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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 $\delta$)

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    $\begingroup$ Is $f$ fixed and you do not care about the dependence on it? If no, (1) seems false: the weighted norm also measures derivatives of $f$. $\endgroup$
    – user378654
    Commented Sep 21, 2021 at 3:22
  • $\begingroup$ ahh i didn't word the question properly. $f$ is a fixed function of $r$ only. I rewrote the question. $\endgroup$
    – Laithy
    Commented Sep 21, 2021 at 5:17
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    $\begingroup$ Question #1 seems to be just a matter of explicit computation, no? The integrals over $\mathbb{S}^2$ you can just use the orthogonality properties of $Y_{lm}$, and you will get a family of linear differential operators on $[1,\infty)$ (let's call them $L_{lm}$) such that the Sobolev norm looks something like $\sum \| L_{lm} c_{lm} \|^2_{L^2}$. $\endgroup$
    – Willie Wong
    Commented Sep 28, 2021 at 14:23
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    $\begingroup$ For Question #2: If you only assume decay of $f_l$ and not its higher derivatives, then generally your function will not be in $H^k_\delta$ for any $k > 0$. With appropriate assumptions of decay of higher derivatives up to order $s'$, then the best $k$ you can choose is $\min(s,s')$, and the best $\delta$ is a matter of explicit computation based on the assumed decay rates. $\endgroup$
    – Willie Wong
    Commented Sep 28, 2021 at 14:28
  • $\begingroup$ Thank you so much @WillieWong for always helping me. I have tried to compute what $L_{lm}$ are but I keep getting stuck. I will try again today. Only if you have time, please provide more details or at least a guess at what the $L_{lm}$ are. I forgot to say that $f_l$ is analytic and so its derivatives satisfy the appropriate assumptions. My initial guess is that $k$ can be chosen to be bigger than $s$. But you are saying thats not correct. So I am a bit confused. I will think about it again. $\endgroup$
    – Laithy
    Commented Sep 28, 2021 at 15:31

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