Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\lVert u \rVert_{\tau, k} < \infty$, where $$\lVert u \rVert_{\tau, k}^2 := \int_{1}^{\infty} r^{-2\tau-1} \lVert u(r)\rVert_{H^k(S^2)}^2 dr$$
Let $g \in H^k(S^2)$ and $F \in L^2_{\delta-2}([1,\infty); H^{k-2}(S^2)$ for $\delta \in (-1,0)$. Suppose that a function $u$ on $M$ satisfies
$$\Delta u = F, \quad \text{on $M$}$$ $$u|_{\partial M} = g$$
What is the best regularity that we can expect for $u$? If $k=2$, then $F\in L^2_{\delta-2}(M)$ and so $u$ lives in the weighted Sobolev space $H^{2+1/2}_{\delta}(M)$.
For arbitrary $k$, I have managed to show that $u$ will satisfy
$$ u \in L^2_{\delta}([1,\infty);H^{k}(S^2)),\quad \partial_r u \in L^2_{\delta-1}([1,\infty);H^{k-1}(S^2)) , \quad \partial^2_{r} u \in L^2_{\delta-2}([1,\infty); H^{k-2}(S^2))$$
Can we say more? In particular, is the following inequality true?
$$\sup_{r\geq 1}{r^{-\delta} \lVert u \rVert_{H^k(S^2)}} + \sup_{r\geq 1}{r^{-\delta+1} \lVert \partial_r u(r) \rVert_{H^{k-1}(S^2)}} \leq C( \lVert F \rVert_{\delta-2, k-2} + \lVert g \rVert_{H^k(S^2)})$$
If so, then $u$ satisfies
$$u \in C_{\delta}([1,\infty);H^k(S^2)), \quad \partial_r u \in C_{\delta-1}([1,\infty); H^{k-1}(S^2))$$
A final question: if we in addition assume that $F$ lives in $C_{\delta-2}([1,\infty);H^{k-2}(S^2))$, how does the regularity of $u$ improve?
