Example 11.31.8B1E195FF817D

Example 11.31.8.B1E195FF817D Let $X=\mathbb C/(\mathbb Z+\mathrm{i}\mathbb Z)$, with coordinate $z=x+\mathrm{i}y$. Take $\theta =\gamma =\eta $, where $\eta =(\mathrm{i}/2)\mathrm{d}z\wedge \mathrm{d}\bar z$, together with $\mathrm{d}V=\eta $ and the flat distance $d$. For integers $k\geq 1$, set

These smooth potentials have mean zero and satisfy

The function $f_ k=\cos (2\pi kx)/(2\pi k)$ has supremum norm at most one and Lipschitz constant at most one. It follows that

Consequently, the exponent $1/2$ in proposition 11.31.7 cannot be replaced by any larger exponent, even for smooth strictly positive potentials on a fixed curve. Nor can the power one in the lower bound be replaced by a smaller positive power: for $0<t\leq 1$, both $\| tu_1\| _{L^1}$ and $d_{\gamma ,d}(tu_1,0)$ equal $t$ times their positive values at $t=1$.