Mildly ill-posed Sobolev q=2 detection rate

Let $m$ be the largest active index, so $A{m}^{2\alpha }<1\le A(m+1{)}^{2\alpha }$ and $(m+1{)}^{-2\alpha }\le A<m^{-2\alpha }$. For $j\le m\mathrm{/}2$, $1-A{j}^{2\alpha }\ge 1-2^{-2\alpha }>0$. For any fixed $p>0$, ${\sum }_{j\le m}{j}^p\le m^{p+1}$ and ${\sum }_{m\mathrm{/}4\le j\le m\mathrm{/}2}{j}^p\ge c_p{m}^{p+1}$ for all sufficiently large $m$, by counting at least $m\mathrm{/}8$ terms each at least $(m\mathrm{/}4{)}^p$. Applying these bounds with $p=4\beta$ and $p=4\beta +2\alpha$ gives $J_1\asymp m^{4\beta +1},J_0\asymp m^{4\beta +1},J_2=A{\sum }_{j\le m}{j}^{4\beta +2\alpha }(1-A{j}^{2\alpha })\asymp m^{4\beta +1}\mathrm{.}$ Thus [F1] yields $r^2\asymp A\asymp m^{-2\alpha }$ and $u_{\epsilon }^2\asymp {\epsilon }^{-4}{r}^4{m}^{-(4\beta +1)}$.

Since $m\asymp r^{-1\mathrm{/}\alpha }$, the last expression becomes $u_{\epsilon }^2\asymp {\epsilon }^{-4}{r}^{4+(4\beta +1)\mathrm{/}\alpha }=(r\mathrm{/}{\epsilon }^{4\alpha \mathrm{/}(4\alpha +4\beta +1)}{)}^{4+(4\beta +1)\mathrm{/}\alpha }$. Hence radii with ratio tending to zero give $u_{\epsilon }\to 0$, and [F1] gives the indistinguishability limits. For a ratio tending to infinity, if $r_{\epsilon }$ does not tend to zero, replace it by ${\bar{r}}_{\epsilon }=\sqrt{r_{\epsilon }{r}_{\epsilon }^{\ast }}$: nonempty alternatives satisfy $r_{\epsilon }\le a_1^{-1}$, so ${\bar{r}}_{\epsilon }\to 0$, ${\bar{r}}_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$, and ${\mathrm{\Theta }}_2(r_{\epsilon })\subseteq {\mathrm{\Theta }}_2({\bar{r}}_{\epsilon })$. The consistent tests from [F1] at ${\bar{r}}_{\epsilon }$ therefore also work at $r_{\epsilon }$. This proves the stated separation rate.