A whole-space Fourier splitting for compactly supported H(curl) fields

Since $U$ is supported in $A$, Holder gives $\mathrm{\parallel }U{\mathrm{\parallel }}_1\le \mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }U{\mathrm{\parallel }}_2$, and therefore $\mathrm{\mid }\widehat{U}(\xi )\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }U{\mathrm{\parallel }}_2$. For $\xi \mathrm{\ne }0$ define $\widehat{p}(\xi )=\frac{\xi \cdot \widehat{U}(\xi )}{2\pi i\mathrm{\mid }\xi {\mathrm{\mid }}^2},\widehat{w}(\xi )=\widehat{U}(\xi )-\frac{\xi (\xi \cdot \widehat{U}(\xi ))}{\mathrm{\mid }\xi {\mathrm{\mid }}^2}\mathrm{.}$ Set both values at zero arbitrarily, for instance zero. The multiplier defining $\widehat{w}$ is an orthogonal projection in ${\mathbb{C}}^3$, so $\mathrm{\parallel }\widehat{w}{\mathrm{\parallel }}_2\le \mathrm{\parallel }\widehat{U}{\mathrm{\parallel }}_2$, and $2\pi i\xi \widehat{p}$ is the complementary orthogonal projection of $\widehat{U}$. At $\mathrm{\mid }\xi \mathrm{\mid }<1$, $\mathrm{\mid }\widehat{p}{\mathrm{\mid }}^2\le (4{\pi }^2{)}^{-1}\mathrm{\mid }A\mathrm{\mid }\mathrm{\parallel }U{\mathrm{\parallel }}_2^2\mathrm{\mid }\xi {\mathrm{\mid }}^{-2}$, whose integral is finite in dimension three; at $\mathrm{\mid }\xi \mathrm{\mid }\ge 1$ it is bounded by $(4{\pi }^2{)}^{-1}\mathrm{\mid }\widehat{U}{\mathrm{\mid }}^2$. Thus $p\in H^1$ and $\mathrm{\parallel }p{\mathrm{\parallel }}_{H^1}\le C_A\mathrm{\parallel }U{\mathrm{\parallel }}_2$ by [F1]. This is the essential low-frequency check; an arbitrary nonsupported $L^2$ field would not give it.

The vector identity $\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{\mid }\widehat{w}{\mathrm{\mid }}^2=\mathrm{\mid }\xi \times \widehat{U}{\mathrm{\mid }}^2$ and the multiplier identity for curl give $\int 4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{\mid }\widehat{w}{\mathrm{\mid }}^2=\mathrm{\parallel }\mathrm{curl}U{\mathrm{\parallel }}_2^2$. Hence $w\in H^1$, with the asserted derivative norm. The displayed Fourier definitions give $U=w+\mathrm{\nabla }p$, $\xi \cdot \widehat{w}=0$, and $\xi \times \widehat{w}=\xi \times \widehat{U}$, so all asserted distributional identities follow by Fourier injectivity. Finally $\widehat{U}(-\xi )=\overline{\widehat{U}(\xi )}$ for real $U$. The two definitions have the corresponding conjugate symmetry (including the sign of $i$ in $\widehat{p}$); uniqueness of the inverse transform therefore gives real $p,w$. Every operation is linear, and [F1] supplies all norm claims.