Two-dimensional Fourier logarithmic velocity bound

In each periodic coefficient integral, one-coordinate integration by parts has boundary term $f(x)e^{-i{k}_j{x}_j}{\mathrm{\mid }}_0^{2\pi }=0$ because both factors are periodic, so $\widehat{{\mathrm{\partial }}_{j}f}(k)=i{k}_j\widehat{f}(k)$. Thus $k\cdot \widehat{u}(k)=0$ for $k\mathrm{\ne }0$ and $\widehat{\omega }(k)=i(k_1{\widehat{u}}_2(k)-k_2{\widehat{u}}_1(k))$. Since a vector in ${\mathbb{C}}^2$ orthogonal to the nonzero real vector $k$ is a complex multiple of $(-k_2,k_1)$, these identities imply $\mathrm{\mid }\widehat{\omega }(k)\mathrm{\mid }=\mathrm{\mid }k\mathrm{\mid }\mathrm{\mid }\widehat{u}(k)\mathrm{\mid }$.

For $K\ge 2$, Cauchy–Schwarz and the definition of $B$ give ${\sum }_{0<\mathrm{\mid }k\mathrm{\mid }\le K}\mathrm{\mid }\widehat{\omega }(k)\mathrm{\mid }\le B{\left({\sum }_{0<\mathrm{\mid }k\mathrm{\mid }\le K}{w}_k^{-2}\right)}^{1\mathrm{/}2}\mathrm{.}$ For each integer $j\ge 1$, the unit squares centered at lattice points with $j\le \mathrm{\mid }k\mathrm{\mid }<j+1$ lie in an annulus of radii $\max (0,j-1)$ and $j+2$; comparing areas shows that this shell contains at most $Cj$ lattice points. Hence ${\sum }_{0<\mathrm{\mid }k\mathrm{\mid }\le K}{w}_k^{-2}\le C{\sum }_{1\le j\le K+1}{j}^{-1}\le C\log (e+K)$, and the low-frequency sum is at most $CB\sqrt{\log (e+K)}$.

For the high frequencies, Cauchy–Schwarz gives ${\sum }_{\mathrm{\mid }k\mathrm{\mid }>K}\mathrm{\mid }k\mathrm{\mid }\mathrm{\mid }\widehat{u}(k)\mathrm{\mid }\le Y{\left({\sum }_{\mathrm{\mid }k\mathrm{\mid }>K}\mathrm{\mid }k{\mathrm{\mid }}^2{w}_k^{-2m}\right)}^{1\mathrm{/}2}\mathrm{.}$ The same shell count bounds the series inside the square root by $C{\sum }_{j>K-1}{j}^{3-2m}\le C_m{K}^{4-2m}$, since $m\ge 4$. Thus the high-frequency sum is at most $C_{m}Y{K}^{2-m}$. The series is finite for every such $u$ by the same estimate and its finite low-frequency part.

Choose $K=2+(e+Y{)}^{1\mathrm{/}(m-2)}$. Then $Y{K}^{2-m}\le Y\mathrm{/}(e+Y)\le 1$, and $\log (e+K)\le C_m\log (e+Y)$ because $\log (e+Y)\ge 1$ and $K\le 2+(e+Y)$. Combining the low- and high-frequency bounds from steps 2.1 and 3.1 proves the asserted estimate.