Proposition 5.27.224BF5A61AFC51

Proof. At a smooth point of a component of multiplicity $m_0$, outside the other components, multiplying the defining function of the component by a holomorphic $m_0$-th root of the remaining unit gives a normal coordinate $w_1$ with $f=w_1^{m_0}$. Coordinate changes with bounded derivatives and Jacobians preserve local first-order Sobolev integrability. With $r=|w_1|$ and $s=m_0\log r$, polar integration of the classical gradient gives, up to fixed positive factors,

Such smooth points occur in every neighborhood of $0$. This proves necessity, since changing the finite upper endpoint does not affect convergence.

For sufficiency, put

For small $\varepsilon >0$, these are $C^2$ plurisubharmonic functions decreasing to $v$. By convexity, $|\chi (s)|\leq C(1+|s|)$ for $s\leq A$, so the local exponential integrability of $\log |f|$, recalled in [SGPT], gives convergence in every local $L^ p$ space with $p<\infty $. By the chain rule,

Apply lemma 5.27.21 with the coarea formula, keeping two powers of $|\partial f|$ for the level estimate and bounding the other $p-2$ powers by the first inequality of the lemma. The local integral of $|\nabla v_\varepsilon |^ p$ over $K\cap \{ 0<|f|<r_ K\} $ is then bounded by

With $s=\tfrac 12\log (r^2+\varepsilon ^2)$, we have $\mathrm ds=r\, \mathrm dr/(r^2+\varepsilon ^2)$ and $r\leq (r^2+\varepsilon ^2)^{1/2}$, so this is at most

The integral over the part where $|f|\geq r_ K$ is uniformly bounded as well. Weak compactness in Sobolev spaces and the local $L^ p$ convergence therefore show that $v\in W^{1,p}_{\mathrm{loc}}$. Its weak gradient agrees with the classical one off $Z$, and $Z$ has Lebesgue measure zero, which proves the final assertion. $\square$