Local solvability of first order linear operators with Lipschitz coefficients in two variables (Q1900179)

Refining previous results [\textit{J. Hounie}, Duke Math. J. 62, No. 2, 467-477 (1991; Zbl 0731.35025); \textit{H. Jacobowitz}, Proc. Am. Math. Soc. 116, No. 3, 787-795 (1992; Zbl 0776.35007)], the authors prove the local solvability in \(L^2\) of a linear differential operator \(L:= {\partial\over \partial t}- \alpha(x, t) {\partial\over \partial x}+ c(x, t)\) provided \(\alpha\) is Lipschitz continuous, \(c\) is measurable and bounded and the so-called condition (P), introduced by [\textit{L. Nirenberg} and \textit{F. Treves}, Commun. Pure Appl. Math. 16, 331-351 (1963; Zbl 0117.06104)], is satisfied. An example, defined by a coefficient \(\alpha\), that satisfies condition (P) but which is not locally solvable is also provided.