Theorem 15.23.5.66FFA0C36433 Fix a smooth positive volume form $\mu $ on $X$ and a nonzero nonnegative $f\in L^ p(X,\mu )$, $p>1$. There exist $\kappa >0$ and a normalized minimal-type solution of
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15.23.6
\begin{equation} \mathrm{MA}^{\mathrm{ext}}_\theta (u)=\kappa f\mu , \qquad u=cL+q\circ \pi +C, \end{equation}
where $q$ is Hölder on $Y$. The divisor coefficient is exactly $c$. In (15.23.4), $u-cL=q(y,0)+C+O(|z|^\alpha )$ on smaller charts for some $\alpha >0$, so its limit is constant along each exceptional fiber.
If $\pi _*(f\mu )=Fh^ n$, then $F\in L^ s(Y,h^ n)$ for every
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15.23.7
\begin{equation} 1<s<\frac{rp}{(r-1)p+1}. \end{equation}
For a point center, data bounded above and below by positive constants near $E$ attain the strict threshold $s<n/(n-1)$, and for finite $p$ no strictly larger universal exponent than $np/((n-1)p+1)$ can be guaranteed. If $h$ is Kähler, the class $[\theta ]$ is big and non-nef, has volume $\int _ Yh^ n$, and $\kappa =\int _ Yh^ n/\int _ Xf\mu $; the normalized solution is unique among all normalized solutions of (15.23.6) with minimal ordinary singularities. Under $\mathrm{dd}^{\mathrm{c}}(h^ j)=0$, $1\leq j\leq n-1$, the scalar of the constructed solution has the same integral formula.
Proof.
Integrability and existence. In (15.23.4) the complex Jacobian is $z^{r-1}$. Put $J=\pi ^*h^ n/\mu $, so $J\asymp |z|^{2(r-1)}$ on nested charts and $F\circ \pi =f/J$ off $E$. For $1<s<p$, change of variables and Hölder's inequality give
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15.23.8
\begin{align} \int _ YF^ s h^ n & =\int _ X f^ sJ^{1-s}\mu \\ & \leq \| f\| _{L^ p(\mu )}^ s \left(\int _ XJ^{-(s-1)p/(p-s)}\mu \right)^{(p-s)/p}. \end{align}
The normal complex integral is finite precisely if $(r-1)(s-1)p/(p-s)<1$, which is (15.23.7). Theorem 14.19.2, Theorem 15.5.1 on $(Y,h)$ supply $q$ and $\kappa $ with $(h+\mathrm{dd}^{\mathrm{c}}q)^ n=\kappa Fh^ n$. Use (15.23.3) off $E$ and theorem 13.14.3 across it. Hölder continuity gives $q(\pi (y,z,w))=q(y,0)+O(|z|^\alpha )$ uniformly on smaller charts.
Sharpness of the exponent. For $f$ bounded above and below by positive constants near the exceptional divisor of a point center, $F$ is comparable to $\operatorname {dist}_ h(\cdot ,Z)^{-2(n-1)}$ near $Z$, giving the strict threshold. More generally, $f\asymp |s_ E|^{-2a}$ with $0\leq a<1/p$ belongs to $L^ p$ upstairs, while its pushforward has threshold $s<n/(n-1+a)$. Letting $a\uparrow 1/p$ proves the stated limitation; it makes no finite-$p$ endpoint assertion.
The Kähler case. When $h$ is Kähler, the strict barrier current $(c+\delta )[E]+\pi ^*h-\delta \Theta '_ E$ constructed in proposition 15.23.2 makes $[\theta ]$ big. Its degree on an exceptional-fiber line is $-c$, so it is not nef. The minimal current $c[E]+\pi ^*h$ has non-pluripolar top mass $\int _ Yh^ n$, and the minimal mass theorem identifies this with $\operatorname {vol}([\theta ])$ as recalled in section 25.18. Full-mass uniqueness and mass integration fix the scalar and normalized current as recalled in section 25.21. A normalized solution of minimal ordinary type, possibly discontinuous, still has bounded $q$ by proposition 15.23.2; on every relatively compact set off $E$, deep truncations equal that competitor, so locality in theorem 13.14.3 gives the bounded target equation off $Z$. Both target measures give $Z$ zero mass, so closed uniqueness applies globally. If only the power conditions hold, theorem 11.1.7 fixes the scalar of the constructed solution. Quantitative Hölder bounds depend on $\kappa F$, not on an unnormalized $F$ alone.
$\square$
Pome
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