11.44.4 A norm defined by the critical scalarA20C27AA8420

Proof. If a potential $v$ of minimal type satisfies $\mathrm{MA}_\theta (v)\geq d g\, \mathrm{d}V$, then $d\leq c(g)$, since otherwise comparison with a critical potential for $g$ contradicts part (1) of corollary 15.13.3. Hence $g\leq h$ implies $c(g)\geq c(h)$. Uniqueness of the critical scalar also gives $c(tg)=c(g)/t$ for $t>0$. These facts prove that $N$ is monotone and absolutely homogeneous, and positivity follows from $0<c(g)<\infty $.

For the triangle inequality, let $a,b\geq 0$ be nonzero, put $A=N(a)$ and $B=N(b)$, and choose critical potentials $v_ a,v_ b$ with measures $(a/A)^ n\mathrm{d}V$ and $(b/B)^ n\mathrm{d}V$. On every relatively compact subset of $\Omega $, (5.28.9) gives

where both potentials are locally bounded. Exhausting $\Omega $ gives the inequality on $X$, since $X\setminus \Omega $ has zero smooth volume. The averaged potential has minimal type, so the first paragraph gives $N(a+b)\leq A+B$. The cases where $a$ or $b$ vanishes are trivial, and for real functions we use $|a+b|\leq |a|+|b|$ and monotonicity.

Next, apply lemma 15.14.1 with $\eta =V_\theta $ to nonnegative densities of $L^ q$ norm at most one. It gives a uniform $m>0$ and subsolutions of minimal type, hence $c(g)\geq m$ for each such nonzero density. For $a\ne 0$, take $g=|a|^ n/\| a\| _\ell ^ n$, which has $L^ q$ norm one. Homogeneity gives $N(a)\leq m^{-1/n}\| a\| _\ell $, which is the first bound in (11.44.6), and the reverse triangle inequality gives the second. Applying the second bound to $g^{1/n},h^{1/n}$ and using $|g^{1/n}-h^{1/n}|^ n\leq |g-h|$ proves the last assertion. $\square$