Question about Fourier transforms of gradient, curl and divergence

Question

Consider a vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$. Denote by $F_u$ the Fourier transform of a scalar or vector field $u$. Can one finds an equality relation between $F_v,F_{\mathrm{curl}v}$ and $F_{\mathrm{div}v}$? The reason that I asked question is that I need to give an upper bound to $F_v$, say in $L^2$, if it is conditioned that $F_{\mathrm{curl}v}$ and $F_{\mathrm{div}v}$ are bounded in $L^2$, that is \begin{equation} \|F_v\|^2\leq C(\|F_{\mathrm{curl}v}\|^2+\|F_{\mathrm{div}v}\|^2) \end{equation} for some constant $C$, but so far I haven't seen how to express $F_v$ in terms of $F_{\mathrm{curl}v}$ and $F_{\mathrm{div}v}$ to get this estimate. Can someone help me? It is also assumed that every term is well defined in related vector spaces if it is needed.

Answer 1

The relations are as follows: $$ F_{\operatorname{div} v}(\vec \xi) = i \vec\xi\cdot F_v,\qquad F_{\operatorname{curl} v}(\vec \xi) = i \vec\xi\times F_v \tag{1} $$ They are easier to remember if one uses the notation $\nabla \cdot v$ and $\nabla \times v$, and notes that on the Fourier side, differentiation (i.e., $\nabla$) corresponds to multiplication by $i\vec \xi$. You can find the calculations here; related posts from the same blog are this and this.

In general, one cannot recover a vector field from curl and divergence, because there exist vector fields with zero curl and zero divergence: e.g., constant fields, and more generally fields of the form $\nabla u$ where $u$ is a harmonic function. The Fourier transform relation $(1)$ expresses this by the fact that multiplication by $\vec\xi$ kills the contribution of the origin (which could be Dirac mass or some of its derivatives).

However, you are probably interested in the case when $v$ vanishes at infinity. Then $v$ can be reconstructed: see Wikipedia article Helmholtz decomposition and the aforementioned blog. For $\vec \xi\ne 0$ the relations $(1)$ imply $$ |F_v (\vec\xi)|^2 = \frac{1}{|\vec\xi |^2}(|F_{\operatorname{div} v}(\vec \xi)|^2 + |F_{\operatorname{curl} v}(\vec \xi)|^2)$$ Note that you still have to worry about small frequencies, i.e., long tail of $v$: they can kill your $L^2$ estimate. For example: the field $\nabla \min(1,|\vec x|^{-1/2})$ decays as $|\vec x|^{-3/2}$, and so is not in $L^2(\mathbb{R}^3)$, while its divergence is in $L^2$ and its curl is zero.