I'm reading the paper: http://projecteuclid.org/euclid.cmp/1103941230.
I cannnot understand a sentence after the equation (9), that is:
Therefore the Fourier transform of $\nabla u$ and $\omega$ satisyfy $(\nabla u)^{\hat{}}(\xi)=S(\xi)\hat{\omega}(\xi)$, where $S$ is a matrix which is bounded independent of $\xi$, ......
(Here $u$ is a vector valued function, and $\omega=\nabla \times u$ is the vorticity of that.)
My questions are as follows:
a) How can we define the Fourier transform of $\nabla u$ and $\omega$ ?
b) How can we derive the relation $(\nabla u)^{\hat{}}(\xi)=S(\xi)\hat{\omega}(\xi)$ from (9) ?
I think $\omega$ is a vector and $\nabla u$ is a matrix. So the equation the relation $(\nabla u)^{\hat{}}(\xi)=S(\xi)\hat{\omega}(\xi)$ doesn't look well-defined.
Please show me the meaning of these.
Thank you.