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I would like to understand the concept of multiplier for vector valued functions and find appropriate references for the multiplier theorems out there.

For instance say that we would like to express $\nabla \times \cdot$ as a Fourier multiplier. Does this make sense ? What I have in mind is that \begin{equation}\mathcal{F}\left[\nabla \times f\right](\xi) = -2\pi i\xi \times \mathcal{F}\left[f\right].\end{equation} Therefore I could make sense of the multiplier of the rotational as some matrix that acts on $\mathcal{F}[f] = (\mathcal{F}[f]_1, \mathcal{F}[f]_2, \mathcal{F}[f]_3)$ with entries \begin{equation}-2\pi i\begin{pmatrix} 0 & -\xi_3 & \xi_2 \\ \xi_3 & 0 & -\xi_1 \\ -\xi_2 & \xi_1 & 0 \end{pmatrix}.\end{equation} Is this how it works ? What about for the multipliers theorem as Hörmander-Mikhlin theorem?

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  • $\begingroup$ If you are trying to construct a fundamental solution for the rotational by just using the Fourier transform, you'll fail: I know since I tried this years ago. Nevertheless this is not due to the fact that this is impossible, but just to the fact that this is not the right way to proceed for system of constant coefficients PDEs (in other words this means the there's no multiplier theorem in this setting). Continue in the following comment... $\endgroup$
    – Daniele Tampieri
    Commented 2 days ago
  • $\begingroup$ ... Continue from the previous comment. I saw some indication on how to do this in Ehrenpreis' Fourier analysis in several complex variables, MR285849, Zbl 0195.10401, chapter VI, §VI.2, pp. 179-187. $\endgroup$
    – Daniele Tampieri
    Commented 2 days ago
  • $\begingroup$ Say then that I want to prove an $L^p$ estimate of something of the type $\nabla (\nabla \wedge (-\Delta)^{-1} f)$. Is there any hope ? How one should then proceed ? $\endgroup$
    – Rundasice
    Commented yesterday
  • $\begingroup$ This is another question, and it is not so easy to find an answer. I can only suggest a path: interpreting $f$ as $f=(f_1, f_2,f_3)$ and $-\Delta ^{-1}$ as the convolution operator with the fundamental solution to the laplacian, you may try to express the vector operators $\nabla \wedge$ and $\nabla$ as integral operators like what is done by Claus Müller (1969)[1957], Foundations of the Mathematical Theory of Electromagnetic Waves, MR0253638, Zbl 0181.57203. $\endgroup$
    – Daniele Tampieri
    Commented yesterday

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