Theorem 23.1.521394B3372728

Proof. Fix $\kappa $ as in lemma 23.1.19, also with $\kappa <1/100$. Choose a small fixed $c>0$ later and set

Use the cutoffs $\chi $ and $\vartheta $ from the earlier heat-time construction. Then $Z_\eta =F_\eta (\varepsilon ,\cdot )$ for $x\leq 0$, $Z_\eta =0$ for $x\geq x_*$, and

The larger value of $q_0$ changes the integrated positivity defect; we estimate it at this scale. Write $r_-=\max (-r,0)$ for the negative part of a real number $r$.

On $[0,b]$ we have $\vartheta =1$. Write $F=F_\eta $ and put $\sigma =(\log t)_ x$, $h=\eta k_ t$ and $\zeta =\sigma t\partial _{\bar w}\log k_ t/\mu $. The identity (23.1.27) applies to $Z_\eta $ on this interval. The contribution involving $h[1+t(\log k_ t)_{w\bar w}]$ is nonnegative. On $[0,a^{-1}]$,

This upper bound for $t$ is smaller than every power of $\eta $. Thus $\kappa q\chi 't\leq \eta $ and $\sigma ^2t\leq \eta ^2$. Use (23.1.20) with $\kappa \chi ^2$ where $\chi >0$, and $F_ t\geq -\eta $ for the $\chi '$ term. It follows that $[Q[Z_\eta ]]_-\leq C\eta $. Where $\chi =0$ also $\chi '=0$, and $Q[Z_\eta ]=\mu -\mu ^2/(1-\eta +h)>0$, since $x\leq a^{-1}=o(1)$.

On $[a^{-1},b]$, $\sigma =2aq$, $\sigma _ x=-\kappa /\eta $, $\mathrm dx=-\mathrm dq/a$. For $t\leq 1$, (23.1.27) and (23.1.20) give

For $t\geq 1$, (23.1.16) gives instead $[Q[Z_\eta ]]_-\leq C\kappa q^2t^2\mathrm e^{-t/4}$. Put $L=\log (1/\eta )$. Integrating the $q^2t$ term uses $\int _{\sqrt L}^\infty q^2\mathrm e^{-q^2}\, \mathrm dq\leq C\sqrt L\eta $. For the exponential term split at $2\sqrt L$: the initial interval has length $O(\sqrt L)$, and on the rest $\eta ^2\mathrm e^{q^2}/q^2\geq \mathrm e^{q^2/4}$ for small $\eta $. Its remaining integral is uniformly bounded even with upper endpoint infinity. For large time use $\mathrm dx=\sqrt{\eta /(2\kappa )}\, \mathrm dt/(tq)$, $q\leq \sqrt L$, and the integrability of $t\mathrm e^{-t/4}$. Including the constant term and initial interval yields

All constants here are independent of $q_0,c,\eta $ with fixed $\kappa $. The bound is small after choosing $c$, although it need not tend to zero.

Also $\sup [(Z_\eta )_ x]_-=O(\sqrt{\eta L})$: for $t\leq 1$ use $F_ t\geq -\eta $ and $t\sqrt{L+\log (1/t)}\leq \sqrt L$; for $t\geq 1$ use (23.1.16). On $[b,x_*]$, $0\leq q\leq 1$ and $(\mathrm e\eta )^{-1}\leq t\leq \eta ^{-1}$, so differentiated cutoffs and (23.1.16) give

Choose $c$ so that $\ell <1/(10\mathrm e)$. Since $Q[0]=\mu (1-\mu )\geq \mathrm e^{-x_*}/4$ here, this implies $Q[Z_\eta ]>0$ on the cutoff interval. For $x\leq 0$ use $Q[Z_\eta ]\geq \mu (1-\mu /(1-\eta ))>0$, and for $x\geq x_*$ use $Z_\eta =0$. On $[-2,-1]$ the same formula gives $Q[Z_\eta ]\geq 2/27$.

The continuous defect $d_\eta (x)=\max _ w[Q[Z_\eta ]]_-$ is therefore supported in $[0,b]$. Choose a smooth nonnegative $r_\eta $, supported in $(-1/2,b+1/2)$, with $r_\eta >d_\eta $ on $[0,b]$ and $m_\eta :=\int r_\eta \leq \int d_\eta +\eta $. Uniform approximation and a positive bump give this choice for each fixed $\eta $; no uniform derivative bound is needed. Fix a smooth probability bump $\psi $ supported in $(-2,-1)$ and put

The support order implies $-m_\eta \leq g_\eta '\leq 0$; $g_\eta =0$ at the negative end and is constant at the positive end. Furthermore $g_\eta \leq 0$ and $|g_\eta |\leq m_\eta (b+3)=O_ c(\eta ^{-1})$. Shrink the fixed $c$ so that (23.1.53) ensures

for all sufficiently small $\eta $. Now $Q[V_\eta ]=Q[Z_\eta ]+r_\eta -m_\eta \psi >0$ everywhere: use the majorant on $[0,b]$, the margin $2/27$ on $(-2,-1)$, and the established positivity elsewhere. The vertical coefficient is above $1-\eta $. For the horizontal coefficient, derivatives vanish for $x\leq -2$; on $[-2,0]$ use $\mu \geq \mu (-2)>m_\eta $; on $[0,\infty )$ use $\mu \geq 1/2$ and $(Z_\eta )_ x\geq -o(1)$. Thus (23.1.8) gives a smooth globally strictly positive potential.

On $x<-3$, $V_\eta =F_\eta (\varepsilon ,\cdot )$ has fiber mean zero, so $0\leq \sup _ XV_\eta \leq C\eta $ by lemma 23.1.51 and $g_\eta \leq 0$. Set $u_\eta =V_\eta -\sup _ XV_\eta $. Where the cutoff is nonzero $t\geq \varepsilon $, giving

Multiplication by a cutoff between zero and one preserves this lower bound for $Z_\eta $, including where $F_\eta $ is positive.

Mass and depth on the concentration region. Let $K_\eta $ consist of all radial angular directions and

using the small lift near zero in $E$. This lies in the linear transport region, with $\vartheta =1$, $x\asymp _ c\eta ^{-1}\to \infty $, $\sigma =2aq\asymp _ c\eta ^{-2}$, and superexponentially small $t$. Lemma 23.1.51 gives, uniformly,

In (23.1.27) the first $h$ contribution is nonnegative, $\sigma _ xtF_ t=-\kappa t(k_ t-1)\geq -3\kappa $, and the completed square tends to one. Its other negative terms tend to zero. Thus $Q[Z_\eta ]\geq 1/4$ for small $\eta $. On the core $\psi =0$ and $g_\eta ''=r_\eta \geq 0$, so $Q[V_\eta ]\geq 1/4$. Also

Indeed $(Z_\eta )_ x=\eta \sigma (tk_ t-t)$ and $\mu +g_\eta '>0$ here; the vertical inequality follows from $1-\eta +\eta k_ t$.

In (23.1.43), $\Delta _{V_\eta }=(1+(V_\eta )_{w\bar w})Q[V_\eta ]$. The fiber annulus has area $3t/2$. Integrating the preceding lower bounds and using $\sigma \, \mathrm dx=-2q\, \mathrm dq$ yields

The annular value estimate also gives

As $\eta q_0^2=c\eta ^{-2}+\eta \log (1/\eta )$, this is bounded above and below by fixed positive multiples of $\eta ^{-2}$ on the core. Together with (23.1.54), this proves the asserted mass and depth bounds. All these potentials are bounded: truncations beyond their individual depth equal them everywhere, so the defining extended product equals (23.1.43). $\square$