In Axler’s Harmonic Function Theory, he and his coauthors develop the theory of harmonic polynomials on $mathbb{R}^n$ 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 polynomial 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 polynomials, factoring through the restriction map to $L^2(S^{n-1})$. By taking polynomial approximations via the Stone-Weierstrass theorem, and verifying some series convergence, I believe one can expand the domain and codomain of the projection to larger, more general function spaces. I have not verified this myself, so I could be wrong.

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 projection; namely it comes with an efficient algorithm for computation 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?