11.31.5 A metric on normalized potentials523982F48F36

Proof. Symmetry and the triangle inequality follow directly from the definition, and $d_{\gamma ,d}\leq 2M_{\theta ,\gamma }$ because both trace measures have mass $M_{\theta ,\gamma }$. Let $\mathcal F$ denote the family of test functions in the supremum. After rescaling, $\mathcal F$ contains a positive multiple of every Lipschitz function, and Lipschitz functions are uniformly dense in $C(X)$. Hence $\mathcal F$ separates finite measures, and together with the injectivity of $\mathcal T_\gamma $ this shows that $d_{\gamma ,d}$ is a metric. The same density, together with the common mass bound, shows that convergence in $d_{\gamma ,d}$ implies weak convergence of the trace measures. Conversely, $\mathcal F$ is uniformly bounded and equicontinuous, so the Arzelà–Ascoli theorem gives a finite uniform $\epsilon $-net in $\mathcal F$ for every $\epsilon >0$. Weak convergence on the finitely many functions of the net, and the bound $2M_{\theta ,\gamma }\epsilon $ for the remaining error, give convergence in $d_{\gamma ,d}$.

Thus $d_{\gamma ,d}$ is the pullback by the homeomorphism of theorem 11.31.2 of a metric for the weak topology. The resulting metric space is compact, hence complete. For two choices of the auxiliary data, both metrics induce the same compact $L^1$ topology, so the identity map is uniformly continuous in both directions. $\square$

Proof. The lower bound follows by duality from the inverse of $A^*$, and the upper bound by smoothing the test functions. Put $w=u-v$, $\sigma =\mathcal T_\gamma (u)-\mathcal T_\gamma (v)$ and $M=M_{\theta ,\gamma }$, and write $\| \sigma \| _{\mathrm{TV}}$ for the total variation of $\sigma $. Then $\int _ Xw\, \mathrm{d}V=0$, $\sigma =\mathrm{dd}^{\mathrm{c}}w\wedge \gamma ^{n-1}$ and $\| \sigma \| _{\mathrm{TV}}\leq 2M$. If $M=0$, theorem 11.31.2 gives $u=v$, and both estimates hold trivially.

For the lower bound, let $g$ be a smooth real function. Since $\ker A=\ker A^*=\mathbb R$, the equation

has a unique smooth solution. For $p>2n$, elliptic $W^{2,p}$ estimates on functions of mean zero and the Sobolev embedding theorem give $\| \phi \| _\infty +\operatorname {Lip}_ d(\phi )\leq C\| g\| _\infty $. Pairing the distributional equation $Aw\, \mathrm{d}V=\sigma $ with $\phi $, and using $\int _ Xw\, \mathrm{d}V=0$, we obtain

Taking the supremum over smooth real $g$ with $\| g\| _\infty \leq 1$ gives $\| w\| _{L^1}\leq C d_{\gamma ,d}(u,v)$.

For the upper bound, let $f$ be any test function in the definition of $d_{\gamma ,d}$. Using a fixed finite coordinate cover, a smooth partition of unity and convolution at scale $0<\epsilon <\epsilon _0$, we obtain smooth functions $f_\epsilon $ satisfying

with constants uniform over this family of test functions. To see the second bound, localize $f$ by the fixed partition of unity. The localized functions have uniformly bounded values and first weak derivatives. In each second derivative of their convolutions, move one derivative onto the mollifier, whose derivative has $L^1$ norm $O(\epsilon ^{-1})$. The lower-order terms of $A^*$ have fixed smooth coefficients and satisfy the same bound. Since $\int _ Xf_\epsilon \, \mathrm{d}\sigma =\int _ XwA^*f_\epsilon \, \mathrm{d}V$, it follows that

If $0<\| w\| _{L^1}<\epsilon _0^2$, choose $\epsilon =\| w\| _{L^1}^{1/2}$. For larger norms use $d_{\gamma ,d}(u,v)\leq 2M$, and for $w=0$ use injectivity. Taking the supremum over $f$ proves the upper bound. $\square$