$L^ p$-boundedness of pseudo-differential operators of class $S_ {0,0}$

Abstract:

We study the ${L^p}$-boundedness of pseudo-differential operators with the support of their symbols being contained in $E \times {{\mathbf {R}}^n}$, where $E$ is a compact subset of ${{\mathbf {R}}^n}$, and their symbols have derivatives with respect to $x$ only up to order $k$, in the Hölder continuous sense, where $k > n/2$ (the case $1 < p \leqslant 2$) and $k > n/p$ (the case $2 < p < \infty$). We also give a new proof of the ${L^p}$-boundedness, $1 < p < \infty$, of pseudo-differential operators of class $S_{0,0}^m$, where $m = m(p) = - n|1/p - 1/2|$, and $a \in S_{0,0}^m$ satisfies $|\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )| \leqslant {C_{\alpha ,\beta }}{\langle \xi \rangle ^m}$ for $(x,\xi ) \in {{\mathbf {R}}^n} \times {{\mathbf {R}}^n},|\alpha | \leqslant k$ and $|\beta | \leqslant k’$, in the Hölder continuous sense, where $k > n/2,k’ > n/p$ (the case $1 < p \leqslant 2$) and $k > n/p,k’ > n/2$ (the case $2 < p < \infty$).