^{\prime },z)$ be a polynomial that is monic in the complex variable $z$. A Hörmander staircase partitions the real ${\xi }^{\prime }$-space into measurable sets ${\Delta }_j$ and assigns bounded heights $c_j$ so that $P({\xi }^{\prime },s+i{c}_j)$ stays uniformly away from zero for ${\xi }^{\prime }\in {\Delta }_j$ and $s\in \mathbb{R}$. Integrating $1/P$ over the resulting union of horizontal contours constructs a fundamental solution of a linear differential operator.

Suppose $P({\xi }^{\prime },z)$ has degree $m$ in $z$ and a nonzero constant leading coefficient $a$. Among the $m+1$ heights $h_{\ell }=3\ell$, at least one satisfiesat each real ${\xi }^{\prime }$. Factor into $m$ linear factors with multiplicity. Each root's imaginary part can be within distance less than one of at most one candidate height. The pigeonhole principle leaves a height whose distance from every root is at least one, proving the product bound. The height can be chosen measurably: the set where a candidate succeeds is the closed intersection of $|P({\xi }^{\prime },q+i{h}_{\ell })|\ge |a|$ over rational $q$, and selecting the first successful candidate gives a Borel set partition. This avoids any need for continuous global root labels.

A retarded fundamental solution is supported in a chosen forward region, such as $[0,\infty )$ for an ordinary differential operator.

For $P(z)=a_N{z}^N+\cdots +a_0$ with $N\ge 1$ and $a_N\ne 0$, the unique retarded fundamental solution is $E=Hu$, where $P(D)u=0$, $u^{(j)}(0)=0$ for $j<N-1$, and $u^{(N-1)}(0)=1/a_N$. The distributional jump formula for a Heaviside product gives $P(D)E={\delta }_0$. Its explicit expression isThe residue theorem gives the initial derivatives from the coefficients at infinity. Equivalently, the inverse Fourier transform is integrated below all its poles, producing support of a distribution in $[0,\infty )$. Uniqueness follows because the difference of two retarded solutions is an analytic homogeneous solution vanishing on the negative half-line.