Fundamental solution

Remarks

Example: on $\R$, the Heaviside function $H$, with $H(x) = 1$ for $x > 0$ and $H(x) = 0$ for $x < 0$, is a fundamental solution of $P = d/dx$. Indeed, for a test function $\varphi$ one has $H(-\varphi') = -\int_0^\infty \varphi'(x) \, dx = \varphi(0)$. Example: on $\R^3$, $E(x) = -1/(4 \pi \abs{x})$ is a fundamental solution of the Laplacian, and on $\R^2$, $E(x) = \frac{1}{2\pi} \log \abs{x}$ is one. Non-example: on $\R$, the constant function $1$ is not a fundamental solution of $d/dx$, since its derivative is $0$ and not $\delta$. A fundamental solution is not unique: adding any distribution $F$ with $P F = 0$ gives another one. If $P$ has constant coefficients and $f$ is a test function, then $u = E * f$ is smooth and solves $P u = f$. Every nonzero operator with constant coefficients has a fundamental solution, by the Malgrange–Ehrenpreis theorem.