H1 logarithmic stability from centered three-dimensional NtD data

Put $N=\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}$ and $\kappa =s-3\mathrm{/}2>0$. For $b\ge 1$, the high-frequency contribution satisfies ${\int }_{\mathrm{\mid }\xi \mathrm{\mid }>b}⟨\xi {⟩}^{-1}\mathrm{\mid }\widehat{Q}(\xi ){\mathrm{\mid }}^{2}d\xi \le C{b}^{-2\kappa }\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{s-2}}^2\le C{b}^{-2\kappa }$. Indeed the ratio of the two Fourier weights is $⟨\xi {⟩}^{-1-2(s-2)}=⟨\xi {⟩}^{-2\kappa }$.

For $t\ge \max \{T,b\}$, [F2] gives the pointwise inequality $\mathrm{\mid }\widehat{Q}(\xi )\mathrm{\mid }\le C\delta e^{Ct}+(C\mathrm{/}t)\mathrm{\parallel }{e}^{-i\xi \cdot x}Q{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}$ for $\mathrm{\mid }\xi \mathrm{\mid }\le b$. The right side is measurable by the nonnegative Fourier integral in [F3]; thus this inequality can be integrated without selecting a measurable family of CGO phases. Square it using $(u+v{)}^2\le 2u^2+2v^2$, multiply by $⟨\xi {⟩}^{-1}$ and integrate. Since ${\int }_{\mathrm{\mid }\xi \mathrm{\mid }\le b}⟨\xi {⟩}^{-1}d\xi \le C{b}^2$, [F3] yields a low-frequency bound $C{b}^2{\delta }^2{e}^{Ct}+C{b}^2{t}^{-2}{N}^2$, after increasing the exponential constant. Combining with step 1.1 gives $N^2\le C{b}^{-2\kappa }+C{b}^2{\delta }^2{e}^{Ct}+C{b}^2{t}^{-2}{N}^2$. This uses the integrated modulated estimate, not an unmodulated pointwise Sobolev pairing.

Choose a fixed $K\ge \max \{T,1\}$ large enough that $C\mathrm{/}{K}^2\le 1\mathrm{/}2$ for the last constant in step 2.1, and set $t=Kb$. Absorb that term into the left side and take square roots. Increasing fixed constants and using $b\le e^b$ for $b\ge 1$ gives $N\le C{b}^{-\kappa }+C\delta e^{Db}$ for some $D>0$ depending only on the fixed class. If $\delta =0$, let $b\to \mathrm{\infty }$ to obtain $N=0$. If $0<\delta <e^{-2D}$, take $b=\log (1\mathrm{/}\delta )\mathrm{/}(2D)\ge 1$. Then $\delta e^{Db}=\sqrt{\delta }$, which is bounded by a constant times $[\log (1\mathrm{/}\delta ){]}^{-\kappa }$ because $e^{-x\mathrm{/}2}{x}^{\kappa }$ is bounded for $x\ge 1$. Hence $N\le C[\log (1\mathrm{/}\delta ){]}^{-\kappa }$ for all sufficiently small positive $\delta$.

By [F1] and the Fourier weight inequality, $\mathrm{\parallel }{\sigma }_1-{\sigma }_2{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1}}\le CN$. This proves the claimed small-noise rate and, if $\delta =0$, equality of the conductivities. For larger positive $\delta$, the uniform bound $\mathrm{\parallel }{a}_1-a_2{\mathrm{\parallel }}_{H^1}\le 2A$ and the positive lower bound for $[\log (3+{\delta }^{-1}){]}^{-\kappa }$ on that region prove the same displayed estimate after increasing $C$. On a sufficiently small noise interval, $\log (3+{\delta }^{-1})$ is comparable to $\log (1\mathrm{/}\delta )$, so the two small-noise formulations agree. All constants came from the fixed class.