In the book “Vorticity and Incompressible Flow” by Majda and Bertozzi, there is the following lemma.
Lemma 3.2. Any smooth incompressible vector field $v$ in $\mathbb{R}^2$ with vorticity $$\omega=\operatorname{curl} v\in L^1(\mathbb R^2)$$ has a radial-energy decomposition.
The above means that (see Definition 3.1)
there exists a smooth radially symmetric vorticity $\overline{\omega}(|x|)$ such that $$v(x)=u(x)+\overline v (x),$$ $$u\in L^2(\mathbb R^2),\qquad \nabla\cdot u=0,$$ where $\overline v$ is determined from $\overline\omega$ by means of the Biot-Savart law: $$\overline v(x)= \begin{pmatrix}-x_2\\x_1\end{pmatrix} |x|^{-2}\int_0^{|x|} s\overline{\omega}(s)ds=[K_2*\overline\omega(|\cdot|)](x) .$$
Here, I denoted $$K_2(x):=\frac{1}{2\pi}\frac{1}{|x|^2}\begin{pmatrix}-x_2\\x_1\end{pmatrix}.$$ In the pages before, they prove that if $\omega\in L^1$ has compact support and $\int \omega=0$, then $K_2*\omega\in L^2$. This proves the lemma in the case where $\omega$ has compact support (and of course with the additional hypothesis that you can recover $v$ from $\omega$, i.e., $v=K_2*\omega$), as one can just subtract a radial test function $\rho$ from $\omega$ in such a way that the difference $\omega-\rho$ has zero mean, then call $u=K_2*(\omega-\rho)$ and $\overline v=K_2*\rho$.
Now…. My question is whether someone can exhibit a proof for the lemma above, provide a reference where the lemma is treated with some more details, or knows whether the lemma is true or not (making eventually the assumption that $v=K_2*\omega$; I think that this assumption is necessary and probably implicitly assumed, just think about the case of a constant velocity field…).
I have some doubts. On one hand, for an arbitrary vorticity in $L^1$ with zero mean it is not true in general that $v=K_2*\omega\in L^2$, without the hypothesis of compact support (the idea is that the mass of the vorticity can be pushed away at infinity in different direction depending on the sign; I could give more details below). So, even though in the book they suggest that the lemma is proved from the previous steps, the proof can’t actually rely on the same argument used for the compactly suported vorticities…. This makes me wonder whether the lemma might be false. On the other hand, I have seen this lemma used in at least a couple of (I think published) papers, so it seems strange to me that it is false (e.g., Bjorland and Niche - On the Decay of Infinite Energy Solutions to the Navier–Stokes Equations in the Plane).
Edit. A very similar question exists, see A decomposition of incompressible vector fields . The problem is — they don’t mention it, but it is stated in the book — that the Taylor expansion at infinity for the Biot–Savart law is proved and works only for compactly supported vorticities in $L^1$. In that particular case, that is in fact enough to prove the lemma. The OP and who answers don’t quite solve the problem of what happens for general $\omega\in L^1(\mathbb R^2)$.
