Tataru and Koch Paper "Well posedness of the Navier- Stokes Equation" Hi I wonder if someone can help me with some inequalities ( [9] in the paper) of this research paper [1].
Suppose $\Pi$ is the Leray operator, this is an operator define as it follows:
$\Pi(f)= f-\nabla (\frac{1}{\Delta}) (\nabla \cdot f)$
And let be $K$ the Heat Kernel, 
$K=\Phi_{\sqrt(4t)}=\frac{1}{(4\cdot \Pi\cdot t)^{\frac{n}{2}}} e^{\frac{-|x|^2}{4t}}$
We need to prove
$| \Pi (\delta_{0}-K)| \leq c t |x|^{-n-2}$
If someone wants to read the paper you can find it easy on the internet.
Thank you guys and sorry for this aweful presentation.
[1]Tataru and Koch : "Well posedness of the Navier- Stokes Equation"
 A: OK, this is my attempt.
Without loss of generality, $|x|=1$ and $x_1 > |x|/\sqrt n$.  Lets look at $\Pi(\delta_0 - \Phi_{4\sqrt t})_{kk}$.
Let $\theta:[0,\infty)\to[0,1]$ be a smooth function that is zero on $[0,1/2]$ and one on $(2,\infty)$.  Suppose $\tau = \min\{t,|x|^{2}\}$.
\begin{align*}
   &\Pi(\delta_0 - \Phi_{4\sqrt t})_{kk} \\
   & = \int_{\mathbb R^n} \theta(\tau|\eta|^2)(1-\frac{\eta_k^2}{|\eta|^2}) (1-\exp(-t|\eta|^2)) \exp(i x\cdot\eta) \, d\eta \\
   & \phantom{{}={}} + \int_{\mathbb R^n} (1-\theta(\tau|\eta|^2))(1-\frac{\eta_k^2}{|\eta|^2}) (1-\exp(-t|\eta|^2)) \exp(i x\cdot\eta) \, d\eta \end{align*}.
For the first term, integrate by parts $n+2$ times with respect to $\eta_1$, and get something bounded by something like
$$ C |x_1|^{-n-2} \left|\int_{\mathbb R^n} F(\tau|\eta|^2)/|\eta|^{n+2} \exp(i x\cdot\eta) \, d\eta\right| $$
where $F$ is uniformly bounded and is zero on $[0,1/2)$.  (Constants may depend up on $n$.)  And this is bounded above by
$$ C |x_1|^{-n-2} \int_{|\eta|>1/\sqrt{\tau}} 1/|\eta|^{n+2} \, d\eta \approx C \tau |x|^{-n-2} \approx C t |x|^{n-2}. $$
For the second term, integrate $n$ times with respect to $\eta_1$, and get something bounded by something like
$$ C |x_1|^{-n} \left|\int_{\mathbb R^n} G(|\eta|)/|\eta|^{n} \exp(i x\cdot\eta) \, d\eta\right| $$
where $G$ is uniformly, is zero on $[2/t,\infty)$, and $G(\eta) = O(t|\eta|^2)$ (that is, very much like $(1-\exp(-\tau|\eta|^2))(1-\theta(t|\eta|^2))$).  This is bounded above by
$$ C t |x_1|^{-n} \int_{|\eta|<1/\sqrt{\tau}} 1/|\eta|^{n-2} \, d\eta \approx C t \tau^{-1} |x|^{-n}. $$
If $t > |x|^2$, then this is bounded by $C t |x|^{-n-2}$.  If $t < |x|^2$, then this is bounded by $C t^{-1} |x|^{-n} \le C t |x|^{-n-2}$.
