# Chain rule for fractional laplacian

Does anyone know a formula of chain rule for fractional laplacian?
say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \to \mathbb{R}$ functional.

Thanks

• Is fractional laplacian a fractional derivative or Reisz transorm? – Uday Apr 13 '12 at 11:27
• Reisz transform( I thought this definition will be consistent with analytical operator theory) – Grant Apr 13 '12 at 17:49
• I would begin by looking for analogues of the product rule. Once you have that, then the chain rule for the case where $g$ is a polynomial will follow, giving some insight into the general situation. – Hans Engler Apr 14 '12 at 14:56

In the case $0 \leq a \leq 1/2$, the rule roughly takes the form
$$(-\Delta)^a g(U) \approx ((-\Delta^a) U) \cdot \nabla g(U) + \ldots$$
where the $\ldots$ error is a paraproduct which is "lower order" than the main term in some sense. One popular way to make this formula precise is the Bony linearisation formula, originally developed in http://www.ams.org/mathscinet-getitem?mr=631751 . This is part of a more general theory known as paradifferential calculus, discussed for instance in Taylor's book http://www.ams.org/mathscinet-getitem?mr=1766415