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

TL;DR

This paper investigates the boundedness of pseudo-differential operators with symbols in a specific class, establishing $L^p$ bounds under certain conditions and employing decomposition, regularity, and interpolation techniques.

Contribution

It demonstrates $L^p$ boundedness for operators with symbols in $S^{n( ho-1)/2}_{ ho,1}$ under mild assumptions, extending previous results.

Findings

Operators are bounded from $L^{ abla}$ to $BMO$ under assumptions.

Established $L^p$ boundedness for $2 \,\leq p<\infty$.

Used decomposition, regularity, and interpolation methods.

Abstract

For symbol $a\in S^{n(\rho-1)/2}_{\rho,1}$ 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 $2\leq p<\infty$. 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.