Let $M = [1,\infty) \times S^2$.

Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm: $$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$ for $u \in H^k_{\delta}(M)$.

Given $u \in H^k_{\delta}(M)$, we can expand it in terms of the spherical harmonics: $$u (r,x) = \sum_{l=0}^{\infty} \sum_{m=0}^{2l+1} c_{lm}(r) Y_{lm}(x)$$ where $r \in [1,\infty)$, $x\in S^2$, and $Y_{lm}$ are the spherical harmonics.

Question #1: How can I write the Sobolev norm $\lVert \cdot \rVert_{k,\delta}$ in terms of the coefficients $c_{lm}$? Is there a Sobolev norm equivalent to $\lVert \cdot \rVert_{k,\delta}$ in terms of the spherical harmonics coefficients?

Let $f_l\in C^{\infty}([1,\infty))$ such that $f_l = O(r^{-2})$. Let $h \in H^s(S^2)$ where $H^s$ is the usual Sobolev space on $S^2$ with the Sobolev norm: $$\lVert h\rVert_{H^s}^2 := \sum_{l=0}^{\infty} \sum_{m=0}^{2l+1} (1+l(l+1))^s |a_{lm}|^2$$ where $s>0$ and $h = \sum_{l=0}^{\infty} \sum_{m=0}^{2l+1} a_{lm} Y_{lm}$.

Define the function $u (r,x) = \sum_{l=0}^{\infty}\sum_{m=0}^{2l+1} f_l(r)a_{lm}Y_{lm}(x)$ for $r \in [1,\infty)$ and $x\in S^2$.

Question #2: In what Sobolev space $H^k_{\delta}(M)$ does $u$ live? (for which $k$ and $\delta$)