On the \(L^ p\)-boundedness of several classes of pseudo-differential operators
Two results are obtained. One of them is the generalization of the sharp \(L^ 2\)-boundedness theorem of \(\psi\) do's given by \textit{R. R. Coifman} and \textit{Y. Meyer} [Astérisque 57, 185 p. (1978; Zbl 0483.35082)] to \(\psi\) do's with amplitudes a(x,\(\xi\),y). Another result is the sharpening of the \(L^ p\)-boundedness. In particular, the authors obtain the following sharp result on the \(L^ p\)-boundedness of \(\psi\) do's for \(2\leq p<\infty:\) Let \(0\leq \delta \leq \rho \leq 1\), \(\delta <1\), \(\rho >0\). If a(x,\(\xi)\) satisfies \[ | D_ x^{\beta} D^{\alpha}_{\xi}a(x,\xi)| \leq C<\xi >^{-m+\delta | \beta | -\rho /\alpha |} \] for all \(| \alpha |\), \(| \beta | \leq [n/2]+1\) and \(0\leq ((1/2)-1/p)n(1-\rho)\leq m\), then the associated \(\psi\) do is \(L^ p\)-bounded. The condition is sharp not only with respect to order m but also with respect to times \([n/2]+1\) of differentiations in \(\xi\). Moreover, combining this result with the \(L^ p\)-boundedness theorem given by \textit{M. Nagase} [Sci. Rep., Osaka 32, No.1, 9-19 (1983; Zbl 0543.35104)], the still better \(L^ p\)-boundedness result can be obtained.
scientific article; zbMATH DE number 3957831