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For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\lambda s)}{4}}d\lambda$ satisfied the followin estimate: $$q_s(z,t)\leq C s^{-n-1} e^{\frac{A}{s}|(z,t)|^{1/2}} $$ with some positive constants $C$ and $A$ and $(z,t)\in \Bbb C^n\times\Bbb R$ with $|(z,t)|=(|z|^4+|t|^2)^{\frac{1}{4}}$.

Now for the one quaternion Heisenberg it is known that the heat kernel $p_s(x,t)=c\int_{\Bbb R^3} e^{-i\lambda .t}\Big( \frac{|\lambda|}{\sinh(|\lambda| s)}\Big)^2 e^{-\frac{ |\lambda||x|^2\coth(|\lambda| s)}{4}}d\lambda$ where $(x,t)\in Q\times \Bbb R^3$ and $Q$ the set of quaternions.

My question can we have the same estimate for $p_s(x,t)$. Thank you in adavance

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