$L^p$ boundedness of pseudo-differential operators with symbols in $S_{\rho ,1}^{n(\rho -1)/2}$

Abstract

For symbol $a\in S_{\rho ,1}^{n(\rho -1)/2}$ the pseudo-differential operator $T_a$ may not be $L^2$ bounded. However, under some mild extra assumptions on $a$, we show that $T_a$ is bounded from $L^{\infty }$ to $BMO$ and on $L^p$ for $2\le p<\infty$. A key ingredient in our proof of the $L^{\infty }$-$BMO$ boundedness is that we decompose a cube, use $x$-regularity of the symbol and combine certain $L^2$, $L^{\infty }$ and $L^{\infty }$-$BMO$ boundedness. We use an almost orthogonality argument to prove an $L^2$ boundedness and then interpolation to obtain the desired $L^p$ boundedness.