Pome71BC8ED98CA7A positive exponent determines a unique solution in the relative class. At exponent zero, the equation determines its positive scalar, but normalized potentials may still differ, as in corollary 15.37.34.
Theorem 10.31.3.2568C6F1B2F7 For every nonzero finite non-pluripolar measure $\mu $ on $X$:
There are $u\in \mathcal E_\varphi $ and a unique $c>0$ such that $\mathrm{MA}_\theta (u)=c\mu $.
For each $\lambda >0$ there is a unique $u\in \mathcal E_\varphi $ such that $\mathrm{MA}_\theta (u)=\mathrm{e}^{\lambda u}\mu $.
If $\mu =f\, \mathrm{d}V$ with $0\leq f\in L^ p(\mathrm{d}V)$ for some $p>1$, the solutions in both parts may be taken with $u\sim \varphi $.
Part (1) is theorem 10.31.45, part (2) is theorem 10.31.44, and the assertion for $L^ p$ densities is corollary 10.31.56.
The standing assumptions also give $v_-(\omega )>0$. Indeed, choose $C>0$ with $\theta \leq C\omega $. Since $\varphi \preceq 0$, theorem 13.17.1 gives
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.