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Does anyone know of some good references for a fractional Leibniz rule for pseudo-differential operators of negative order? As a specific example, I would like to compute $\partial_{x}^{-1}(uv)$, assuming $\xi \neq 0$ in phase space and mean-zero $u,v$.

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First, what does "$\partial_x^{-1}$" mean? Second, given a pseudo-differential operator $P$ of negative order, you might be able to get the Leibniz rule by using the standard symbol calculus for the composition of $P$ with the operator "multiply by $u$" applied to the function $v$. –  Deane Yang Jul 17 '12 at 14:54
    
Try Osler's papers on the general Leibniz formula: rowan.edu/open/depts/math/osler/my_papersl.htm –  Tom Copeland Jul 17 '12 at 20:55
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Let $P$ be a pseudodifferential operator with symbol $p(x,\xi)$ belonging to $S^m_{1,0}$ and let $a(x)$ be a $C^\infty$ function. Then the operator $Pa$ defined by $Pa u=P(au)$ is a pseudodifferential operator of order $m$ with symbol $q(x,\xi)$ such that $$ q-\sum_{\vert \alpha\vert\le N} \frac{1}{i^{\vert \alpha\vert}\alpha !}(\partial_\xi^\alpha p)(x,\xi) (\partial_x^\alpha a)(x)\in S^{m-N-1}_{1,0}, $$ which provides an asymptotic expansion of the symbol of the operator Pa, following simply from the composition formula.

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