{\displaystyle
\;\!
\mathbb{F}\:\![\;\!f\;\!]
\equiv
\hat{f}
\;\!
} $ a transformada de Fourier de uma dada função $ \!\;\!f \!\;\!\in L^{1}(\mathbb{R}^{3}) $, viz.,\
\
$$\tag{2.6}
\mathbb{F}\:\![\;\!f\;\!]\:\!(k)
\:\equiv\;
\hat{f}(k)
\,:=\;
(\:\!2\:\!\pi)^{-\,3/2} \!\!
\int_{\mbox{}_{\scriptstyle \!\:\!\mathbb{R}^{3}}}
\!\!\:\!
e^{\mbox{\scriptsize $- \!\;\!
\stackrel{\mbox{\tiny $\circ$}}{\mbox{\sf \i$\!\;\!$\i}}
\!\:\! k \!\;\!\cdot\!\;\! x$}}
f(x) \: dx,
\qquad
k \in \mathbb{R}^{3}
$$\
(onde $ {\displaystyle
\:\!
\stackrel{\mbox{\tiny $\circ$}}{\mbox{\sf \i$\!\;\!$\i}}
\mbox{}\!\!\;\!\mbox{}^{2}
\!\;\!=\;\! - \;\!1
} $). Dada $ {\displaystyle
\;\!
\mbox{\bf v}(\cdot,s) =
(\:\! \mbox{v}_{\mbox{}_{\!\;\!1}}\!\;\!(\cdot,s),
      \mbox{v}_{\mbox{}_{\!\;\!2}}\!\;\!(\cdot,s),
      \mbox{v}_{\mbox{}_{\!\;\!3}}\!\;\!(\cdot,s)\:\! )
\in
L^{1}(\mathbb{R}^{3}) \cap L^{2}(\mathbb{R}^{3})
\:\!
} $ arbitrária, obtém-se, pela identidade de Parseval,\
\
$$\notag
\begin{split}
\|\:  e^{\;\!\nu \:\!\Delta \mbox{\scriptsize $(t-s)$}} \;\!
\mbox{\bf v}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{\:\!2}
\;\!&=\;\;\!
\|\; \mathbb{F}\;\![\, e^{\;\! \nu \:\!\Delta \mbox{\scriptsize $(t-s)$}}
\;\!\mbox{\bf v}(\cdot,s) \: ] \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{\:\!2} \\
&=\,
\int_{\mathbb{R}^{3}} \!\!
e^{-\,2\,\nu \;\!|\,\mbox{\scriptsize $k$}\,
|_{\mbox{}_{\!\;\!2}}^{\:\!\scriptstyle 2} \:\!(t \,-\, s)}
\;\!
|\, \hat{\mbox{\bf v}}(k,s) \,
|_{\mbox{}_{\!\;\!2}}^{\:\!2}
\, dk \\
&\leq\;
\|\, \hat{\mbox{\bf v}}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle \infty}}^{\:\!2} \!\!\;\!
\int_{\mathbb{R}^{3}} \!\!
e^{-\,2\,\nu \;\!|\,\mbox{\scriptsize $k$}\,
|_{\mbox{}_{\!\;\!2}}^{\:\!\scriptstyle 2} \:\!(t \,-\, s)}
\, dk \\
&=\;
\Bigl(\;\! \frac{\;\!\pi\;\!}{2} \,\Bigr)^{\!\!\;\!3/2}
\!\;\!
\nu^{-\,3/2}
\,( t - s )^{- \,3/2}
\,
\|\, \hat{\mbox{\bf v}}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle \infty}}^{\:\!2}
\!\;\!,
\end{split}
$$\
ou seja,\
\
$$\tag{2.7}
\|\:  e^{\;\!\nu\:\!\Delta \mbox{\scriptsize $(t-s)$}} \;\!
\mbox{\bf v}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\;\!\leq\;\;\!
\Bigl(\;\! \frac{\;\!\pi\;\!}{2} \,\Bigr)^{\!\!\;\!3/4}
