Lewy's example

Remarks

The key step concerns right-hand sides that depend only on $t$. If $\psi$ is a real-valued $C^1$ function of $t$ and $Lu = \psi'(t)$ has a $C^1$ solution near the origin, then $\psi$ is real-analytic near $t = 0$. Choosing $\psi$ smooth but not real-analytic gives an equation with no solution near the origin; a Baire category argument then produces one $f$ that works at every point. The operator $L$ is, up to a constant factor, the tangential Cauchy–Riemann operator of the hypersurface $\set{(z, w) \in \C^2 : \operatorname{Im} w = \abs{z}^2}$, written in the coordinates $z = x + iy$ and $t = \operatorname{Re} w$. This contrasts with the Cauchy–Kovalevskaya theorem, which gives local solutions when the coefficients and the data are all real-analytic. Hörmander later explained the phenomenon by a necessary condition for local solvability involving the commutator of $L$ and its complex conjugate $\bar L$.