On the \(L^ 2\)-boundedness of pseudo-differential operators (Q908443)

It is known that pseudo-differential operators with symbols of class \(S_{\rho,1}^{-n(1-\rho)/2}\) (0\(\leq \rho \leq 1)\) are not always bounded in \(L^ 2(R^ n)\). In particular if \(0<\rho <1\), \textit{L. Rodino} [Proc. Am. Math. Soc. 58, 211-215 (1976; Zbl 0309.47039)] gives a counterexample of pseudo-differential operators, which are not bounded in \(L^ 2({\mathbb{R}}^ n)\) but are defined by symbols in \(S_{\rho,1}^{- n(1-\rho)/2}.\) In the present paper an example of such operators, which is a little different from the one by Rodino (loc. cit.), is given. The example is similar to the one which is obtained by \textit{Chin-Hung-Ching} [J. Differ. Equations 11, 436-447 (1972; Zbl 0248.35106)] for the case \(\rho =1\). Moreover the paper shows an \(L^ 2\)-boundedness theorem of pseudo- differential operators with symbols which has a little larger decreasing order than n(1-\(\rho)\)/2, without using the smoothness of symbols in the space variables x.