In Axler's Harmonic Function Theory, he and his coauthors develop the theory of harmonic functions on spheres and discs by considering the restrictions of arbitrary polynomials on the sphere $S^{n-1} = \{x \in \mathbb{R}^n : ||x||^2 = 1 \}$ and taking the Poisson integral to get a harmonic polynomial in the interior ball. One can then take the Kelvin transform to get a harmonic function on the exterior of the sphere. This process yields a canonical projection $\mathscr{P}(\mathbb{R}^n) \to \mathscr{H}(\mathbb{R}^n)$, from the space of polynomials to the space of harmonic functions, factoring through the restriction map to $L^2(S^{n-1})$.

Does this theory generalize to knot complements? Say we have a knot $K \subseteq \mathbb{R}^3$, and we take a small tubular neighborhood $V$ around $K$, whose boundary is topologically a torus $T$. Given a function on the knot complement, one could restrict to $T$ and then solve the Dirichlet problem on the knot complement to get a projection like the one above. However, in the sphere case, there are many nice properties of the harmonic function theory; namely it comes with an efficient algorithm for computation of a harmonic polynomial basis of $L^2(S^{n-1})$ which involves repeatedly differentiating the function $f(x) = |x|^{2-n}$.

Is anyone aware of any theory along this vein? Are there any obstacles to generalizing what happens in the sphere case?