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
Does anyone know a formula of chain rule for fractional laplacian? Thanks 


In the fractional case, it turns out that approximate chain rules are more useful than exact formulae (at least for applications to the analysis of PDE). See http://wiki.math.toronto.edu/TorontoMathWiki/index.php/Fractional_Derivative 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/mathscinetgetitem?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/mathscinetgetitem?mr=1766415 


I showed that the Riesz potential is an integral formulation of general fractional twoside derivative, that is much more general in the sense that is valid for any order greater than 1 for a broad class of funtions. I can give copies by sending a mail to mdo@fct.unl.pt 

