Let \(u\) and \(v\) be smooth functions on the sphere \(\mathbb{S}^d_R\) with radius \(R>0\).
Then
\[ \langle \nabla^5 u, \nabla^5 v\rangle _{L^2(\mathcal{T}^5(\mathbb{S}^d_R))} = \langle u, P_5(-\Delta )v\rangle_{L^2(\mathbb{S}^d_R)} \]with
\begin{align} P_5(t)&=P_4(t)\Bigl(t-\frac{4(d-1)}{R^2}\Bigr) + \frac{c_3}{R^4} P_3(t) + \frac{c_2}{R^6} P_2(t) + \frac{c_1}{R^8}P_1(t), \end{align}where \(P_1\), \(P_2\), \(P_3\) and \(P_4\) are defined in the previous cases (\(c_1\), \(c_2\) and \(c_3\) need to be calculated).
Using the computer, I got
- \(c_3=28\),
- \(c_2=12(d-1)-88\),
- \(c_1=-36(d-1)^2+752(d-1)-832\).