Pseudodifferential calculus for oscillating symbols (Q1917003)

The author considers the operator in \(\mathbb{R}^2\), \[ Pu= u_{xx}- u_{yy}+ a_1(x, y)u_x+ a_2(x, y)u_y+ a_0(x, y)u \] regarded as a map \(P: {\mathcal S}'(\mathbb{R}^2)\to {\mathcal S}'(\mathbb{R}^2)\). Under the assumption \[ | a_1(x,y)\pm a_2(x,y)|\geq c(1+ | x|+| y|)^r,\quad r>-1, \] results of global solvability and regularity are given. In particular, \(u\in C^\infty(\mathbb{R}^2)\) if \(Pu\in C^\infty(\mathbb{R}^2)\), and \(u\in L^p_{\text{loc}}(\mathbb{R}^2)\) if \(Pu\in L^p_{\text{loc}}(\mathbb{R}^2)\). The proofs are based on a suitable calculus for superposition of pseudodifferential operators. The underlying idea is to consider the solution in \({\mathcal S}'(\mathbb{R}^2)\) of the equation \(\partial_j u+ au= 0\), with \(a= a_1\pm a_2\). As the author observes, the case \(a_1\equiv a_2\equiv a\equiv 0\) of the wave operator is clearly outside the hypothesis.