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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 $A\in L^p_{\rm loc}(\mathbb{R}^n)$?

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  • $\begingroup$ Do you mean that you want a bound on the norm of $f(x)\,g(-i\nabla + A)$? Did you try the proof in the cases $p=2$ and $p=\infty$? $\endgroup$
    – LL 3.14
    Commented Dec 14, 2022 at 23:01

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