Solvability of Lewy equation in the space of the tempered ultrahyperfunctions (Q1941967)

It is shown that the Lewy equation has a solution in spaces of tempered ultrahyperfunctions which are smaller than the space of the Fourier ultrahyperfunctions \({\mathcal G}'(\mathbb{R})\) (see [\textit{S.-Y. Chung, D. Kim} and \textit{S. K. Kim}, J. Fac. Sci., Univ. Tokyo, Sect. I A, 40, No. 1, 63--71 (1993; Zbl 0796.46027)]). In particular, it is demonstrated that the Lewy equation for a source function \(f \in C_0^1 (\mathbb{R}_{x,y}^2; L^1(\mathbb{R}_t))\) has a solution in the space \(C^1 (\mathbb{R}_{x,y}^2; (\tilde{S}^1)^{\prime})(\mathbb{R}_t),\) where the space \((\tilde{S}^1)^{\prime}(\mathbb{R})\) is the space of tempered ultrahyperfunctions.