Example 5.27.244B607387D3F1

Proof. Here $\chi (s)=-(-s)^ a$ is of class $C^2$ on $(-\infty ,-1]$, so the preceding propositions apply with $A=-1$. Moreover, $\chi '(s)=a(-s)^{a-1}$ and $\chi ''(s)=a(1-a)(-s)^{a-2}$, so $\ell =0$. The two propositions give the assertion about the Laplacian and the first-order thresholds. At $a=1/2$, the first-order integral diverges logarithmically; for $p>2$, its exponential factor forces divergence.

For second derivatives, take a smooth point of a component of multiplicity $m$, outside the other components, with a normal coordinate $w_1$ such that $f=w_1^ m$. Write $F(r)=\chi (m\log r)$ with $r=|w_1|$. The real second derivative of $v_ a$ in the unit direction tangent to the circles $|w_1|=r$ equals $F'(r)/r=m\chi '(m\log r)/r^2$. Thus the integral of its absolute value over the normal disk is bounded below, up to a fixed positive factor, by

Hence the classical Hessian of $v_ a$ off the divisor is not locally integrable, whereas local integrability is necessary for $W^{2,1}_{\mathrm{loc}}$. Since smooth coordinate changes preserve this Sobolev class, the argument is complete. $\square$