I accidentally met such question. Let's start from easy ones.
Let $\Omega$ be an open convex domain in $\mathbb{R}^2$ and $u(x)$ satisfies that $$u(x) = \int_{\partial\Omega} \nabla_y G(|x-y|)\cdot n_y f(y) d\sigma(y)$$ with $$G(r) = -\int_{r}^{\infty} \frac{1}{\rho} d\rho$$ then $u(x)$ has the weak unique continuation property (wucp) since $\Delta u = 0$ inside $\Omega$.
So I wonder if there is a general theory to deal with the unique continuation for integral operators. For example, if we change the integral with $$u(x) = \int_{\partial\Omega} \nabla_y H(|x-y|)\cdot n_y f(y) d\sigma(y)$$ where $$H(r) = -\int_{r}^{\infty} \frac{e^{-\rho}}{\rho} d\rho$$ do we still have the wucp? Now $u$ no longer satisfies a LOCAL PDE anymore.
Any advice or reference will be helpful.
Thanks.