Quadratic transport in negative Sobolev space

Put $w_k=(1+L^2\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^r$. For any sequence $c$ with ${\sum }_k{w}_k^{-1}\mathrm{\mid }{c}_k{\mathrm{\mid }}^2<\mathrm{\infty }$, Cauchy–Schwarz gives $\mathrm{\mid }{\sum }_k{c}_k\cdot \widehat{\phi }(-k)\mathrm{\mid }\le \mathrm{\parallel }c{\mathrm{\parallel }}_{H^{-r}}({\sum }_k{w}_k\mathrm{\mid }\widehat{\phi }(-k){\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Repeated integration by parts bounds the last factor by a constant times $p_M(\phi )$ for an integer $M>r+d\mathrm{/}2$, because smooth Fourier coefficients decay as $(1+\mathrm{\mid }k\mathrm{\mid }{)}^{-M}$. Thus the displayed pairing is absolutely convergent and continuous, so defines a periodic distribution; testing on one character recovers each $c_k$, proving injectivity. It agrees with the regular distribution of an $L^2$ field by Fourier truncation and Parseval. Since $\mathrm{\parallel }{P}_k\mathrm{\parallel }\le 1$, coefficient multiplication by $P_k$ is a contraction on $H^{-r}$, preserves real conjugate symmetry, and agrees on $L^2$ with the supplied projection. For $A=a\otimes a\in L^1$, its row divergence is the continuous functional $⟨\mathrm{div}A,\phi ⟩=-{\sum }_{i,j}\int A_{ij}{\mathrm{\partial }}_j{\phi }_{i}d\mu$, bounded by ${\sum }_{i,j}\mathrm{\parallel }{A}_{ij}{\mathrm{\parallel }}_1{p}_1(\phi )$. Now $a_i{a}_j\in L^1$ and Cauchy–Schwarz gives $\mathrm{\mid }\widehat{a_i{a}_j}(k)\mathrm{\mid }\le \mathrm{\parallel }{a}_i{\mathrm{\parallel }}_2\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_2$, hence the Euclidean tensor norm of this coefficient is at most $C_d\mathrm{\parallel }a{\mathrm{\parallel }}_2^2$. The unprojected divergence coefficient is $b_k=i{\sum }_j{\kappa }_{k,j}\widehat{a_{\cdot }{a}_j}(k)$ and has size at most $C_d\mathrm{\mid }{\kappa }_k\mathrm{\mid }\mathrm{\parallel }a{\mathrm{\parallel }}_2^2$.

The square of the unprojected negative norm is bounded by $C_d\mathrm{\parallel }a{\mathrm{\parallel }}_2^4{\sum }_{k\in {\mathbb{Z}}^d}(1+L^2\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^{-r}\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2$. For lattice shell $m\le \mathrm{\mid }k\mathrm{\mid }<m+1$, the number of points is at most $C_d{m}^{d-1}$, while each summand is at most $C_{L,r}{m}^{2-2r}$ for $m\ge 1$; the shell series is dominated by ${\sum }_m{m}^{d+1-2r}$, which converges because $r>d\mathrm{/}2+1$. The zero mode contributes zero. Therefore $b$ defines an element of $H^{-r}$. Its distribution agrees with the original divergence: rectangular Fourier sums of a smooth test converge with their first derivatives uniformly, so integration against the $L^1$ tensor passes to that limit and gives precisely the coefficient pairing of step 1.1. The extension $P$ defined there applies to this distribution and is a contraction; taking square roots proves the stated projected bound.