It is well known that in the case of integer order differentiation the formula $\partial_{x}f(x,u(x))=\partial_{u}f\cdot \partial_{x}u+\partial_{x}f\cdot u$ holds. If we define fractional derivative via Fourier transform F as $D_{x}^{\alpha}u(x)=F^{-1}[{(i\xi)^{\alpha}\widehat{u}(\xi)}]$, where ^ denotes the Fourier transform, is there a similarly formula for $D_{x}^{\alpha}f(x,u(x))$?
[Math] Chain rule for fractional derivative defined via Fourier transform
fractional calculus
Related Solutions
Regarding your first question, the fractional derivative operators are in fact DEFINED by how they act on the Fourier transform side. If the Fourier Transform of $f(x)$ is $\hat f(\xi)$ then the Fourier transform of its derivative $f'(x) = Df(x)$ is $2\pi i \xi \hat f(\xi)$ and hence for a positive integer $k$, the Fourier Transform of its $k$-th derivative $D^kf(x)$ is $(2\pi i \xi)^k \hat f(\xi)$. To define the $k$-th derivative operator when $k$ is a fraction or even an arbitrary real number, that same formula is used---i.e., Fourier Transform, multiply the resulting function of $\xi$ by $(2\pi i \xi)^k$, and then inverse Fourier Transform.
As for your second question, I do not know of anything that is in the nature of a generalization of the Plancherel Theorem from $L^2$ to other $L^p$ spaces, and certainly the obvious generalization is false.
I found a document by Mauro Bologna from Chile, "Short Introduction to Fractional Calculus" (PDF link), which shows this on p.52:
where $\cos_\mu$ and $\sin_\mu$ are "generalized" sine and cosine functions (p.50 of that document).
(I post this without penetrating the mysteries of this topic.)
Best Answer
If you are interested in estimates (rather than pointwise equalities), you might check out this paper of Christ and Weinstein:
http://deepblue.lib.umich.edu/bitstream/handle/2027.42/29171/0000217.pdf?sequence=1
See in particular Proposition 3.1 therein.