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The asymptotics for large $n$ is $$J(n,\kappa) := \Big(\frac{2}{\pi}\Big)^2 \int_{-\pi}^\pi \int_{-\pi}^\pi \exp{(i\,n(x+y))}\frac{\sin^2x\,\sin^2y} {2\kappa - (\cos{x}+\cos{y}) }\, dx \,dy \sim$$ $$\sim \frac{8}{\sqrt{\pi \kappa n}}(\kappa^2-1)^{7/4} (\kappa - \sqrt{\kappa^2-1})^{2n}\quad, \quad (\kappa>1)$$ The proof consists of 5 parts. The first ...

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Yes. see:The symplectic camel and phase space quantization, by Maurice De Gosson. Journal of Physics A: Mathematical and General, Volume 34, Number 47

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the Plancherel density is derived from the Plancherel measure, see arXiv:1812.00047 for the precise definition:

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You have $I(\lambda, x)=x\cdot\int_{\mathbb S^{n-1}} ye^{i \lambda x\cdot y} d\sigma(y)=x\cdot J(x,\lambda)$ and you claim that for $\vert x\vert \lambda \ge 1$, you have $$J(x,\lambda)=O((\vert x\vert \lambda)^{-\frac{n-1}{2}}).$$ Indeed, using coordinate charts and a finite partition of unity, you are reduced to the case where  J(x,\lambda)=\int_{\...

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Let $u$ be a smooth function on $\mathbb R^n\backslash\{0\}$ homogeneous with degree $\lambda$ (on $\mathbb R^n\backslash\{0\}$). If $\lambda$ is not an integer $\le -n$, then $u$ can be uniquely extended to a tempered distribution homogeneous with degree $\lambda$. Moreover, the Fourier transform of an homogeneous distribution with degree $\lambda$ is an ...

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The answer is: Weyl, H. 1919, Annalen der Physik, 365, 481 doi: 10.1002/andp.19193652104 Though it's difficult to find in there if you don't understand German

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