=== The Oseen kernel The following Lemma are taken from Chapter 11 in [Lem02]. **Lemma 2 (The Oseen kernel)** For $1\leq j,k\leq n$, the operator $O_{j,k,t}=\Delta^{-1}\p_j\p_k e^{t\Delta}$ is a convolution operator $O_{j,k,t}f=K_{j,k,t}*f$, where the kernel $K_{j,k,t}$ is called the Oseen kernel and satisfies $K_{j,k,t}(x) = t^{-n/2}K_{j,k}(x/\sqrt{t})$ for a smooth function $K_{j,k}$ such that, for all $\alpha\in \mathbb{N}^n$: $$(*)\hspace{32pt}( 1 + \abs{x} )^{n+\abs{\alpha}} \p_\alpha K_{j,k} \in L^\infty(\mathbb{R}^n).$$ **//Proof://** We have $\widehat K_{j,k}=\f{\xi_j\xi_k}{\abs{\xi}^2}e^{-\abs{\xi}^2}$, thus for all $\alpha$, $\p_\alpha K_{j,k} \in L^\infty$ since $e^{-\abs{\xi}^2}$ belongs to the Schwarz space $\mathcal{S}(\mathbb{R}^n)$. Hence, we only need to show (*) for $\abs{x}\geq 1$. For $\abs{x}\geq 1$, we use the Littlewood-Paley decomposition (see Chapter 3 in [Lem02] or [[http://www.math.ucla.edu/~tao/254a.1.01w/|T. Tao's notes]]). We introduce $\varphi\in\mathcal{D}(\mathbb{R}^n)$ such that \begin{equation} \varphi(\xi)= \begin{cases} 1,\hspace{6pt}\forall\,\abs{\xi}\leq\f{1}{2},~\\ 0,\hspace{6pt}\forall\,\abs{\xi}\geq 1,~\\ 0\leq \varphi\leq 1. \end{cases} \end{equation} We then set $\psi(\xi)=\varphi(\xi/2)-\varphi(\xi)$, and define $$\widehat{S_j f} = \varphi(\f{\xi}{2^j})\hat f,\hspace{6pt}\widehat{\Delta_j f}=\psi( \f{\xi}{2^j})\hat f.$$ clearly, by definition, we have $S_{j+1}f - S_j f=\Delta_j f$. **Claim 5:** $S_0K_{j,k} = \sum_{l<0}\Delta_lK_{j,k}$. Note that we only need to show that $\lim_{N\rightarrow -\infty} S_N K_{j,k} = 0$. By definition \begin{equation} S_N K_{j,k} = \int e^{ix\cdot\xi} \varphi(\f{\xi}{2^N}\f{\xi_j\xi_k}{\abs{\xi}^2}e^{-\abs{\xi}^2} d\xi,\hspace{6pt}\text{with} \begin{cases} supp\,\varphi(\f{\xi}{2^N})\subset \{ \abs{\xi}<2^N \} \rightarrow \emptyset (N\rightarrow 0),~\\ \f{\xi_j\xi_k}{\abs{\xi}^2}e^{-\abs{\xi}^2}\in L^1(\mathbb{R}^n), \end{cases} \end{equation} which converges to $0$ by [[http://en.wikipedia.org/wiki/Dominated_convergence_theorem|Dominated convergence theorem]]. **claim 6:** $(id - S_0)K_{j,k}$ belongs to $\mathcal{S}(\mathbb{R}^n)$. It suffices to prove that the Fourier transform of $(id - S_0)K_{j,k}$ belongs to $\mathcal{S}(\mathbb{R}^n)$. $$\mathcal{F}( (id - S_0)K_{j,k} )= (1 - \varphi(\xi) )\widehat{K_{j,k}} = (1 - \varphi(\xi) )\f{\xi_j\xi_k}{\abs{\xi}^2}e^{-\abs{\xi}^2} \in \mathcal{S}(\mathbb{R}^n), $$ since $1-\varphi$ is supported away from the origin. Now \begin{equation} \begin{cases} \mathcal{F}(\Delta_l K_{j,k} )(\xi) = \varphi(\f{\xi}{2^l}) \f{\xi_j\xi_k}{\abs{\xi}^2}e^{-\abs{\xi}^2}, ~\\ \hat w_{j,k,l}(\xi) = \varphi(\xi) \f{\xi_j\xi_k}{\abs{\xi}^2}e^{-\abs{2^l\xi}^2} , \text{ (def. of }w_{j,k,l}), \end{cases} \end{equation} which implies that $\mathcal{F}(\Delta_l K_{j,k} = \hat w_{j,k,l}(\f{\xi}{2^l})$. Therefore, we find $\Delta_l K_{j,k}(x) = 2^{ln}w_{j,k,l}(2^lx)$. These functions $w_{j,k,l}(x) (l<0)$ are uniformly bounded in $\mathbb{R}^n$. Then for every $N\in \mathbb{N}$, \begin{equation} \begin{split} (1+2^l\abs{x})^N 2^{-l(n+\abs{\alpha}} \abs{ \p_\alpha (\Delta K_{j,k})(x) } ~\\ =(1+2^l\abs{x})^N 2^{-l(n+\abs{\alpha}} 2^{ln} \abs{ \p_\alpha( w_{j,k,l}(2^lx))} ~\\ =(1+2^l\abs{x})^N \abs{ \p_\alpha w_{j,k,l}(2^lx)} \leq C_{N,\alpha}. \end{split} \end{equation} This gives for $N>n+\abs{\alpha}$, \begin{equation} \begin{split} \abs{ \p_\alpha S_0K_{j,k} } &\leq C \sum_{ 2^l\abs{x}\leq 1}2^{l(n+\abs{\alpha})} + \sum_{2^l\abs{x}>1}2^{l(n+\abs{\alpha}-N)} \abs{x}^{-N} ~\\ &\leq C\abs{x}^{-n-\abs{\alpha}}, \end{split} \end{equation} which shows that for $\abs{x}\geq 1$: $$(1 + \abs{x})^{n+\abs{\alpha}} \lesssim \f{(1 + \abs{x})^{n+\abs{\alpha}}}{\abs{x}^{n+\abs{\alpha}}} \leq 2^{n+\abs{\alpha}}. $$ **Remark:** Let $m\in\mathcal{C}^\infty(\mathbb{R}^d\backslash \{0\})$ be homogeneous of degree $0$, and set $O_{m,t}=m(D_x)e^{t\Delta}$ which is a convolution operator with kernel $K(t,x)$. Then $K(t,x) = \f{1}{t^{n/2}}K(x/\sqrt{t})$ for some smooth function satisfying: $$\text{for all }\alpha\in \mathbb{N}^n,\hspace{6pt} (1 + \abs{x})^{n+\abs{\alpha}} \p_\alpha K(x)\in L^\infty(\mathbb{R}^n). $$ === References [Lem02] P.G. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics **431**. Chapman & Hall/CRC, Boca Raton, FL, 2002.