I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the most appropriately titled chapter I think I've ever seen).

Here is the simplest example: If $f\in L^p(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n),$ with $2\leq p<\infty$, then $f(x)g(-i\nabla)$ is a Schatten-class operator of order $p$ and $$ ||f(x)g(-i\nabla)||_p\leq (2\pi)^{-\frac{n}{p}}||f||_p||g||_p $$ with equality if $p=2$. My question is, do inequalities such as the one above, exist for operators of the form $-i\nabla+A$ for $A\in L^p_{\rm loc}(\mathbb{R}^n)$?

I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the most appropriately titled chapter I think I've ever seen).

Here is the simplest example: If $f\in L^p(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n),$ with $2\leq p<\infty$, then $f(x)g(-i\nabla)$ is a Schatten-class operator of order $p$ and $$ ||f(x)g(-i\nabla)||_p\leq (2\pi)^{-\frac{n}{p}}||f||_p||g||_p $$ with equality if $p=2$. My question is, do inequalities such as the one above exist for operators of the form $-i\nabla+A$ for $A\in L^p_{\rm loc}(\mathbb{R}^n)$?