Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states that $$ u_1(x) = -\frac1{2\pi}\int_{\mathbb R^2}\frac{x_2-y_2}{|x-y|^2}\omega(y) \, dy $$ and $$ u_2(x) = \frac1{2\pi}\int_{\mathbb R^2}\frac{x_1-y_1}{|x-y|^2}\omega(y) \, dy. $$
I am interested in a local version of the law in $B(x_o, r) \subset \mathbb R^2$. More precisely, suppose $u: \Omega \to \mathbb R^2$, where $\Omega \subset \mathbb R^2$ is bounded, and $u, \omega \in L_{loc}^2(\Omega)$ with $\nabla \cdot u = 0$. Is it true that $$ u_1(x) = -\frac1{2\pi}\int_{B(x_o,r)}\frac{x_2-y_2}{|x-y|^2}\omega(y) \, dy+A_1^{x_o}(x) $$ and $$ u_2(x) = \frac1{2\pi}\int_{B(x_o,r)}\frac{x_1-y_1}{|x-y|^2}\omega(y) \, dy+A_2^{x_o}(x). $$ for all $x \in B(x_o,r) \subset \Omega$ and for some smooth-ish function $A^{x_o}$? If not this, is there something similar which is true?
Finally, I am also interested in the general case of $\mathbb R^d$ for $d >2$, but for simplicity we may first restrict to the case $d=2$.