Severely ill-posed Sobolev q=2 detection rate

Let $m$ be the largest active index, so $(m+1{)}^{-2\alpha }\le A<m^{-2\alpha }$, and write $\mathrm{\ell }=m-j$. For $1\le \mathrm{\ell }\le m\mathrm{/}2$, the mean-value theorem applied to $x^{2\alpha }$ on $[1\mathrm{/}2,1]$ gives $c_{\alpha }\frac{\mathrm{\ell }+1}{m}\le 1-A(m-\mathrm{\ell }{)}^{2\alpha }\le C_{\alpha }\frac{\mathrm{\ell }+1}{m}\mathrm{.}$ For $\mathrm{\ell }>m\mathrm{/}2$, the profile factor is at most one. Summing against ${\sigma }_j^4=e^{4\beta m}{e}^{-4\beta \mathrm{\ell }}$, the near terms are bounded by the convergent sums ${\sum }_{\mathrm{\ell }\ge 0}(\mathrm{\ell }+1)e^{-4\beta \mathrm{\ell }}$ and ${\sum }_{\mathrm{\ell }\ge 0}(\mathrm{\ell }+1{)}^2{e}^{-4\beta \mathrm{\ell }}$, and the far tail is exponentially smaller than $m^{-2}$. The $\mathrm{\ell }=1$ term supplies matching lower bounds. Thus $J_1\asymp J_2\asymp e^{4\beta m}\mathrm{/}m,J_0\asymp e^{4\beta m}\mathrm{/}{m}^2\mathrm{.}$ For $J_2$, its extra factor $A{a}_j^2$ is at most one and is bounded below by a positive constant at $j=m-1$. Consequently [F1] gives $r^2\asymp A\asymp m^{-2\alpha }$ and $u_{\epsilon }^2\asymp {\epsilon }^{-4}{r}^4{e}^{-4\beta m}$.

The cutoff relation implies $m\asymp r^{-1\mathrm{/}\alpha }$. For $r_{\epsilon }^{\ast }=(\log {\epsilon }^{-1}\mathrm{/}\beta {)}^{-\alpha }$, if $r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to 0$, then $m\mathrm{/}(\log {\epsilon }^{-1}\mathrm{/}\beta )\to \mathrm{\infty }$, so the expression in step 1.1 tends to zero. If the ratio tends to infinity while $r_{\epsilon }\to 0$, then $m=o(\log {\epsilon }^{-1})$ and $\log u_{\epsilon }^2=4\log {\epsilon }^{-1}-4\beta m+4\log r_{\epsilon }+O(1)\to \mathrm{\infty }$, because $\log (1\mathrm{/}{r}_{\epsilon })=O(\log \log {\epsilon }^{-1})$. If $r_{\epsilon }$ does not tend to zero, reduce to ${\bar{r}}_{\epsilon }=\sqrt{r_{\epsilon }{r}_{\epsilon }^{\ast }}$. Nonempty alternatives satisfy $r_{\epsilon }^2\le a_1^{-2}$ by the two q=2 constraints, so ${\bar{r}}_{\epsilon }\to 0$ and ${\bar{r}}_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$. The inclusion ${\mathrm{\Theta }}_2(r_{\epsilon })\subseteq {\mathrm{\Theta }}_2({\bar{r}}_{\epsilon })$ transfers consistent tests to the original class. By [F1], these limits prove the displayed rate.