Tame transport commutator

For a trigonometric polynomial approximation, the $k$th coefficient of the commutator is ${\sum }_{v+\mathrm{\ell }=k}(w_k^m-w_{\mathrm{\ell }}^m)\widehat{f}(v)i{\mathrm{\ell }}_j\widehat{g}(\mathrm{\ell })$. The mean-value theorem applied to $x\mapsto (1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{m\mathrm{/}2}$ gives $\mathrm{\mid }{w}_{\mathrm{\ell }+v}^m-w_{\mathrm{\ell }}^m\mathrm{\mid }\le C_m\mathrm{\mid }v\mathrm{\mid }(w_v^{m-1}+w_{\mathrm{\ell }}^{m-1})$, because $w_{\mathrm{\ell }+\theta v}^{m-1}\le C_m(w_{\mathrm{\ell }}^{m-1}+w_v^{m-1})$.

The first resulting convolution is bounded in ${\mathrm{\ell }}^2$ by $C\mathrm{\parallel }{w}^m\widehat{f}{\mathrm{\parallel }}_2{\sum }_{\mathrm{\ell }}\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }\mathrm{\mid }\widehat{g}(\mathrm{\ell })\mathrm{\mid }$; the second is bounded by $C({\sum }_v\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }\widehat{f}(v)\mathrm{\mid })\mathrm{\parallel }{w}^m\widehat{g}{\mathrm{\parallel }}_2$. Cauchy–Schwarz and ${\sum }_k\mathrm{\mid }k{\mathrm{\mid }}^2{w}_k^{-2s}<\mathrm{\infty }$ for $s>1+d\mathrm{/}2$ bound these sums by $C\mathrm{\parallel }g{\mathrm{\parallel }}_{H^s}$ and $C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^s}$, respectively. The required strict inequality holds since $s\ge 4$ and $d\le 3$. Discrete Young then gives the claimed bound for polynomials.

Smooth Fourier truncations converge in each displayed Sobolev norm, so the polynomial estimate passes to smooth $f,g$ by continuity of multiplication and Fourier multipliers. The strict $s\ge 4>d\mathrm{/}2+1$ condition also agrees with the $C^1$ embedding in [F2]; no endpoint commutator estimate is used.