11.44 Selection by a vanishing exponential parameterDB96849B14F8

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. $\square$

Proof. The formula for $\kappa $ follows from $N(a\mathrm{e}^{r/n})=c(\mathrm{e}^ rf)^{-1/n}$. The Hahn–Banach theorem gives a linear functional that norms $N$ at $a$. By (11.44.6) it is continuous on $L^\ell $, and hence is represented by some $\eta _0\in L^{\ell '}$. For $b\geq 0$, testing on $\operatorname {sgn}(\eta _0)b$ gives

Since $a\geq 0$ and $\int _ Xa\eta _0\, \mathrm{d}V=A=N(a)$, the same equality holds with $|\eta _0|$ in place of $\eta _0$. Monotonicity of $N$ extends the domination to real $b$, so $|\eta _0|\in \mathcal H_ a$. The defining conditions of $\mathcal H_ a$ are weakly closed and give $\| \eta \| _{\ell '}\leq C_ q$, so reflexivity proves weak compactness.

For nonzero nonnegative $b_0,b_1$, the weighted arithmetic–geometric mean inequality applied to $b_ j/N(b_ j)$, together with monotonicity and the triangle inequality, gives

Taking $b_ j=a\mathrm{e}^{r_ j/n}$ proves convexity of $\kappa $. Monotonicity and the translation rule follow from the corresponding properties of $N$, and together they imply the Lipschitz bound.

Every $\nu _\eta $ is a probability and satisfies

Jensen's inequality gives $\kappa (r)\geq \int _ Xr\, \mathrm{d}\nu _\eta $, which proves one inclusion in (11.44.10). The map $\eta \mapsto \nu _\eta $ is weakly continuous, since $ar\in L^\ell $ for $r\in C(X)$.

Conversely, let $\nu \in \mathcal P_ f$ and put $L(r)=\int _ Xr\, \mathrm{d}\nu $. Differentiating the curve $t\mapsto a\mathrm{e}^{tr/n}$ in $L^\ell $ at $t=0$ gives

Here the one-sided derivative $N'_+(a;\cdot )$ of the continuous convex function $N$ exists, and the error estimate follows from its Lipschitz bound. The inequality $L\leq \kappa $, applied to $tr$ and $-tr$, therefore gives

Thus $T(ar)=A L(r)$ defines a continuous linear functional on the space of products $ar$, $r\in C(X)$. These products are dense in

indeed, it suffices to approximate $b/a$ in $L^\ell (a^\ell \mathrm{d}V)$ by continuous functions. If $ar\geq 0$, then $ar=ar_+$ almost everywhere, and the first inequality in (11.44.12) gives $L(r-r_+)=0$, so $T(ar)\geq 0$. Approximating nonnegative elements of $S_ a$ by products $ar$ with $r\geq 0$ continuous shows that the continuous extension of $T$ to $S_ a$ is positive.

Represent this extension by some $\eta \geq 0$ in $L^{\ell '}$ vanishing on $\{ a=0\} $. Since $T(a)=A$, extending the second inequality in (11.44.12) to $S_ a$ and applying it to $b-a$ gives $T(b)\leq N(b)$ for $b\in S_ a$. For general $b\in L^\ell $,

Thus $\eta \in \mathcal H_ a$, and $\nu _\eta $ agrees with $\nu $ on all continuous functions. This proves the representation (11.44.10), and weak compactness of $\mathcal P_ f$ follows from that of $\mathcal H_ a$.

The bound (11.44.11) follows from Hölder's inequality and $\| \eta \| _{\ell '}\leq C_ q$. Under the last hypothesis, the norming inequality gives $\int _ X\mathrm{e}^{tz/n}\, \mathrm{d}\nu \leq 1$. Since $z$ is $\nu $-integrable by (11.44.11), Jensen's inequality implies $\int _ Xz\, \mathrm{d}\nu \leq 0$. $\square$

Proof. The centered potential satisfies $\mathrm{MA}_\theta (w_\lambda )=c\mathrm{e}^{\lambda w_\lambda }f\, \mathrm{d}V$, while $\psi \leq 0$ gives $\mathrm{MA}_\theta (\psi )\geq c\mathrm{e}^{\lambda \psi }f\, \mathrm{d}V$. Part (2) of corollary 15.13.3 therefore gives $\psi \leq w_\lambda $, hence $k_\lambda \geq 0$. For each fixed $\lambda $, the density $\mathrm{e}^{\lambda w_\lambda }f$ lies in $L^ p$, and its critical scalar is $c$. Therefore

The moment estimate above gives $aw_\lambda \in L^\ell $, so proposition 11.44.9, with $z=w_\lambda $ and $t=\lambda $, gives the asserted integral inequalities. Fix one $\nu \in \mathcal P_ f$. Then

by (11.44.13), with $K$ independent of $\lambda $. The normalization $\sup _ Xv_\lambda =0$ gives $v_\lambda \leq V_\theta $, and subtracting $k_\lambda $ from the lower bound $\psi \leq w_\lambda $ proves (11.44.16).

The potentials $v_\lambda =u_\lambda -\sup _ Xu_\lambda $ form the normalized exponential family of proposition 15.17.3, which gives subsequences converging locally uniformly on $\Omega $ and in $L^1(X)$ to potentials of minimal type. Extracting a convergent subsequence of $k_\lambda \in [0,K]$ gives the same compactness for $w_\lambda =v_\lambda +k_\lambda $. The bound $|w_\lambda |\leq |\psi |+K$ also gives $L^1$ convergence along every subsequence converging locally uniformly on $\Omega $, since $X\setminus \Omega $ has zero volume. $\square$

Proof. By part (2) of corollary 15.13.3, $w_\lambda $ is the greatest admissible potential for $r=0$. If $m=\min _ Xr$ and $M=\max _ Xr$, then $w_\lambda -M/\lambda $ is admissible for $r$, and translating any potential admissible for $r$ by $m/\lambda $ makes it admissible for $r=0$. Hence

The mixed density inequality on $\Omega $ and the arithmetic–geometric mean inequality show that a convex combination of potentials admissible for $r_0$ and $r_1$ is admissible for the same convex combination of the forcing terms. Approximating the two suprema at $x$ proves concavity. Inclusion of the admissible classes proves monotonicity, and translation by constants proves the translation rule and the Lipschitz bound.

Now apply the Hahn–Banach construction in the proof of theorem 11.34.2 to $-\lambda \mathcal F_{\lambda ,x}$ at zero. This functional is finite, continuous, convex and increasing, and it satisfies $-\lambda \mathcal F_{\lambda ,x}(r+b)=-\lambda \mathcal F_{\lambda ,x}(r)+b$ for constants $b$. The same tests with nonnegative functions and constants show that its subgradient is a probability measure $\nu _{\lambda ,x}$. The resulting supporting inequality, combined with $s(x)\leq \mathcal F_{\lambda ,x}(r)$, gives (11.44.20); it is not necessary that the supremum be attained. $\square$

of theorem 11.44.2. By proposition 11.44.9 and the moment bound following it, $\mathcal P_ f$ has the stated properties, and lemma 11.44.15 gives precompactness of the centered potentials. It remains to identify every cluster potential.

Step 1: limits of the forcing probabilities. Fix $x\in \Omega $. For $r\in C(X)$, choose a critical potential $\psi _ r$ for $\mathrm{e}^ r f$ with supremum zero. By the definition of $\kappa $,

Since $\psi _ r\leq 0$, the potential $\psi _ r-\kappa (r)/\lambda $ is admissible in (11.44.18). Thus

For fixed $x$ and $r$, the values $w_\lambda (x)$ are uniformly bounded by lemma 11.44.15, and $\psi _ r(x)$ is finite. Hence every weak cluster point of $\nu _{\lambda ,x}$ as $\lambda \downarrow 0$ belongs to $\mathcal P_ f$.

Step 2: a pointwise comparison. Let $s$ satisfy (11.44.3). The function $-\lambda s$ is lower semicontinuous and bounded below on $X$, with the value $+\infty $ allowed, so metric inf-convolution and truncation give continuous functions $r_ j\uparrow -\lambda s$. Since $r_ j\leq -\lambda s$, the potential $s$ is admissible for every $r_ j$, and (11.44.20) bounds the integrals of the $r_ j$ from above. Monotone convergence, after subtracting a common lower bound, shows that $s$ is integrable against $\nu _{\lambda ,x}$ and gives

Step 3: cluster potentials belong to the class (11.44.3). Take a sequence $\lambda _ j\downarrow 0$ with $w_{\lambda _ j}\to z$ locally uniformly on $\Omega $ and in $L^1(X)$. The bounds (11.44.16) show that $z$ has minimal type, and $z$ is continuous on $\Omega $. Since $c\mathrm{e}^{\lambda _ j w_{\lambda _ j}}f\to cf$ in $L^ p$, by dominated convergence and the common bound $w_\lambda \leq K$, continuity of local bounded products gives $\mathrm{MA}_\theta (z)=cf\, \mathrm{d}V$ on $\Omega $, and then on $X$ because neither side charges $X\setminus \Omega $.

For each $\nu \in \mathcal P_ f$, the bound $|w_\lambda |\leq |\psi |+K$ has a $\nu $-integrable right-hand side, and $\nu $ charges only $\Omega $. Dominated convergence in $\int _ Xw_\lambda \, \mathrm{d}\nu \leq 0$ therefore gives $\int _ Xz\, \mathrm{d}\nu \leq 0$. Hence $z$ belongs to the class (11.44.3).

Step 4: $z$ is the greatest element of the class. For any $s$ in the class and fixed $x\in \Omega $, pass to a further subsequence with $\nu _{\lambda _ j,x}\rightharpoonup \nu $, so that $\nu \in \mathcal P_ f$ by Step 1. The potential $s$ is upper semicontinuous and bounded above on $X$, so

For an unbounded $s$, this follows from the portmanteau theorem applied to $\max (s,-M)$, followed by $M\to \infty $; the final integral is finite by (11.44.11). Passing to the limit in (11.44.22) gives $z(x)\geq s(x)$. This holds for every $x\in \Omega $, and hence on $X$ by lemma 5.24.3. Thus every cluster potential is the greatest element $w$ of the class, and precompactness gives convergence of the whole family.

Step 5: the normalizations. The maximum of $\int _ Xw\, \mathrm{d}\nu $ over $\mathcal P_ f$ is attained, because under (11.44.10) it is the maximum of the weakly continuous functional $\eta \mapsto A^{-1}\int _ Xaw\eta \, \mathrm{d}V$ on $\mathcal H_ a$; note that $aw\in L^\ell $. If the maximum were negative, a small positive constant translate of $w$ would still satisfy (11.44.3), contrary to greatestness. Thus it is zero.

Finally, $k_\lambda \to \sup _ Xw$. Indeed, from any subsequence choose a further one with $k_\lambda \to k$. Then $v_\lambda =w_\lambda -k_\lambda $ converges in $L^1$ to $w-k$, and theorem 5.25.2 gives $\sup _ X(w-k)=0$. Thus $k=\sup _ Xw$ for every cluster point of $(k_\lambda )$. Subtracting the constants proves the convergence of $u_\lambda -\sup _ Xu_\lambda $. If $\theta $ is Hermitian, we may take $\rho =0$ and $\Omega =X$, so that local uniform convergence is uniform convergence. $\square$