@Dolii The convergence needs to be in $L^2(D)$. This amounts to the boundedness of the the harmonic extension from $L^2(\partial D)$ to $L^2(D)$. I believe this to be true, at least assuming $\partial D$ is smooth enough, but I have yet to find a proof. There is a Springer book The Dirichlet Problem with L2-Boundary Data for Elliptic Linear Equations on that particular problem, which may be helpful to you.

@Dolii What is needed is that $h_n$ converges to something in $L^2(D)$. But this is more difficult than what I have thought.

Yes, and your attempt showed how to approximate when the boundary value is 0. Then you only need to approximate the boundary value in $L^2$, while keeping the Laplacian 0 at the boundary, which can be done using the hint.

Hint: to approximate the boundary value in $L^2$, use a smooth function and then solve the Dirichlet problem with that as the boundary value.

Take a simple example with maximal degree $d^n$ (for example, $x_1^d=2$ and $x_{i+1}^d=x_i$, $i=1,\dots,n-1$). Then make the change of coordinates $x_i'=x_i+x_n$, $i=1,\dots,n-1$.

Moreover if you define $f(1) = 0$ then $f$ is not even continuous at $z = 1$.

Do you really mean to put weights in the physical space instead of the frequency space? Then no such bound is possible.

This is not an answer.

Do you mean the Polish "dark l"?

On the other hand, the conclusion is true if $R$ is itself a local ring, in which $R_m$ is identical to $R$.

Try the simplest example: $R=\mathbb Z$. Then every maximal ideal is of the form $m=(p)$ where $p$ is a prime number. Then $R_m={\mathbb Z}_{(p)}$ is a DVR, so every two ideals are related by containment. On the other hand, there are plenty of pairs of integers $(x,y)$ such that $x\not\mid y$ and $y\not\mid x$.

I guess they both goes to 0 as $x$ goes to 0, and the property is characterizing the rate at which $F$ goes to 0 as $x$ goes to 0.

I think it's fitting and proper to invite Kevin Buzzard to this question.