The transport term is orthogonal to itself and the tame commutator estimate $\mathrm{\parallel }[{\mathrm{\Lambda }}^m,u\cdot \mathrm{\nabla }]u{\mathrm{\parallel }}_{L^2}\le C_m\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$

For trigonometric polynomials $u,w$ with $\mathrm{div}u=0$, integration by parts [F4] gives $\int ((u\cdot \mathrm{\nabla })w)\cdot wdx={\sum }_{i,j}\int u_j({\mathrm{\partial }}_j{w}_i)w_{i}dx=\frac{1}{2}{\sum }_j\int u_j{\mathrm{\partial }}_j\mathrm{\mid }w{\mathrm{\mid }}^{2}dx=-\frac{1}{2}\int (\mathrm{div}u)\mathrm{\mid }w{\mathrm{\mid }}^{2}dx=0$. The map $(u,w)\mapsto \int ((u\cdot \mathrm{\nabla })w)\cdot wdx$ is continuous on $H^m\times H^m$: indeed, $\mid \int ((u\cdot \mathrm{\nabla })w)\cdot wdx\mid \le \mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}\le C_m\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\mathrm{\parallel }w{\mathrm{\parallel }}_{H^m}^2$ by the embedding bound [F5] and [F3]; since $P_{N}u$ is divergence-free and $P_{N}u\to u$, $P_{N}w\to w$ in $H^m$ by [F4], the identity extends from polynomials to all $u,w\in H^m$, proving claim 1.

Weight comparison. Extend $w$ to ${\mathbb{R}}^3$ by $w(x)=(1+4{\pi }^2\mathrm{\mid }x{\mathrm{\mid }}_2^2{)}^{1\mathrm{/}2}$. Its gradient has norm at most $2\pi$, and $w(l+tv)\le w(l)+2\pi \mathrm{\mid }v{\mathrm{\mid }}_2\le w_l+w_v$ for $0\le t\le 1$. Since here $m>5\mathrm{/}2$, the convexity inequality $(a+b{)}^{m-1}\le 2^{m-2}(a^{m-1}+b^{m-1})$ applies. Integrating the derivative of $w(l+tv{)}^m$ for $v=k-l$ gives $\mathrm{\mid }{w}_k^m-w_l^m\mathrm{\mid }\le m2\pi 2^{m-2}\mathrm{\mid }v{\mathrm{\mid }}_2(w_v^{m-1}+w_l^{m-1})$.

Meaning and coefficients of the commutator. Put $u_N=P_{N}u$, the Fourier truncation, which converges to $u$ in $H^m$ and $H^s$ by [F4]. For polynomials the commutator has coefficients $c_k={\sum }_l(w_k^m-w_l^m)(\widehat{u}(k-l)\cdot 2\pi il)\widehat{u}(l)$ by finite convolution. The same series for general $u$ is absolutely convergent: the convolution bounds below apply also to the sum of absolute values, and each convolution of an ${\mathrm{\ell }}^2$ family with an ${\mathrm{\ell }}^1$ family is finite at each coordinate by [F2]. The two terms in the displayed commutator need not separately lie in $L^2$. They are interpreted in $H^{-1}$: the first lies there because $(u\cdot \mathrm{\nabla })u\in H^{m-1}$ by the transport product bound [F2], and the second is ${\sum }_j{\mathrm{\partial }}_j(u_j{\mathrm{\Lambda }}^{m}u)$ since $\mathrm{div}u=0$. Here $u_j$ is bounded by [F5], ${\mathrm{\Lambda }}^{m}u\in L^2$, and differentiation maps $L^2$ to $H^{-1}$ coefficientwise because $2\pi \mathrm{\mid }{k}_j\mathrm{\mid }\mathrm{/}{w}_k\le 1$. Products and these derivatives are defined by limits of the truncated fields; $u_N\to u$ uniformly and ${\mathrm{\Lambda }}^m{u}_N\to {\mathrm{\Lambda }}^{m}u$ in $L^2$, so the limits exist and agree with the stated products. Thus the difference is well defined in $H^{-1}$ and its coefficients are the series just displayed.

For $v=k-l$ define $P_v:=\mathrm{\mid }v{\mathrm{\mid }}_2{w}_v^{m-1}\mathrm{\mid }\widehat{u}(v){\mathrm{\mid }}_2$, $Q_l:=2\pi \mathrm{\mid }l{\mathrm{\mid }}_2\mathrm{\mid }\widehat{u}(l){\mathrm{\mid }}_2$, $A_v:=\mathrm{\mid }v{\mathrm{\mid }}_2\mathrm{\mid }\widehat{u}(v){\mathrm{\mid }}_2$ and $B_l:=2\pi \mathrm{\mid }l{\mathrm{\mid }}_2{w}_l^{m-1}\mathrm{\mid }\widehat{u}(l){\mathrm{\mid }}_2$. Step 1.2 bounds $\mathrm{\mid }{c}_k\mathrm{\mid }$ by $m2\pi 2^{m-2}[(P\ast Q)(k)+(A\ast B)(k)]$. Since $2\pi \mathrm{\mid }k{\mathrm{\mid }}_2\le w_k$, $\mathrm{\parallel }P{\mathrm{\parallel }}_2\le (2\pi {)}^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}$ and $\mathrm{\parallel }B{\mathrm{\parallel }}_2\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}$; [F1] gives $\mathrm{\parallel }Q{\mathrm{\parallel }}_1\le W_s^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$ and $\mathrm{\parallel }A{\mathrm{\parallel }}_1\le (2\pi {)}^{-1}{W}_s^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$. Young [F2] therefore gives $\mathrm{\parallel }c{\mathrm{\parallel }}_2\le m{2}^{m-1}{W}_s^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}\le C_m\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$ with the unchanged $C_m=4{\pi }^{2}m{2}^{m-2}(W_s^{\mathrm{\prime }}+1)$. The same calculation for the bilinear coefficient expression with different inputs $a,b$ bounds it by $m{2}^{m-2}{W}_s^{\mathrm{\prime }}(\mathrm{\parallel }a{\mathrm{\parallel }}_{H^m}\mathrm{\parallel }b{\mathrm{\parallel }}_{H^s}+\mathrm{\parallel }a{\mathrm{\parallel }}_{H^s}\mathrm{\parallel }b{\mathrm{\parallel }}_{H^m})$. Applying this to $u_N-u$ and $u_N,u$ shows convergence of the truncated commutators in ${\mathrm{\ell }}^2$. Parseval [F3] supplies their $L^2$ limit, which is the $H^{-1}$ difference of step 2.1. This proves claim 2 without strengthening its $H^m$ hypothesis.

For claim 3, apply step 1.1 with $w:={\mathrm{\Lambda }}^{m}u$: $⟨(u\cdot \mathrm{\nabla }){\mathrm{\Lambda }}^{m}u,{\mathrm{\Lambda }}^{m}u⟩=0$, so $⟨(u\cdot \mathrm{\nabla })u,{\mathrm{\Lambda }}^{2m}u⟩=⟨{\mathrm{\Lambda }}^m((u\cdot \mathrm{\nabla })u),{\mathrm{\Lambda }}^{m}u⟩=⟨(u\cdot \mathrm{\nabla }){\mathrm{\Lambda }}^{m}u,{\mathrm{\Lambda }}^{m}u⟩+⟨[{\mathrm{\Lambda }}^m,u\cdot \mathrm{\nabla }]u,{\mathrm{\Lambda }}^{m}u⟩=⟨[{\mathrm{\Lambda }}^m,u\cdot \mathrm{\nabla }]u,{\mathrm{\Lambda }}^{m}u⟩$, using that ${\mathrm{\Lambda }}^{2m}$ and ${\mathrm{\Lambda }}^m$ are self-adjoint and ${\mathrm{\Lambda }}^{2m}={\mathrm{\Lambda }}^m{\mathrm{\Lambda }}^m$ [F3].