PomeProposition 11.46.17.F4F876167D51 The $C^1$ threshold $\delta =1/2$ and $C^2$ threshold $\delta =1$ are sharp. When $2\delta <1$, the stated Hölder exponent is also sharp. Nontrivial probability-density perturbations with smoothly flat residuals exist.
Proposition 11.46.17F4F876167D51
Proof. Choose smooth nonnegative radial cutoffs $\chi _0,\chi _1$ with disjoint supports: $\chi _0=1$ near $r=0$ and vanishes outside a finite small ball; $\chi _1$ is nonzero in a finite annulus. Put
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11.46.18
\begin{equation} B=\frac{\int r^\delta \chi _0q_\alpha \kappa ^ k}{\int \chi _1q_\alpha \kappa ^ k},\qquad q_ c=q_\alpha \bigl [1+c(r^\delta \chi _0-B\chi _1)\bigr ]. \end{equation}
For sufficiently small $c>0$, $q_ c$ is positive, smooth off $o$ and along the hyperplane, and has total mass exactly one. It is comparable to $q_\alpha $ near $o$, hence is in no $L^ p$, $p>1$. Theorem 11.46.8 gives a compact solution in the prescribed ordinary type. Near $o$ its central flux difference is exactly $c\int _0^ r s^\delta (Q_\alpha ^ k)'\, \mathrm{d}s$. The substitution $s=r\mathrm{e}^{-y}$ and dominated convergence give
\[ \int _0^ r s^{\delta -1}(1-\log s)^{-1-b_\alpha }\, \mathrm{d}s \sim \delta ^{-1}r^\delta L^{-1-b_\alpha }. \]
Using (11.46.6) and the $k$th-root expansion yields
\begin{align*} R_ c'(r)& \sim \frac{c\alpha (1-\alpha )}\delta r^{\delta -1}L^{\alpha -2},\\ R_ c(r)-R_ c(0)& \sim \frac{c\alpha (1-\alpha )}{\delta ^2} r^\delta L^{\alpha -2}. \end{align*}
For $\delta <\frac12$, the radial gradient $2\sqrt r\, R_ c'$ diverges, ruling out $C^1$ extension. For $\delta <1$, a tangential real Hessian eigenvalue $2R_ c'$ diverges, ruling out $C^2$ extension. The annular compensation changes no central flux. For $2\delta <1$ the displayed asymptotic also rules out any larger Hölder exponent. The critical derivative thresholds in Theorem 11.46.8 are therefore sharp for this family.
Replacing $r^\delta \chi _0$ in (11.46.18) by $\mathrm{e}^{-1/r}\chi _0$ and adjusting $B$ by the same integral formula gives a nonzero mass-preserving perturbation satisfying every flat-error hypothesis. Proposition 11.46.15 then supplies a smoothly flat residual across the pole. $\square$
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