Radial Carleman estimate

Given: the integer $s\ge 1$, $\tau =s+1$, and $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3\setminus \{0\})$.

Spherical decomposition: $(Y_{lm})$ is a complete orthonormal family of $L^2(S^2,\sigma )$ and the polar unitary $\psi \mapsto (u_{lm})$, $u_{lm}(r)=r{\int }_{S^2}\psi (r\mathrm{\Omega })\overline{Y_{lm}(\mathrm{\Omega })}d\sigma (\mathrm{\Omega })$, identifies $L^2({\mathbb{R}}^3)$ with ${⨁}_{l,m}{L}^2((0,\mathrm{\infty }),dr)$; in the $(l,m)$ channel the Laplacian acts as $-\mathrm{\Delta }{u}_{lm}=-u_{lm}^{\mathrm{\prime }\mathrm{\prime }}+l(l+1)r^{-2}{u}_{lm}$, and the Laplace-Beltrami operator has eigenvalue $-l(l+1)$ on $Y_{lm}$ (Sphere ONB, Sphere Laplacian diagonal domain, Angular radial unitary reduction clauses (a)-(d)).

Write the physical spherical coefficient as ${\varphi }_{lm}(r)=u_{lm}(r)\mathrm{/}r$. Under $t=-\log r$, put $v_{lm}(t)=r^{1\mathrm{/}2}{\varphi }_{lm}(r)=r^{-1\mathrm{/}2}{u}_{lm}(r)$ and $w_{lm}(t)=e^{st}{v}_{lm}(t)$. Direct differentiation of the radial expression gives $\mathrm{\Delta }{\varphi }_{lm}=r^{-5\mathrm{/}2}[{\mathrm{\partial }}_t^2-(l+1\mathrm{/}2{)}^2]v_{lm}$. Since the physical radial measure is $r^{2}dr=r^{3}dt$ after reversing the bounds and $\tau =s+1$, the exact weighted norms are $\mathrm{\parallel }{r}^{-\tau }u{\mathrm{\parallel }}_2^2={\sum }_{l,m}\mathrm{\parallel }{w}_{lm}{\mathrm{\parallel }}_{L^2(dt)}^2$ and $\mathrm{\parallel }{r}^{-\tau +2}\mathrm{\Delta }u{\mathrm{\parallel }}_2^2={\sum }_{l,m}\mathrm{\parallel }[({\mathrm{\partial }}_t-s{)}^2-(l+1\mathrm{/}2{)}^2]w_{lm}{\mathrm{\parallel }}_{L^2(dt)}^2$. Equivalently the first radial integral in the reduced coefficient is $\int r^{-2\tau }\mathrm{\mid }{u}_{lm}(r){\mathrm{\mid }}^{2}dr$, with no second radial Jacobian. The sums follow from angular Parseval in [F1] and nonnegative integration; they need not be finite (Plancherel theorem, algebra).

One-dimensional Fourier calculus: for $w\in C_c^{\mathrm{\infty }}(\mathbb{R})$ the Fourier transform of $({\mathrm{\partial }}_t-s{)}^{2}w$ is $(i\xi -s{)}^2\widehat{w}(\xi )$, and Plancherel gives $\mathrm{\parallel }({\mathrm{\partial }}_t-s{)}^{2}w{\mathrm{\parallel }}_{L^2(dt)}^2={\int }_{\mathbb{R}}\mathrm{\mid }i\xi -s{\mathrm{\mid }}^4\mathrm{\mid }\widehat{w}(\xi ){\mathrm{\mid }}^{2}d\xi$; the algebraic identity $\mathrm{\mid }(i\xi -s{)}^2-(l+\frac{1}{2}{)}^2{\mathrm{\mid }}^2=[{\xi }^2+(s-l-\frac{1}{2}{)}^2][{\xi }^2+(s+l+\frac{1}{2}{)}^2]$ holds for real $\xi$ (Plancherel theorem, algebra).

Compactly supported $H^2({\mathbb{R}}^3;\mathbb{C})$ classes admit smooth compact-support mollifications converging in $H^2$, with support in $K+{\bar{B}}_{\epsilon }(0)$ for a mollifier supported in ${\bar{B}}_1(0)$; this is the explicit construction in Compactly supported smooth functions are dense in W^{k,p}(R^n) Proof, fourth and fifth steps, under Countable Choice.