N^{2(1-\sigma)}\sum_{k}\int_{0}^{s_{0}}\int_{0}^{s_{0}}\langle\partial\bar{F}_{k}(s_{1}),\partial\bar{F}_{k}(s_{2})\rangle_{\dot{H}^{\sigma}}\,\mathrm{d}s_{1}\,\mathrm{d}s_{2}$
$\displaystyle\lesssim\sum_{k}\int_{0}^{s_{0}}\int_{0}^{s_{0}}\frac{(s_{1}^{\frac{1}{2}}2^{k})^{\sigma}}{\langle
s_{1}^{\frac{1}{2}}2^{k}\rangle}\frac{(s_{2}^{\frac{1}{2}}2^{k})^{\sigma}}{\langle
s_{2}^{\frac{1}{2}}2^{k}\rangle}\alpha_{k}^{2}\frac{\,\mathrm{d}s_{1}}{s_{1}}\frac{\,\mathrm{d}s_{2}}{s_{2}}$
(7.6) $\displaystyle\lesssim\sum_{k}\alpha_{k}^{2}\lesssim\delta^{2}.$

The remaining term

$\tilde{A}_{i}^{rem}\vcentcolon=\int_{0}^{s_{0}}[\tilde{A}^{\ell},\tilde{F}_{i\ell}](s^{\prime})\,\mathrm{d}s^{\prime}$

can in fact be estimated easily in $N^{-\frac{1}{2}}H^{1}(B)$ placing
$\tilde{A}(s^{\prime}),\partial_{x}\tilde{A}(s^{\prime})$ in $L^{\infty}(B)$
and using (7.4).

###  7.2. Proof of Proposition 1.10 Part (2)

Step 1. (Second change of gauge)

Since $U=U(x)$ is independent of $t$ and $s$, $\tilde{A}$ remains in the
temporal-caloric gauge $\tilde{A}_{0}(t,x,0)=0,\tilde{A}_{s}(t,x,s)=0$. In
particular

$\partial_{t}\tilde{A}_{i}(t_{0},0)=\int_{0}^{s_{0}}\partial_{t}\tilde{\mathbf{D}}^{\ell}\tilde{F}_{i\ell}(t_{0},s^{\prime})\,\mathrm{d}s^{\prime}+\partial_{t}\tilde{A}_{i}(t_{0},s_{0}).$

We want to use the bounds

$(Ns^{\frac{1}{2}})^{1-\sigma}\|(s^{\frac{1}{2}}\tilde{\mathbf{D}}_{x})^{m}\tilde{F}_{\alpha\beta}(t_{0},s)\|_{L^{2}(B)}\lesssim\delta$
(7.7)

to place $\partial_{t}\tilde{A}_{i}(t_{0},s_{0})$ in $L^{2}(B)$, but
$\tilde{A}_{\alpha}(\cdot,s_{0})$ is not in the temporal gauge. So we change
the gauge to make it so, letting

$\accentset{\approx}{A}_{\alpha}=V\tilde{A}_{\alpha}V^{-1}-\partial_{\alpha}VV^{-1},\qquad\accentset{\approx}{F}_{\alpha\beta}=V\tilde{F}_{\alpha\beta}V^{-1}$

for $V=V(t,x)$ which makes $\accentset{\approx}{A}_{\alpha}(\cdot,s_{0})$
temporal i.e. by solving the ODE

$\partial_{t}V=V\tilde{A}_{0}(\cdot,s_{0}),\qquad V(t_{0})=Id.$

Now we have

$\accentset{\approx}{A}_{0}(t,s_{0})=0,\qquad\accentset{\approx}{A}_{s}=0$
(7.8)

and, since $V(t_{0})=Id$,

$\accentset{\approx}{A}_{i}(t_{0},s)=\tilde{A}_{i}(t_{0},s),\qquad\accentset{\approx}{F}_{\alpha\beta}(t_{0},s)=\tilde{F}_{\alpha\beta}(t_{0},s)$
(7.9)

Step 2. (Bounds on $\accentset{\approx}{A}_{0}$) From
$\accentset{\approx}{F}_{s0}=\accentset{\approx}{\mathbf{D}}^{\ell}\accentset{\approx}{F}_{\ell
0}$ and (7.8), (7.9) one has

$\accentset{\approx}{A}_{0}(t_{0},s)=\int_{s}^{s_{0}}\tilde{\mathbf{D}}^{\ell}\tilde{F}_{0\ell}(t_{0},s^{\prime})\,\mathrm{d}s^{\prime}$

From this and (2.13), (7.7) one obtains

$(Ns^{\frac{1}{2}})^{1-\sigma}s^{\frac{1}{4}}\|\accentset{\approx}{A}_{0}(t_{0},s)\|_{L^{\infty}(B)}\lesssim\delta.$
(7.10)

From Step 2 in Part (1) we have (7.3) at $t=t_{0}$ for $m=0,1$ which allows to
pass from (7.7) to

$(Ns^{\frac{1}{2}})^{1-\sigma}\|(s^{\frac{1}{2}}\partial_{x})^{m}\tilde{F}_{\alpha\beta}(t_{0},s)\|_{L^{2}(B)}\lesssim\delta,\qquad\
m\leq 2.$ (7.11)

We claim

$\displaystyle\partial_{i}\accentset{\approx}{A}_{0}(t_{0},0)$
$\displaystyle=\int_{0}^{s_{0}}\partial_{i}\partial^{\ell}\tilde{F}_{0\ell}(t_{0},s^{\prime})+\partial_{i}[\tilde{A}^{\ell},\tilde{F}_{0\ell}](t_{0},s^{\prime})\,\mathrm{d}s^{\prime}$
(7.12) $\displaystyle\in N^{\sigma-1}H^{\sigma-1}(B)+L^{2}(B).$

Indeed, the second integral is in $N^{-\frac{1}{2}}L^{2}(B)$ using (7.3) and
(7.11).

To bound the first integral we proceed like in Step 3 from Part (1). Extend
$\tilde{F}$ by $\bar{F}$ from $B\times[0,s_{0}]$ to
$\mathbb{R}^{3}\times[0,s_{0}]$ such that (7.11) holds on $\mathbb{R}^{3}$ for
$\bar{F}$. Then

$(Ns^{\frac{1}{2}})^{1-\sigma}s\|P_{k}\partial_{x}^{2}\bar{F}(s)\|_{\dot{H}^{\sigma-1}}\lesssim\frac{s^{\frac{1}{2}}2^{k\sigma}}{\left\langle
s^{\frac{1}{2}}2^{k}\right\rangle}\alpha_{k}$

for some $\|(\alpha_{k})\|_{\ell^{2}}\lesssim\delta$. Then

$\displaystyle
N^{2(1-\sigma)}\|\int_{0}^{s_{0}}\partial^{2}\tilde{F}(t_{0},s^{\prime})\,\mathrm{d}s^{\prime}\|_{H^{\sigma-1}(B)}^{2}$
$\displaystyle\lesssim$ (7.13) $\displaystyle
N^{2(1-\sigma)}\sum_{k}\int_{0}^{s_{0}}$
$\displaystyle\int_{0}^{s_{0}}\langle\partial^{2}\bar{F}_{k}(s_{1}),\partial^{2}\bar{F}_{k}(s_{2})\rangle_{\dot{H}^{\sigma-1}}\,\mathrm{d}s_{1}\,\mathrm{d}s_{2}$

This is bounded by (7.6) and thus $\lesssim\delta^{2}$.

Step 3. (Estimate $\partial_{t}\accentset{\approx}{A}_{i}$) Using (7.8) and
$\partial_{t}\accentset{\approx}{A}_{i}(t_{0},s_{0})=\accentset{\approx}{F}_{0i}(t_{0},s_{0})$
write

$\partial_{t}\accentset{\approx}{A}_{i}(t_{0},0)=\int_{0}^{s_{0}}\accentset{\approx}{\mathbf{D}}_{t}\accentset{\approx}{\mathbf{D}}^{\ell}\accentset{\approx}{F}_{i\ell}(t_{0},s^{\prime})-[\accentset{\approx}{A}_{0},\accentset{\approx}{\mathbf{D}}^{\ell}\accentset{\approx}{F}_{i\ell}](t_{0},s^{\prime})\,\mathrm{d}s^{\prime}+\accentset{\approx}{F}_{0i}(t_{0},s_{0}).$

The last term $\accentset{\approx}{F}_{0i}(t_{0},s_{0})$ is in $\delta
L^{2}(B)$. We rewrite the integral, by commuting and using Bianchi (2.3) and
(7.9), as

$\int_{0}^{s_{0}}\tilde{\mathbf{D}}_{x}^{2}\tilde{F}_{0x}+[\tilde{F}_{0\ell},\tilde{F}_{\ell
i}]-[\accentset{\approx}{A}_{0},\tilde{\mathbf{D}}^{\ell}\tilde{F}_{i\ell}]\,\mathrm{d}s^{\prime},\qquad\quad
t=t_{0}$ (7.14)

We bound the integral of $[\tilde{F}_{0\ell},\tilde{F}_{\ell i}]$ and
$[\accentset{\approx}{A}_{0},\tilde{\mathbf{D}}^{\ell}\tilde{F}_{i\ell}]$ in
$N^{-\frac{1}{2}}L^{2}(B)$ using (7.7) together with (2.13), and (7.10)
together with (7.7). The same holds for the integral of
$\tilde{\mathbf{D}}_{x}^{2}\tilde{F}_{0x}-\partial_{x}^{2}\tilde{F}_{0x}$. The
integral of $\partial_{x}^{2}\tilde{F}_{0x}$ is estimated exactly by (7.13).

This proves $\partial_{t}\accentset{\approx}{A}_{i}(t_{0},0)\in
L^{2}(B)+N^{\sigma-1}H^{\sigma-1}(B)$.

Step 4. (Return to $\tilde{A}_{\alpha}$) From the temporal condition
$\tilde{A}_{0}(t,0)=0$ and (7.9) one has

$\partial_{t}\tilde{A}_{i}(t_{0},0)=\tilde{F}_{0i}(t_{0},0)=\accentset{\approx}{F}_{0i}(t_{0},0)=\partial_{t}\accentset{\approx}{A}_{i}(t_{0},0)-\partial_{i}\accentset{\approx}{A}_{0}(t_{0},0)+[\tilde{A}_{i},\accentset{\approx}{A}_{0}](t_{0},0)$

We claim this is in $L^{2}(B)+N^{\sigma-1}H^{\sigma-1}(B)$: the term
$\partial_{t}\accentset{\approx}{A}_{i}(t_{0},0)$ was treated in Step 3, the
term $\partial_{i}\accentset{\approx}{A}_{0}(t_{0},0)$ in (7.12) in Step 2,
while for $[\tilde{A}_{i},\accentset{\approx}{A}_{0}]$ we place both inputs in
$L^{4}(B)$ by Sobolev embedding.

##  8\. Local differences of heat flows which coincide on a ball

The goal of this section is to compare two caloric heat flow solutions which
coincide somewhere initially. Due to infinite speed of propagation they will
not coincide in the future, but one can obtain quantitative local $L^{2}$
estimates for the difference.

Consider two balls $B^{(1)}\subset B$ of radius $\simeq 1$.

######  Proposition 8.1.

Let $A_{t,x,s},A^{\prime}_{t,x,s}:[t_{0},t_{1}]\times
B\times[0,s_{0}]\to\mathfrak{g}$ be two smooth caloric solutions to (dYMHF)
which coincide on a smaller initial cylinder $[t_{0},t_{1}]\times
B^{(1)}\times\\{0\\}$. Let $\chi$ be a spatial bump function supported in the
ball $B^{(1)}$. Let $N^{2}s_{0}=1$. We assume, for all $t\in[t_{0},t_{1}]$ and
$0\leq m\leq 3$, $\varepsilon\ll 1$, the following bounds:

$\displaystyle\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F_{\alpha\beta}(t,s)\|_{L^{\infty}_{s}L^{2}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}([0,s_{0}]\times B)}\lesssim\varepsilon$
(8.1)
$\displaystyle\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x}^{\prime})^{m}F_{\alpha\beta}^{\prime}(t,s)\|_{L^{\infty}_{s}L^{2}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}([0,s_{0}]\times B)}\lesssim\varepsilon$
(8.2) $\displaystyle A^{\prime}_{x}\in\varepsilon
L^{\infty}_{t}L^{\infty}_{s}\left[H^{1}(B)+N^{\sigma-1}H^{\sigma}(B)\right]$
(8.3)

and

$\displaystyle\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}w_{x}(t,s)\|_{L^{\infty}_{s}L^{2}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}([0,s_{0}]\times B)}\lesssim\varepsilon$
(8.4)
$\displaystyle\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x}^{\prime})^{m}w_{x}^{\prime}(t,s)\|_{L^{\infty}_{s}L^{2}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}([0,s_{0}]\times B)}\lesssim\varepsilon$
(8.5)

Then, for any $s\in[0,s_{0}]$, $t\in[t_{0},t_{1}]$ one has

$\displaystyle(Ns^{\frac{1}{2}})^{9(1-\sigma)}\|\chi[F_{\alpha\beta}(t,s)-F^{\prime}_{\alpha\beta}(t,s)]\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim s^{2}\varepsilon$ (8.6)
$\displaystyle(Ns^{\frac{1}{2}})^{8(1-\sigma)}\|\chi[w_{\ell}(t,s)-w^{\prime}_{\ell}(t,s)]\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim s^{\frac{5}{4}}\varepsilon$ (8.7)

It will be clear from the proof that one can in fact obtain arbitrary powers
of $s$ in the conclusion.

Heuristically, these bounds are clear from the decay of the Gaussian kernels
on the spatial scales involved.

Main equations. By caloric solutions we mean that they satisfy
$A_{s}=A^{\prime}_{s}=0$, which implies

$\partial_{s}A_{\ell}=F_{s\ell}=\mathbf{D}^{j}F_{j\ell}$ (8.8)

for each $t\in[t_{0},t_{1}]$. Taking divergence and using the identity
$\mathbf{D}^{\ell}\mathbf{D}^{j}F_{j\ell}=0$ one has

$\partial_{s}\partial^{\ell}A_{\ell}=-[A^{\ell},F_{s\ell}]$ (8.9)

From (6.1) and $A_{s}=0$ we get the schematic equation

$\left(\partial_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\left(\chi
F\right)=2\chi[F,F]-\Delta\chi F-2\nabla\chi\cdot\mathbf{D}_{x}F$ (8.10)

and from (6.2)

$\left(\partial_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\left(\chi\mathbf{D}_{x}F\right)=\chi\mathcal{O}(F,\mathbf{D}_{x}F)-\Delta\chi\mathbf{D}_{x}F-2\nabla\chi\cdot\mathbf{D}^{2}_{x}F$
(8.11)

while from (6.3) we obtain

$\left(\partial_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\left(\chi
w\right)=2\chi[F,w]+\chi\mathcal{O}(F,\mathbf{D}_{x}F)-\Delta\chi
w-2\nabla\chi\cdot\mathbf{D}_{x}w$ (8.12)

Preliminary bounds. From the covariant Gagliardo-Nirenberg inequalities
(2.12), (2.13) and (8.1), (8.2), for $p\in[2,\infty]$ and $m=0,1$ one has

$(Ns^{\frac{1}{2}})^{1-\sigma}\big{(}\|\mathbf{D}_{x}^{m}F(s)\|_{L^{p}(B)}+\|\mathbf{D}_{x}^{{}^{\prime}m}F^{\prime}(s)\|_{L^{p}(B)}\big{)}\lesssim\varepsilon
s^{-\frac{3}{2}(\frac{1}{2}-\frac{1}{p})}s^{-\frac{m}{2}}$ (8.13)

Similarly from (8.4), (8.5)

$(N^{2}s)^{1-\sigma}\big{(}\|\mathbf{D}_{x}^{m}w_{x}(s)\|_{L^{p}(B)}+\|\mathbf{D}_{x}^{{}^{\prime}m}w^{\prime}_{x}(s)\|_{L^{p}(B)}\big{)}\lesssim\varepsilon
s^{-\frac{1}{4}}s^{-\frac{3}{2}(\frac{1}{2}-\frac{1}{p})}s^{-\frac{m}{2}}$
(8.14)

From (8.3) one has

$A^{\prime}_{x}\in\varepsilon
L^{\infty}_{t}L^{\infty}_{s}\left[L^{6}(B)+N^{\sigma-1}L^{q}(B)\right],\qquad\frac{1}{2}-\frac{1}{q}=\frac{\sigma}{3}.$
(8.15)

Notation. Below we use the following notations and identities:

$\displaystyle\delta
A_{x}=A_{x}-A^{\prime}_{x}\qquad\delta\partial^{\ell}A_{\ell}=\partial^{\ell}A_{\ell}-\partial^{\ell}A^{\prime}_{\ell}$
$\displaystyle\delta
F=F-F^{\prime}\qquad\delta\mathbf{D}_{x}^{m}F=\mathbf{D}_{x}^{m}F-(\mathbf{D}_{x}^{{}^{\prime}})^{m}F^{\prime}\qquad\mathbf{D}_{x}\delta
F=\mathbf{D}_{x}(F-F^{\prime})$ $\displaystyle\delta
w=w_{x}-w^{\prime}_{x}\qquad\delta\mathbf{D}_{x}w=\mathbf{D}_{x}w-\mathbf{D}^{\prime}_{x}w^{\prime}$
$\displaystyle\delta\mathbf{D}^{\ell}\mathbf{D}_{\ell}=\mathbf{D}^{\ell}\mathbf{D}_{\ell}-\mathbf{D}^{{}^{\prime},\ell}\mathbf{D}_{\ell}^{\prime}=2[\delta
A^{\ell},\partial_{\ell}\cdot]+[\delta\partial^{\ell}A_{\ell},\cdot]+[A^{\ell},[A_{\ell},\cdot]]-[A^{{}^{\prime}\ell},[A_{\ell}^{\prime},\cdot]]$
(8.16)

The following lemma shows that by slightly reducing the support of the cutoff
functions we can improve the exponents of powers of $s$ for differences of
solutions that coincide at the initial time on a domain.

######  Lemma 8.2.

Let $A_{t,x,s},A^{\prime}_{t,x,s}$ be two solutions like in Proposition 8.1.
Let $\chi_{k},\chi_{k+1}$ be bump functions supported in $B^{(1)}$ such that
$\chi_{k}\equiv 1$ on ${\mathrm{supp}}\chi_{k+1}$. Fix $t\in[t_{0},t_{1}]$.
Let $k\geq 0$ and suppose that for $m=0,1$, $p\in[2,4]$ one has

$\displaystyle(Ns^{\frac{1}{2}})^{(k+1)(1-\sigma)}s^{\frac{m}{2}}\|\chi_{k}\delta\mathbf{D}_{x}^{m}F(s)\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon s^{\frac{k}{4}}$ (8.17)
$\displaystyle\|(Ns^{\frac{1}{2}})^{(k+1)(1-\sigma)}s^{\frac{m+1}{2}}\chi_{k}\delta\mathbf{D}_{x}^{m+1}F\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}[0,s_{1}]}$
$\displaystyle\lesssim\varepsilon s_{1}^{\frac{k}{4}}$ (8.18)
$\displaystyle(Ns^{\frac{1}{2}})^{(k+1)(1-\sigma)}\|\chi_{k}(A_{x}-A_{x}^{\prime})(s)\|_{L^{p}}$
$\displaystyle\lesssim\varepsilon
s^{\frac{k+2}{4}}s^{-\frac{3}{2}(\frac{1}{2}-\frac{1}{p})}$ (8.19)
$\displaystyle(Ns^{\frac{1}{2}})^{(k+2)(1-\sigma)}\|\chi_{k}(\partial^{\ell}A_{\ell}-\partial^{\ell}A^{\prime}_{\ell})(s)\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon s^{\frac{k+1}{4}}$ (8.20)

for any $s,s_{1}\in[0,s_{0}]$. Then one has

$\displaystyle(Ns^{\frac{1}{2}})^{(k+2)(1-\sigma)}s^{\frac{m}{2}}\|\chi_{k+1}\delta\mathbf{D}_{x}^{m}F(s)\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon s^{\frac{k+1}{4}}$ (8.21)
$\displaystyle\|(Ns^{\frac{1}{2}})^{(k+2)(1-\sigma)}s^{\frac{m+1}{2}}\chi_{k+1}\delta\mathbf{D}_{x}^{m+1}F\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}[0,s_{1}]}$
$\displaystyle\lesssim\varepsilon s_{1}^{\frac{k+1}{4}}$ (8.22)
$\displaystyle(Ns^{\frac{1}{2}})^{(k+2)(1-\sigma)}\|\chi_{k+1}(A_{x}-A_{x}^{\prime})(s)\|_{L^{p}}$
$\displaystyle\lesssim\varepsilon
s^{\frac{k+3}{4}}s^{-\frac{3}{2}(\frac{1}{2}-\frac{1}{p})}$ (8.23)
$\displaystyle(Ns^{\frac{1}{2}})^{(k+3)(1-\sigma)}\|\chi_{k+1}(\partial^{\ell}A_{\ell}-\partial^{\ell}A^{\prime}_{\ell})(s)\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon s^{\frac{k+2}{4}}$ (8.24)

Moreover, if one assumes $k\geq 2$ and

$(N^{2}s_{1})^{1-\sigma}s_{1}^{\frac{1}{4}}\|\chi_{k}\delta
w(s_{1})\|_{L^{2}}+\|(N^{2}s)^{1-\sigma}s^{\frac{3}{4}}\chi_{k}\delta\mathbf{D}_{x}w\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}[0,s_{1}]}\lesssim\frac{\varepsilon
s_{1}^{\frac{k-2}{4}}}{(Ns_{1}^{\frac{1}{2}})^{(k-2)(1-\sigma)}}$ (8.25)

for all $s_{1}\in[0,s_{0}]$ then one also has

$(N^{2}s_{1})^{1-\sigma}s_{1}^{\frac{1}{4}}\|\chi_{k+1}\delta
w(s_{1})\|_{L^{2}}+\|(N^{2}s)^{1-\sigma}s^{\frac{3}{4}}\chi_{k+1}\delta\mathbf{D}_{x}w\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}[0,s_{1}]}\lesssim\frac{\varepsilon
s_{1}^{\frac{k-1}{4}}}{(Ns_{1}^{\frac{1}{2}})^{(k-1)(1-\sigma)}}$ (8.26)

Assuming this Lemma we can can now prove the main estimates of this section.

###  8.1. Proof of Proposition 8.1

Fix a time $t\in[t_{0},t_{1}]$. By integrating equation (8.8) point-wise and
the fact that the solutions coincide at $s=0$ on $B^{(1)}$, we have

$A_{\ell}(s_{1})-A_{\ell}^{\prime}(s_{1})=\int_{0}^{s_{1}}\mathbf{D}^{j}F_{j\ell}(s)-\mathbf{D}^{{}^{\prime}j}F^{\prime}_{j\ell}(s)\,\mathrm{d}s,\qquad
x\in B^{(1)}$

which implies, by estimating $\mathbf{D}^{j}F_{j\ell}$ and
$\mathbf{D}^{{}^{\prime}j}F^{\prime}_{j\ell}$ separately using (8.1), (8.2),
(8.13), that

$(Ns^{\frac{1}{2}})^{1-\sigma}\|A_{x}(s)-A_{x}^{\prime}(s)\|_{L^{p}(B^{(1)})}\lesssim\varepsilon
s^{\frac{1}{2}}s^{-\frac{3}{2}(\frac{1}{2}-\frac{1}{p})},\quad p\in[2,4]$
(8.27)

By integrating equation (8.9), on $B^{(1)}$ we have

$\partial^{\ell}A_{\ell}(s_{1})-\partial^{\ell}A^{\prime}_{\ell}(s_{1})=-\int_{0}^{s_{1}}[A_{x}-A_{x}^{\prime},\mathbf{D}_{x}F]+[A^{\prime},\mathbf{D}_{x}F-\mathbf{D}^{\prime}_{x}F^{\prime}]\,\mathrm{d}s$

We bound the integrand in $L^{2}_{x}$. We use (8.27) with $p=2$ and (8.13)
with $p=\infty$ for the first term. The second term we bound by
$(L^{6}+N^{\sigma-1}L^{q})\times(L^{3}\cap L^{3/\sigma})\to L^{2}$ using
(8.15) and (8.13) with $p=3$ and $p=3/\sigma$, for both $\mathbf{D}_{x}F$ and
$\mathbf{D}^{\prime}_{x}F^{\prime}$ separately. We obtain

$(Ns^{\frac{1}{2}})^{2(1-\sigma)}\|\partial^{\ell}A_{\ell}(s)-\partial^{\ell}A^{\prime}_{\ell}(s)\|_{L^{2}(B^{(1)})}\lesssim\varepsilon
s^{\frac{1}{4}}$ (8.28)

We will apply Lemma 8.2 several times with different cutoff functions to
obtain (8.6) and (8.7), each time getting an extra power of $s^{1/4}$ and
slightly reducing the support.

Given $\chi$ supported in $B^{(1)}$, we consider a sequence of bump functions
$\chi_{0},\chi_{1},\dots,\chi_{8}=\chi$ all supported in $B^{(1)}$ such that
$\chi_{k}\equiv 1$ on ${\mathrm{supp}}\chi_{k+1}$, for each $k\in\\{0,\dots
7\\}$.

Based on (8.1), (8.2), (8.27), (8.28) we first apply Lemma 8.2 with $k=0$ and
$\chi_{0},\chi_{1}$. We obtain (8.17)-(8.20) with $k=1$ for $\chi_{1}$. We
repeatedly apply Lemma 8.2 with $\chi_{k},\chi_{k+1}$ until we obtain (8.6).
Starting from $k\geq 2$ we also obtain bounds on $\delta w$: in the $k=2$ step
the bound (8.25) follows trivially from (8.4), (8.5) by the triangle
inequality. When $k=7$ we conclude (8.7).

###  8.2. Proof of Lemma 8.2

For brevity, in the course of this proof we denote
$\tilde{\chi}\vcentcolon=\chi_{k}$ and $\chi\vcentcolon=\chi_{k+1}$. Fix a
time $t\in[t_{0},t_{1}]$.

####  8.2.1. Difference equation for F

We first prove (8.21), (8.22) for $m=0$.

Subtracting (8.10) for $F$ and $F^{\prime}$ and using the form of
$\delta\mathbf{D}^{\ell}\mathbf{D}_{\ell}$ from (8.16) we obtain the schematic
equation (where we drop some constants and irrelevant signs)

$\displaystyle\left(\partial_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\left(\chi\delta
F\right)=$ $\displaystyle\ \chi[\delta F,F]+\chi[\delta F,F^{\prime}]$ (8.29)
$\displaystyle+\nabla\chi\cdot\delta\mathbf{D}_{x}F+\Delta\chi\delta F$ (8.30)
$\displaystyle+[\delta A^{\ell},\partial_{\ell}(\chi
F^{\prime})]+\chi[\delta\partial^{\ell}A_{\ell},F^{\prime}]$ (8.31)
$\displaystyle+\chi[A^{\ell},[A_{\ell},F^{\prime}]]-\chi[A^{{}^{\prime}\ell},[A_{\ell}^{\prime},F^{\prime}]]$
(8.32)

Note that $\chi\delta F(0)=0$. Using the energy estimate (6.5) with $p=0$ we
obtain

$\|\chi\delta
F\|_{L^{\infty}_{s}L^{2}_{x}[0,s_{1}]}+\|s^{\frac{1}{2}}\mathbf{D}_{x}\left(\chi\delta
F(s)\right)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}[0,s_{1}]}\lesssim\varepsilon
s_{1}^{\frac{k+1}{4}}(Ns_{1}^{\frac{1}{2}})^{-(k+2)(1-\sigma)}$ (8.33)

provided we estimate (8.29) - (8.32) in $L^{1}_{s}L^{2}_{x}[0,s_{1}]$. Then,
note that by bounding $s^{\frac{1}{2}}\nabla\chi\cdot\delta F(s)$ and
$s^{\frac{1}{2}}\chi[\delta A,F](s)$ using (8.17), (8.19) and (8.13) we would
also have

$\|s^{\frac{1}{2}}\chi\mathbf{D}_{x}\left(\delta
F\right)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}[0,s_{1}]}+\|s^{\frac{1}{2}}\chi\delta\mathbf{D}_{x}F\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}[0,s_{1}]}\lesssim\varepsilon
s_{1}^{\frac{k+1}{4}}(Ns_{1}^{\frac{1}{2}})^{-(k+2)(1-\sigma)}$ (8.34)

which will give (8.21), (8.22) for $m=0$.

We estimate the nonlinear terms in $L^{1}_{s}L^{2}_{x}[0,s_{1}]$ as follows:

For (8.29) we use (8.17) with $m=0$ and (8.13) with $p=\infty,\ m=0$;

For (8.30) we use (8.17), (8.18) with $m=0$, recalling that
$\tilde{\chi}\equiv 1$ on ${\mathrm{supp}}\chi$;

For (8.31) we use (8.20), (8.13) with $p=\infty$; to estimate the second term.
Then use (8.19), (8.13) with $m=1,0,\ p=\infty$ to estimate $\chi[\delta
A^{\ell},\mathbf{D}_{\ell}^{\prime}F^{\prime}]$ and $[\delta
A^{\ell},\partial_{\ell}\chi F^{\prime}]$.

For (8.32) and the remainder from (8.31), write everything in terms of

$\mathcal{O}(A^{\prime},\chi\delta A,F^{\prime})\quad\text{and}\
\qquad\mathcal{O}(\chi\delta A,\tilde{\chi}\delta A,F^{\prime})$ (8.35)

The first term is estimated as $(L^{6}+N^{\sigma-1}L^{q})\times(L^{3}\cap
L^{p})\times L^{\infty}\to L^{2}$ using (8.15), (8.19) with $p=3/\sigma$ and
(8.13). The second term is bounded as $L^{4}\times L^{4}\times L^{\infty}\to
L^{2}$ from (8.19), (8.13).

####  8.2.2. Difference equation for $\mathbf{D}_{x}F$

Similarly one writes the parabolic equation for differences from (8.11)

$\displaystyle\left(\partial_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\left(\chi\delta\mathbf{D}_{x}F\right)=$
$\displaystyle\ \chi\mathcal{O}(\delta
F,\mathbf{D}_{x}F)+\chi\mathcal{O}(F^{\prime},\delta\mathbf{D}_{x}F)$ (8.36)
$\displaystyle+\nabla\chi\cdot\delta\mathbf{D}_{x}^{2}F+\Delta\chi\delta\mathbf{D}_{x}F$
(8.37) $\displaystyle+[\delta
A^{\ell},\partial_{\ell}(\chi\mathbf{D}^{\prime}_{x}F^{\prime})]+\chi[\delta\partial^{\ell}A_{\ell},\mathbf{D}^{\prime}_{x}F^{\prime}]$
(8.38)
$\displaystyle+\chi[A^{\ell},[A_{\ell},\mathbf{D}^{\prime}_{x}F^{\prime}]]-\chi[A^{{}^{\prime}\ell},[A_{\ell}^{\prime},\mathbf{D}^{\prime}_{x}F^{\prime}]]$
(8.39)

Denote the nonlinearity by $\mathcal{M}$. Using the energy estimate (6.5) with
with $p=\frac{1}{2}$ one obtains

$\displaystyle\|\chi
s^{\frac{1}{2}}\delta\mathbf{D}_{x}F\|_{L^{\infty}_{s}L^{2}_{x}[0,s_{1}]}+\|s\mathbf{D}_{x}\left(\chi\delta\mathbf{D}_{x}F(s)\right)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}[0,s_{1}]}\lesssim$
(8.40) $\displaystyle\|\chi
s^{\frac{1}{2}}\delta\mathbf{D}_{x}F\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}[0,s_{1}]}+\int_{0}^{s_{1}}s^{\frac{1}{2}}\|\mathcal{M}(s)\|_{L^{2}_{x}}\,\mathrm{d}s$
(8.41)

We prove

$\eqref{EnergyDiffDx2}\lesssim\varepsilon
s_{1}^{\frac{k+1}{4}}(Ns_{1}^{\frac{1}{2}})^{-(k+2)(1-\sigma)}$ (8.42)

The first term in (8.41) was estimated in (8.34). To estimate $\mathcal{M}$ in
$L^{1}_{s}L^{2}_{x}[0,s_{1}]$ we proceed exactly as in the previous step, with
the following difference: for the terms which have $\mathbf{D}_{x}F$ instead
of $F$, when applying (8.17), (8.18), (8.13) we use $m=1$ instead of $m=0$.

As before, the second term in (8.40) can be replaced by
$s\chi\delta\mathbf{D}_{x}^{2}F(s)$, which completes the proof of (8.21),
(8.22) also for $m=1$.

####  8.2.3. Bound on $A$ and $divA$

By integrating equations (8.8), (8.9) and the fact that the solutions coincide
at $s=0$ on the support of $\chi$ we have

$\chi\left(A_{\ell}(s_{1})-A_{\ell}^{\prime}(s_{1})\right)=\chi\int_{0}^{s_{1}}\delta\mathbf{D}_{x}F(s)\,\mathrm{d}s$
(8.43)
$\chi\left(\partial^{\ell}A_{\ell}(s_{1})-\partial^{\ell}A^{\prime}_{\ell}(s_{1})\right)=-\chi\int_{0}^{s_{1}}[\delta
A,\mathbf{D}_{x}F]+[A^{\prime},\delta\mathbf{D}_{x}F]\,\mathrm{d}s$ (8.44)

By the covariant Gagliardo-Nirenberg inequalities (2.11) we have

$\|\chi\delta\mathbf{D}_{x}F(s)\|_{L^{p}_{x}}\lesssim\|\chi\delta\mathbf{D}_{x}F(s)\|_{L^{2}_{x}}^{1-\theta}\|\mathbf{D}_{x}\left(\chi\delta\mathbf{D}_{x}F(s)\right)\|_{L^{2}_{x}}^{\theta},\quad\theta=3(2^{-1}-p^{-1})$
(8.45)

Integrating this and using Holder in $\frac{\,\mathrm{d}s}{s}$ we obtain
(8.23) from (8.40).

For (8.24) we estimate the integral (8.44) as follows: for the first term use
(8.23) with $p=2$ and (8.13) with $p=\infty,\ m=1$; the second term is
estimated as $(L^{6}+N^{\sigma-1}L^{q})\times(L^{3}\cap L^{p})\to L^{2}$ using
(8.15) and (8.45) for $L^{3}$ and $L^{p}$ with $p=3/\sigma$, together with
(8.40).

####  8.2.4. Bound on $w$

Subtracting (8.12) for $w$ and $w^{\prime}$ and using the form of
$\delta\mathbf{D}^{\ell}\mathbf{D}_{\ell}$ from (8.16) we obtain the schematic
equation

$\displaystyle\left(\partial_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\left(\chi\delta
w\right)=$ $\displaystyle\ \chi[\delta F,w]+\chi[F,\delta w]$ (8.46)
$\displaystyle+\chi\mathcal{O}(\delta
F,\mathbf{D}_{x}F)+\chi\mathcal{O}(F^{\prime},\delta\mathbf{D}_{x}F)$ (8.47)
$\displaystyle+\nabla\chi\cdot\delta\mathbf{D}_{x}w+\Delta\chi\delta w$ (8.48)
$\displaystyle+[\delta A^{\ell},\partial_{\ell}(\chi
w^{\prime})]+\chi[\delta\partial^{\ell}A_{\ell},w^{\prime}]$ (8.49)
$\displaystyle+\chi[A^{\ell},[A_{\ell},w^{\prime}]]-\chi[A^{{}^{\prime}\ell},[A_{\ell}^{\prime},w^{\prime}]]$
(8.50)

Using the energy estimate (6.5) with $p=0$ we obtain

$\mathcal{T}\vcentcolon=\|\chi\delta
w\|_{L^{\infty}_{s}L^{2}_{x}[0,s_{1}]}+\|s^{\frac{1}{2}}\mathbf{D}_{x}\left(\chi\delta
w\right)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}[0,s_{1}]}\lesssim\varepsilon\mathcal{T}+\frac{\varepsilon
s_{1}^{\frac{k-1}{4}}}{(Ns_{1}^{\frac{1}{2}})^{(k+2)(1-\sigma)}}$ (8.51)

provided we bound (8.46) - (8.50) in $L^{1}_{s}L^{2}_{x}[0,s_{1}]$ by the RHS
of (8.51). Then we absorb the $\varepsilon\mathcal{T}$ to the LHS and, replace
the $\mathbf{D}_{x}\left(\chi\delta w\right)$ by $\chi\delta\mathbf{D}_{x}w$
by bounding $s^{\frac{1}{2}}\nabla\chi\cdot\delta w$ and
$s^{\frac{1}{2}}\chi[\delta A,w]$ using (8.25), (8.19) and (8.14) obtaining

$\|\chi\delta
w(s_{1})\|_{L^{2}_{x}}+\|s^{\frac{1}{2}}\chi\delta\mathbf{D}_{x}w\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}[0,s_{1}]}\lesssim\frac{\varepsilon
s_{1}^{\frac{k-1}{4}}}{(Ns_{1}^{\frac{1}{2}})^{(k+2)(1-\sigma)}}$

This is more than enough for (8.26). We estimate the nonlinear terms in
$L^{1}_{s}L^{2}_{x}[0,s_{1}]$ very similarly to the case of $\chi\delta F$:

For (8.46) we first use (8.17) with $m=0$ and (8.14) with $p=\infty,\ m=0$;
then for $\chi[F,\delta w]$ we use (8.13) with $p=\infty,\ m=0$, which
contributes the term $\varepsilon
s_{1}^{\frac{1}{4}}\mathcal{T}\ll\varepsilon\mathcal{T}$.

The terms in (8.47) were already estimated in (8.36).

For (8.48) we use (8.25), recalling that $\tilde{\chi}\equiv 1$ on
${\mathrm{supp}}\chi$;

For (8.49) we use (8.20), (8.14) with $p=\infty$; to estimate the second term.
Then use (8.19), (8.14) with $m=1,0,\ p=\infty$ to estimate $\chi[\delta
A^{\ell},\mathbf{D}_{\ell}^{\prime}w^{\prime}]$ and $[\delta
A^{\ell},\partial_{\ell}\chi w^{\prime}]$.

For (8.50) and the remainder from (8.49), write everything in terms of

$\mathcal{O}(A^{\prime},\chi\delta A,w^{\prime})\quad\text{and}\
\qquad\mathcal{O}(\chi\delta A,\tilde{\chi}\delta A,w^{\prime})$

and then proceed exactly like in (8.35), using (8.14) instead of (8.13).

##  9\. Decompositions and estimates in DeTurck’s gauge

In this section we consider solutions $A_{t,x,s}$ to (YM) and (dYMHF) on
$[t_{0},t_{1}]\times\mathbb{R}^{3}\times[0,s_{0}]$, $t_{1}=t_{0}+1$ in:

  1. (1)

The temporal gauge at $s=0$, i.e. $A_{0}(t,x,0)=0\ $.

  2. (2)

The DeTurck gauge $A_{s}=\partial^{\ell}A_{\ell}$ on $[t_{0},t_{1}]\times\
\mathbb{R}^{3}\times[0,s_{0}]$.

  3. (3)

The Coulomb gauge at $s=0,\ t=t_{0}$, i.e.
$\partial^{\ell}A_{\ell}(t_{0},x,0)=0$.

Let $\sigma>\frac{5}{6}$. We assume $A_{i}(t,x,0)$ obeys the bounds in
Proposition 4.1 part (2) and therefore also (4.12) - (4.14). Then Proposition
5.1 holds uniformly in $t\in[t_{0},t_{1}]$, and as a consequence (5.13),
(5.14).

We assume $\mathcal{IE}(t)\lesssim\varepsilon^{2}$ control of the modified
energy (1.6) for all $t\in[t_{0},t_{1}]$. In particular, by Proposition 6.3
and 6.4 the curvature $F_{\alpha\beta}$ and the tension field $w_{i}$ defined
in (1.9) obey the bounds

$\displaystyle\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F_{\alpha\beta}(t,s)\|_{L^{\infty}_{s}L^{2}_{x}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\lesssim\varepsilon$
(9.1)
$\displaystyle\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}w_{x}(t,s)\|_{L^{\infty}_{s}L^{2}_{x}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\lesssim\varepsilon$
(9.2)

for all $0\leq m\leq 8$ and $t\in[t_{0},t_{1}]$.

Under these assumptions we control the norms (9.14) - (9.23).

###  9.1. Decompositions in DeTurck’s gauge

####  9.1.1. Decomposition of $F$

Expanding (6.1) and canceling the $[A_{s},F]$ term with the
$[\partial^{\ell}A_{\ell},F]$ term one obtains

$(\partial_{s}-\Delta)F_{\alpha\beta}=2[F_{\alpha}^{\
\ell},F_{\ell\beta}]+2[A^{\ell},\partial_{\ell}F_{\alpha\beta}]+[A^{\ell},[A_{\ell},F_{\alpha\beta}]]$
(9.3)

and one decomposes

$F_{\alpha\beta}(s)=e^{s\Delta}F_{\alpha\beta}(0)+F^{bil}_{\alpha\beta}(s)$
(9.4)

where the bilinear term

$F^{bil}_{\alpha\beta}(s_{1})\vcentcolon=\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}\left(2[F_{\alpha}^{\
\ell},F_{\ell\beta}]+2[A^{\ell},\partial_{\ell}F_{\alpha\beta}]+[A^{\ell},[A_{\ell},F_{\alpha\beta}]]\right)(s)\,\mathrm{d}s$
(9.5)

obeys better estimates, such as

######  Lemma 9.1.

Under the assumptions above, one has:

$\|F^{bil}(s)\|_{L^{\infty}_{t}L^{2}_{x}}+s^{\frac{1}{2}}\|\partial_{x}F^{bil}(s)\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim
s^{\frac{1}{4}}/(N^{2}s)^{1-\sigma}\varepsilon.$ (9.6)

###### Proof.

Fix a time $t\in[t_{0},t_{1}]$. It suffices to bound the three terms in the
bracket in (9.5) in $L^{2}_{x}$ by
$s^{-\frac{3}{4}}/(N^{2}s)^{1-\sigma}\varepsilon$. For the second estimate in
(9.6) one begins with the fact that
$(s_{1}-s)^{\frac{1}{2}}\partial_{x}e^{(s_{1}-s)\Delta}:L^{2}_{x}\to
L^{2}_{x}$.

For $[F,F]$ and $[A^{\ell},\mathbf{D}_{\ell}F_{\alpha\beta}]$ we use Holder,
placing the first term in $L^{\infty}_{x}$ and the second in $L^{2}_{x}$,
recalling (5.13). For $[A,[A,F]]$ we do $L^{\infty}_{x}\times
L^{\infty}_{x}\times L^{2}_{x}\to L^{2}_{x}$. ∎

####  9.1.2. Decomposition of $w$

Expanding (6.3), plugging in (9.4) and canceling the $[A_{s},w]$ term with the
$[\partial^{\ell}A_{\ell},w]$ term one obtains

$(\partial_{s}-\Delta)w_{k}=\mathcal{N}^{(2)}+\mathcal{N}^{(3)}+\mathcal{N}^{self}$

where

$\displaystyle\mathcal{N}^{(2)}(s)$
$\displaystyle\vcentcolon=2[e^{s\Delta}\partial_{t}A^{\ell}(0),e^{s\Delta}(\partial_{k}\partial_{t}A_{\ell}-2\partial_{\ell}\partial_{t}A_{k})(0)]$
(9.7) $\displaystyle\mathcal{N}^{(3)}(s)$
$\displaystyle\vcentcolon=[F^{bil}_{0\ell},\mathbf{D}_{x}F_{0x}]+2[e^{s\Delta}\partial_{t}A^{\ell}(0),\partial_{x}F^{bil}_{0x}+[A_{x},F_{0x}]]$
$\displaystyle\mathcal{N}^{self}(s)$ $\displaystyle\vcentcolon=2[F_{k}^{\
\ell},w_{\ell}]+2[A^{\ell},\partial_{\ell}w_{k}]+[A^{\ell},[A_{\ell},w_{k}]]$

where $\mathcal{N}^{(2)}$ contains the dominant part from
$2\left[F^{0\ell},\mathbf{D}_{k}F_{0\ell}+2\mathbf{D}_{\ell}F_{k0}\right]$
(recall the temporal gauge condition implies $F_{0\ell}=\partial_{t}A_{\ell}$
at $s=0$ ). The term $\mathcal{N}^{(3)}$ consists of the remainder and
$\mathcal{N}^{self}$ contains the terms involving $w$.

Since $w(0)=0$, this gives rise to the following decomposition which isolates
the leading quadratic part of $w$:

$w(s)=w^{(2)}(s)+w^{(3)}(s)$ (9.8)

where

$w^{(2)}(s_{1})\vcentcolon=\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}\mathcal{N}^{(2)}(s)\,\mathrm{d}s,\quad
w^{(3)}(s_{1})\vcentcolon=\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}(\mathcal{N}^{(3)}(s)+\mathcal{N}^{self}(s))\,\mathrm{d}s$

The term $w^{(2)}(s)$ can be written, following [39], as

$w^{(2)}_{i}(s)=\mathbf{W}(\partial_{t}A^{\ell}(0),\partial_{i}\partial_{t}A_{\ell}(0)-2\partial_{\ell}\partial_{t}A_{i}(0))$
(9.9)

where $\mathbf{W}$ is a symmetric bilinear form with symbol

$\displaystyle\mathbf{W}(\xi,\eta,s)$
$\displaystyle=\int_{0}^{s}e^{-\left(s-s^{\prime}\right)|\xi+\eta|^{2}}e^{-s^{\prime}\left(|\xi|^{2}+|\eta|^{2}\right)}\,\mathrm{d}s^{\prime}$
(9.10)
$\displaystyle=-\frac{1}{2\xi\cdot\eta}e^{-s|\xi+\eta|^{2}}(1-e^{2s(\xi\cdot\eta)})$

which enjoys the following favorable frequency concentration:

######  Lemma 9.2 ([41] \- Lemma 8.2).

For any $k,k_{1},k_{2}\in\mathbb{Z}$ and $s>0$, denoting
$k_{\max}=\max(k,k_{1},k_{2})$, the bilinear operator

$\left\langle s2^{2k}\right\rangle^{10}\left\langle
s^{-1}2^{-2k_{\max}}\right\rangle
2^{2k_{\max}}P_{k}\mathbf{W}\left(P_{k_{1}}(\cdot),P_{k_{2}}(\cdot),s\right)$

is disposable, i.e. its kernel has bounded mass, see (2.5).

Using this structure of $\mathbf{W}$, one can obtain more precise estimates
for $w^{(2)}$, such as

######  Lemma 9.3.

Under the assumptions above, one has:

$\|P_{k}w^{(2)}_{x}(s)\|_{L^{2}_{t}L^{2}_{x}}\lesssim\varepsilon
2^{(0+)k}\frac{(2^{k}s^{\frac{1}{2}})^{\frac{1}{2}-}}{\left\langle
2^{2k}s\right\rangle^{8}}\frac{1}{(N^{2}s)^{1-\sigma}}\left(1+\frac{1}{2^{\frac{k}{2}-}}\frac{1}{(Ns^{\frac{1}{2}})^{1-\sigma}}\right)$
(9.11)

Therefore, after summing in $k$ one obtains:

$\|w^{(2)}_{x}(s)\|_{L^{2}_{t}L^{2}_{x}}\lesssim\varepsilon
s^{0-}/(N^{2}s)^{1-\sigma}$ (9.12)

######  Remark 9.4.

One can decompose the inputs of $w^{(2)}_{x}(s)$ into
$\partial_{t}A^{cf}+\partial_{t}A^{df}$. It is clear from the proof that the
bound (9.11) is true also for the separate components of $w^{(2)}_{x}(s)$, in
particular for $P_{k}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{df},s)$.

###### Proof.

Doing a Littlewood-Paley decomposition and invoking Lemma 9.2 we need to bound

$P_{k}\mathcal{O}(\partial_{t}A_{k_{1}}(0),\partial_{t}A_{k_{2}}(0))\cdot
2^{-k_{\max}}\left\langle s2^{2k}\right\rangle^{-10}\left\langle
s^{-1}2^{-2k_{\max}}\right\rangle^{-1}$ (9.13)

in the two cases, without loss of generality:

  1. (1)

low-high $2^{k_{1}}\lesssim 2^{k}\simeq 2^{k_{2}}\simeq 2^{k_{\max}}$. Bound
$\partial_{t}A_{k_{1}}(0)$ in $L^{2}L^{\infty}$ using (4.14) and
$\partial_{t}A_{k_{2}}(0)$ in $L^{\infty}L^{2}$. Then one sums in $k_{1}$,
obtaining the bound

$\varepsilon 2^{(0+)k}\frac{2^{2k}s}{\left\langle
2^{2k}s\right\rangle^{10}}\frac{1}{(N^{2}s)^{1-\sigma}}.$

  2. (2)

high-high to low $2^{k}\lesssim 2^{k_{1}}\simeq 2^{k_{2}}\simeq 2^{k_{\max}}$.
Split both inputs into $\partial_{t}A^{cf}+\partial_{t}A^{df}$. When at least
one input is $\partial_{t}A^{cf}$, place that input in $L^{2}L^{\infty}$ using
(4.12) and the other one in $L^{\infty}L^{2}$. After summing the high
frequencies one obtains the bounds

$\varepsilon\frac{1}{2^{\frac{k}{2}-}}\left(\frac{2^{k}}{N}\right)^{3(1-\sigma)}\frac{1}{(2^{2k}s)^{10}}\qquad\text{or}\qquad\varepsilon\frac{s^{\frac{1}{4}-}}{(Ns^{\frac{1}{2}})^{3(1-\sigma)}}$

depending on whether $s^{-\frac{1}{2}}=k(s)\leq k$ or $k\leq k(s)$.

The remaining case
$P_{k}\mathcal{O}(\partial_{t}A^{df}_{k_{1}}(0),\partial_{t}A^{df}_{k_{2}}(0))$
is bounded using the Bilinear Strichartz estimate (2.31). After summation one
can obtain either

$\varepsilon 2^{(0+)k}\frac{2^{2k}s}{\left\langle
2^{2k}s\right\rangle^{10}}\frac{1}{(N^{2}s)^{1-\sigma}}\qquad\text{or}\qquad\varepsilon
2^{(0+)k}\frac{(2^{k}s^{\frac{1}{2}})^{\frac{1}{2}-}}{\left\langle
2^{2k}s\right\rangle^{10}}\frac{1}{(N^{2}s)^{1-\sigma}}$

depending on whether $s^{-\frac{1}{2}}=k(s)\leq k$ or $k\leq k(s)$.

∎

###  9.2. Estimates for the curvature and the tension field

####  9.2.1. Preparations

For easy reference we record here the bounds on
$[t_{0},t_{1}]\times\mathbb{R}^{3}$ which we will use below. These follow from
(4.8), (4.7), (2.30), (4.14), (5.13), (9.1), (2.11).

$\displaystyle\|P_{k}A^{cf}(0)\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon
2^{-\frac{5}{4}k}\left\langle\frac{2^{k}}{N}\right\rangle^{1-\sigma}$ (9.14)
$\displaystyle s^{\frac{1}{8}}\|e^{s\Delta}A^{cf}(0)\|_{L^{\infty}_{t,x}}$
$\displaystyle\lesssim\|e^{s\Delta}A^{cf}(0)\|_{L^{\infty}_{t}L^{12}_{x}}\lesssim\frac{\varepsilon}{(Ns^{\frac{1}{2}})^{1-\sigma}}$
(9.15) $\displaystyle\|\partial_{t}P_{k}A^{cf}(0)\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon
2^{-\frac{k}{2}}\left\langle\frac{2^{k}}{N}\right\rangle^{2(1-\sigma)}$ (9.16)
$\displaystyle\|\partial_{t}P_{k}A^{df}(0)\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon\left\langle\frac{2^{k}}{N}\right\rangle^{1-\sigma}$
(9.17) $\displaystyle\|P_{k}\nabla_{t,x}A^{df}(0)\|_{L^{2}_{t}L^{\infty}_{x}}$
$\displaystyle\lesssim
2^{(1+)k}\left\langle\frac{2^{k}}{N}\right\rangle^{1-\sigma}$ (9.18)
$\displaystyle\|e^{s\Delta}A^{df}(0)\|_{L^{2}_{t}L^{\infty}_{x}}$
$\displaystyle\lesssim s^{0-}/(Ns^{\frac{1}{2}})^{1-\sigma}$ (9.19)
$\displaystyle\|e^{s\Delta}\partial_{t}A_{i}(0)\|_{L^{2}_{t}L^{\infty}_{x}}$
$\displaystyle\lesssim s^{-\frac{1}{2}+}/(Ns^{\frac{1}{2}})^{1-\sigma}$ (9.20)
$\displaystyle\|A_{i}(s)\|_{L^{\infty}_{t}L^{\infty}_{x}}$
$\displaystyle\lesssim s^{-\frac{1}{4}}/(Ns^{\frac{1}{2}})^{1-\sigma}$ (9.21)
$\displaystyle\|F(s)\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon/(Ns^{\frac{1}{2}})^{1-\sigma}$ (9.22)
$\displaystyle\|F(s)\|_{L^{\infty}_{t}L^{\infty}_{x}}$
$\displaystyle\lesssim\varepsilon
s^{-\frac{3}{4}}/(Ns^{\frac{1}{2}})^{1-\sigma}$ (9.23)

####  9.2.2. Bounds on the curvature

######  Proposition 9.5.

Under the assumptions above, one has:

$\displaystyle\|F^{bil}(s)\|_{L^{2}_{t}L^{\infty}_{x}}\lesssim
s^{-\frac{1}{4}+}/(Ns^{\frac{1}{2}})^{3(1-\sigma)}\varepsilon$ (9.24)
$\displaystyle\|F^{bil}(s)\|_{L^{2}_{t}L^{2}_{x}}+s^{\frac{1}{2}}\|\partial_{x}F^{bil}(s)\|_{L^{2}_{t}L^{2}_{x}}\lesssim
s^{\frac{1}{2}-\frac{1}{8}}/(Ns^{\frac{1}{2}})^{2(1-\sigma)}\varepsilon$
(9.25)

###### Proof.

Part 1. We begin with (9.24). If $\mathcal{T}$ is one the three terms in the
bracket in (9.5), then we bound either

$\|\int_{0}^{s}e^{(s-s^{\prime})\Delta}\mathcal{T}(s^{\prime})\,\mathrm{d}s^{\prime}\|_{L^{2}L^{\infty}}\quad\text{or}\quad\int_{0}^{s}\frac{1}{(s-s^{\prime})^{\frac{3}{4}}}\|\mathcal{T}(s^{\prime})\|_{L^{2}L^{2}}\,\mathrm{d}s^{\prime}$

since one has
$(s-s^{\prime})^{\frac{3}{4}}e^{(s-s^{\prime})\Delta}:L^{2}_{x}\to
L^{\infty}_{x}$. Let

  1. (1)

$\mathcal{T}=[F,F]$, which we place in $L^{2}L^{2}$. For:

$[F^{bil},F](s^{\prime})$ we use (9.6) and (9.23), while for

$[e^{s^{\prime}\Delta}\partial_{t}A_{\ell}(0),F]$,
$[e^{s^{\prime}\Delta}\partial_{x}A^{df}(0),F]$ we use (9.20) or
Strichartz/(9.19) and (9.22).

  2. (2)

$\mathcal{T}=[A^{\ell},\partial_{\ell}F_{\alpha\beta}]$ which we separate as
follows:

$[e^{s^{\prime}\Delta}A^{df}(0)+A^{bil}(s^{\prime}),\mathbf{D}_{\ell}F]$ as
$L^{2}L^{\infty}\times L^{\infty}L^{2}\to L^{2}L^{2}$ by (9.19), (5.14).

$[e^{s^{\prime}\Delta}A^{cf}(0),\partial_{x}F^{bil}+e^{s^{\prime}\Delta}\partial_{x}\partial_{t,x}A^{cf}(0)+e^{s^{\prime}\Delta}\partial_{x}[A,A](0)]$
as $L^{\infty}L^{\infty}\times L^{\infty}L^{2}\to L^{2}L^{2}$ using (9.15),
(9.16), (9.6).

$[e^{s^{\prime}\Delta}A^{cf}(0),e^{s^{\prime}\Delta}\partial_{x}\partial_{t,x}A^{df}(0)]$
which after the $s^{\prime}$ integration takes the form

$\mathbf{W}(A^{cf}(0),\partial_{x}\partial_{t,x}A^{df}(0))$ (9.26)

where $\mathbf{W}$ is defined in (9.10). We use Lemma 9.2 and the Littlewood
Paley trichotomy to bound this term in $L^{2}L^{\infty}$. In the low-high and
high-low input frequencies cases we place $A^{cf}$ in $L^{\infty}L^{\infty}$
and $\partial_{t,x}A^{df}$ in $L^{2}L^{\infty}$ using Strichartz. In the high-
high to low case use Bernstein on the output low frequency and
$L^{\infty}L^{2}\times L^{2}L^{\infty}\to L^{2}L^{2}$ with (9.14) and (9.18).

  3. (3)

$\mathcal{T}=[A^{\ell},[A_{\ell},F]]$ which is placed in $L^{2}L^{2}$ as
follows:

For $[e^{s^{\prime}\Delta}A^{df}(0)+A^{bil}(s^{\prime}),[A,F](s^{\prime})]$ as
$L^{2}L^{\infty}\times L^{\infty}L^{\infty}\times L^{\infty}L^{2}$ using
(9.19), (5.14), (9.21), (9.22).

Similarly one deals with
$[e^{s^{\prime}\Delta}A^{cf}(0),[e^{s^{\prime}\Delta}A^{df}(0)+A^{bil}(s^{\prime}),F(s^{\prime})]].$

The remaining term
$[e^{s^{\prime}\Delta}A^{cf}(0),[e^{s^{\prime}\Delta}A^{cf}(0),F(s^{\prime})]]$
is bounded as $L^{\infty}L^{12}\times L^{\infty}L^{12}\times L^{\infty}L^{3}$
using (9.15) and interpolating between (9.22) and (9.23).

Part 2. We continue with (9.25). We bound the bracket from (9.5) in
$L^{2}L^{2}$. This suffices also for the $\partial_{x}F^{bil}_{0i}$ bound
because
$\partial_{x}e^{(s_{1}-s)\Delta}:L^{2}_{x}\to(s_{1}-s)^{-\frac{1}{2}}L^{2}_{x}$.
The majority of terms in (9.5) were already estimated in $L^{2}L^{2}$ in Part
1. The term that remains to be estimated in $L^{2}L^{2}$ is (9.26).

Again we use Lemma 9.2 and the Littlewood Paley trichotomy. In the high-high
to low and high-low cases use $L^{\infty}L^{2}\times L^{2}L^{\infty}\to
L^{2}L^{2}$ with (9.14) and (9.18). In the low-high case ($A^{cf}$ \- low
frequency) place $A^{cf}(0)$ in $L^{\infty}L^{12+}$ and
$\partial_{t,x}A^{df}(0)$ in $L^{\infty}L^{\frac{12}{5}-}$ using Bernstein.
This last case is responsible for the loss of $s^{-\frac{1}{8}}$ in (9.25),
which can probably be removed. ∎

######  Corollary 9.6.

One has:

$\|F_{\alpha\beta}(s)\|_{L^{2}_{t}L^{\infty}_{x}}\lesssim\varepsilon
s^{-\frac{1}{2}+}/(Ns^{\frac{1}{2}})^{(1-\sigma)}$ (9.27)

###### Proof.

From (9.4) and the temporal condition $A_{0}(s=0)=0$ one writes

$F_{0i}(s)=e^{s\Delta}\partial_{t}A_{i}(0)+F^{bil}_{0i}(s)$

and the bound follows from (9.20) and (9.24). Similarly

$\displaystyle F_{k\ell}(s)$
$\displaystyle=e^{s\Delta}(\partial_{k}A_{\ell}-\partial_{\ell}A_{k}+[A_{\ell},A_{k}])(0)+F^{bil}_{k\ell}(s)$
$\displaystyle=e^{s\Delta}(\partial_{k}A_{\ell}^{df}-\partial_{\ell}A_{k}^{df})(0)+e^{s\Delta}[A_{\ell},A_{k}](0)+F^{bil}_{k\ell}(s)$

Now use (9.18) and (9.24) again. For $e^{s\Delta}[A_{\ell},A_{k}](0)$ use
either Strichartz when we have $A^{df}$ or use the better regularity of
$A^{cf}$, together with Hölder and Bernstein. ∎

####  9.2.3. Bounds on the tension field $w$

######  Lemma 9.7.

Under the assumptions above, one has:

$\|w_{x}(s)\|_{L^{\infty}_{t}L^{1}_{x}}+s^{\frac{1}{2}}\|\partial_{x}w_{x}(s)\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim
s^{\frac{1}{2}}/(N^{2}s)^{1-\sigma}\varepsilon$ (9.28)

###### Proof.

Fix a time $t\in[t_{0},t_{1}]$. The first inequality was already proved in
(6.7).

For the second inequality we write schematically

$\partial_{x}w(s_{1})=\int_{0}^{s_{1}}\partial_{x}e^{(s_{1}-s)\Delta}\left([F,\mathbf{D}_{x}F]+[F,w]+[A,[A,w]]+[A,\partial_{x}w]\right)\,\mathrm{d}s$

We use the fact that
$\partial_{x}e^{(s_{1}-s)\Delta}:L^{1}_{x}\to(s_{1}-s)^{-\frac{1}{2}}L^{1}_{x}$
so bound the round bracket in $L^{1}_{x}$. The main term is

$\int_{0}^{s_{1}}(s_{1}-s)^{-\frac{1}{2}}\|F\|_{L^{2}_{x}}\|\mathbf{D}_{x}F\|_{L^{2}_{x}}\,\mathrm{d}s\lesssim\varepsilon/(N^{2}s_{1})^{1-\sigma}$

Similarly for $[F,w]$ and $[A,[A,w]]$: place $F(s)$, respectively the
$A(s)$’s, in $L^{\infty}_{x}$ and place $w_{x}(s)$ in $L^{1}_{x}$ using the
first part from (9.28). Finally, for $[A,\partial_{x}w]$, we place $A(s)$ in
$L^{\infty}_{x}$ by (9.21) and $\partial_{x}w$ in $L^{1}_{x}$. The resulting
term is absorbed to the LHS similarly to (6.8). ∎

######  Lemma 9.8.

Under the assumptions above, one has:

$\|w^{(3)}_{x}(s)\|_{L^{2}_{t}L^{1}_{x}}\lesssim
s^{1-\frac{1}{8}}/(Ns^{\frac{1}{2}})^{3(1-\sigma)}\varepsilon$ (9.29)

###### Proof.

Recall

$w^{(3)}(s_{1})\vcentcolon=\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}(\mathcal{N}^{(3)}(s)+\mathcal{N}^{self}(s))\,\mathrm{d}s$

where $\mathcal{N}^{(3)}$ and $\mathcal{N}^{self}$ are defined in (9.7).

We begin with $\mathcal{N}^{(3)}$:

$[F^{bil}_{0\ell},\mathbf{D}_{x}F_{0x}]$ is estimated as $L^{2}L^{2}\times
L^{\infty}L^{2}$ using Proposition 9.5.

$[e^{s\Delta}\partial_{t}A(0),\partial_{x}F^{bil}_{0x}+[A_{x},F_{0x}]]$ is
estimated as $L^{\infty}L^{2}\times L^{2}L^{2}$ using Prop. 9.5.

We continue with $\mathcal{N}^{self}$:

$[F_{k}^{\ \ell},w_{\ell}]$ is estimated as $L^{2}L^{\infty}\times
L^{\infty}L^{1}$ using Corollary 9.6 and Lemma 9.7.

$[e^{s\Delta}A^{df}+A^{bil},\partial_{x}w]$ seen as $L^{2}L^{\infty}\times
L^{\infty}L^{1}$ using (9.19), (5.14) and Lemma 9.7.

$[e^{s\Delta}A^{cf},\partial_{x}w]$ is estimated as $L^{\infty}L^{12}\times
L^{\infty}L^{\frac{12}{11}}$ using (9.15) and

$\|\partial_{x}w(s)\|_{L^{\infty}L^{\frac{12}{11}}}\lesssim\varepsilon
s^{-\frac{1}{8}}/(Ns^{\frac{1}{2}})^{2(1-\sigma)}$

obtained by interpolating between (9.28) and
$(N^{2}s)^{1-\sigma}s^{\frac{3}{4}}\partial_{x}w(s)\in\varepsilon
L^{\infty}L^{2}$.

$[A^{\ell},[A_{\ell},w_{k}]]$ is bounded in several ways, e.g. as
$L^{\infty}L^{\infty}\times L^{\infty}L^{\infty}\times L^{\infty}L^{1}$. ∎

##  10\. Trilinear estimates

This section is devoted to the proof of (1.32).

We consider a solution which is in:

  1. (1)

The temporal gauge at $s=0$, i.e. $A_{0}(t,x,0)=0\ $.

  2. (2)

The DeTurck gauge $A_{s}=\partial^{\ell}A_{\ell}$ on $[t_{0},t_{1}]\times\
\mathbb{R}^{3}\times[0,s_{0}]$.

  3. (3)

The Coulomb gauge at $s=0,\ t=t_{0}$, i.e.
$\partial^{\ell}A_{\ell}(t_{0},x,0)=0$.

We assume $A_{i}(t,x,0)$ obeys the conditions in Prop. 4.1 part (2) and Prop.
5.1 uniformly in $t$. We assume $\mathcal{IE}(t)\lesssim\varepsilon^{2}$ and
therefore (9.1), (9.2).

######  Proposition 10.1.

Let $\sigma>\frac{5}{6}$ and $t_{1}\leq t_{0}+1$. Let
$A_{t,x,s}:[t_{0},t_{1}]\times\mathbb{R}^{3}\times[0,s_{0}]\to\mathfrak{g}$ be
a regular solution to (YM) at $s=0$ and to (dYMHF) obeying the gauge
conditions $(1),(2),(3)$ above. Assume the bounds stated above and
consequently we assume all the estimates from Section 9.

Then, if $\varepsilon$ is small enough, one has

$(N^{2}s)^{1-\sigma}\left|\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\chi(x)(w^{\ell}(s),F_{0\ell}(s))\,\mathrm{d}x\,\mathrm{d}t\right|\lesssim\frac{s^{\frac{1}{4}-}}{(Ns^{\frac{1}{2}})^{(1-\sigma)}}\varepsilon^{2}$
(10.1)

After peeling off some terms, the main contribution to (10.1) will come from a
null form term. The key estimates needed for this main term are encapsulated
by:

######  Proposition 10.2.

Denoting $k_{\max}=\max(k_{1},k_{2},k_{3})$ where $k_{1},k_{2},k_{3}\geq 0$
are dyadic frequencies, one has the following inequality for frequency
localized functions:

$\left|\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}{\bf
N}(f_{k_{1}},g_{k_{2}})h_{k_{3}}\,\mathrm{d}x\,\mathrm{d}t\right|\lesssim
2^{(\frac{3}{2}+)k_{\max}}\|f_{k_{1}}\|_{X^{0,\frac{1}{2}+}}\|g_{k_{2}}\|_{X^{0,\frac{1}{2}+}}\|h_{k_{3}}\|_{X^{0,\frac{1}{2}+}}$
(10.2)

By translation invariance, the same estimate holds for $\mathcal{O}{\bf N}$ if
$\mathcal{O}$ is a disposable operator.

Recall the definitions of a null form from (2.8), (2.9). Moreover, if
$\mathcal{O}$ is a bilinear operator, we denote by $\mathcal{O}{\bf N}(f,g)$ a
linear combination of

$\Delta^{-1}\nabla^{i}\left(\mathcal{O}(\partial_{i}f,\partial_{j}g)-\mathcal{O}(\partial_{j}f,\partial_{i}g)\right)\quad\text{and}\quad\mathcal{O}(\Delta^{-1}\nabla^{i}\partial_{i}f,\partial_{j}g)-\mathcal{O}(\Delta^{-1}\nabla^{i}\partial_{j}f,\partial_{i}g)$

######  Remark 10.3.

Proposition 10.2 states a classical type of multilinear estimate, except for
the presence of the sharp time cutoff $1_{[t_{0},t_{1}]}$. The proof is
essentially contained in [22].

###  10.1. Proof of Proposition 10.1

Splitting $w(s)$ and $F_{0\ell}(s)$ by (9.4) and (9.8) we decompose

$\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\chi(x)\big{(}w^{\ell}(s),F_{0\ell}(s)\big{)}\,\mathrm{d}x\,\mathrm{d}t=\mathcal{T}_{1}+\mathcal{T}_{2}+\mathcal{T}$

into two quadrilinear terms and a trilinear term

$\displaystyle\mathcal{T}_{1}\vcentcolon=\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\chi(x)\big{(}w^{(3)\ell}(s),F_{0\ell}(s)\big{)}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\mathcal{T}_{2}\vcentcolon=\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\chi(x)\big{(}w^{(2)\ell}(s),F_{0\ell}^{bil}(s)\big{)}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\mathcal{T}\vcentcolon=\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\chi(x)\big{(}w^{(2)\ell}(s),e^{s\Delta}\partial_{t}A_{\ell}\big{)}\,\mathrm{d}x\,\mathrm{d}t$

We first estimate the quadrilinear terms by spacetime norms:

$\left|\mathcal{T}_{1}\right|\lesssim\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\left|\chi\big{(}w^{(3)}(s),F(s)\big{)}\right|\,\mathrm{d}x\,\mathrm{d}t\lesssim\|w^{(3)}(s)\|_{L^{2}_{t}L^{1}_{x}}\|F(s)\|_{L^{2}_{t}L^{\infty}_{x}}$
(10.3)

which is bounded by $s^{\frac{3}{8}-}/(N^{2}s)^{2(1-\sigma)}\varepsilon^{2}$
using Lemma 9.8 and Corollary 9.6,

$\left|\mathcal{T}_{2}\right|\lesssim\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\left|\chi\big{(}w^{(2)}(s),F^{bil}(s)\big{)}\right|\,\mathrm{d}x\,\mathrm{d}t\lesssim\|w^{(2)}(s)\|_{L^{2}_{t}L^{2}_{x}}\|F^{bil}(s)\|_{L^{2}_{t}L^{2}_{x}}$
(10.4)

which is also bounded by
$s^{\frac{3}{8}-}/(N^{2}s)^{2(1-\sigma)}\varepsilon^{2}$ using (9.12) and
(9.25). Since $\sigma>\frac{5}{6}$, this is more than enough.

It remains to consider the main term $\mathcal{T}$. Performing a Littlewood-
Paley decomposition we need to bound

$\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)\ell}(s),\tilde{P}_{k}(\chi
e^{s\Delta}\partial_{t}A_{\ell})\big{)}\,\mathrm{d}x\,\mathrm{d}t$

To pass the frequency localization onto $\partial_{t}A_{\ell}$ write

$\tilde{P}_{k}(\chi e^{s\Delta}\partial_{t}A_{\ell})=\tilde{P}_{k}(\chi
e^{s\Delta}P_{k}\partial_{t}A_{\ell})+\tilde{P}_{k}([\chi,P_{k}]e^{s\Delta}\partial_{t}A_{\ell})$

The commutator term is bounded in (10.20). Then main term is

$\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)\ell}(s),\chi
e^{s\Delta}\partial_{t}P_{k}A_{\ell}\big{)}\,\mathrm{d}x\,\mathrm{d}t.$

We split all inputs as $\partial_{t}A=\partial_{t}A^{df}+\partial_{t}A^{cf}$.
For the dominant term, with $3$ divergence-free inputs, we further split
$w^{(2)}={\bf P}w^{(2)}+{\bf P}^{\perp}w^{(2)}$ obtaining:

$\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}{\bf
P}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{df},s),\chi
e^{s\Delta}\partial_{t}A^{df}_{k}\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.5)

which contains a null form, and the following commutator-type error term

$\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}{\bf
P}^{\perp}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{df},s),\tilde{P}_{k}(\chi
e^{s\Delta}\partial_{t}A^{df}_{k})\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.6)

Then we have terms with $1$ curl-free input and $2$ divergence-free inputs:

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{df},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{cf}\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.7)
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)}(\partial_{t}A^{cf},\partial_{t}A^{df},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{df}\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.8)
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{cf},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{df}\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.9)

terms with $2$ curl-free inputs and $1$ divergence-free input:

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)}(\partial_{t}A^{cf},\partial_{t}A^{cf},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{df}\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.10)
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)}(\partial_{t}A^{cf},\partial_{t}A^{df},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{cf}\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.11)
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{cf},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{cf}\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (10.12)

and finally a term with $3$ curl-free inputs

$\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}P_{k}w^{(2)}(\partial_{t}A^{cf},\partial_{t}A^{cf},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{cf}\big{)}\,\mathrm{d}x\,\mathrm{d}t.$ (10.13)

####  10.1.1. The main term (10.5)

We proceed to specifying the null form, recalling (9.9). From the identities
(2.7) and Lemma 9.2 one can write

$P_{k}{\bf P}\mathbf{W}({\bf
P}\partial_{t}A^{\ell}_{k_{1}},\partial_{x}\partial_{t}A^{df}_{k_{2},\ell}-2\partial_{\ell}\partial_{t}A_{k_{2},x}^{df})$
(10.14)

as

$\left\langle s2^{2k}\right\rangle^{-10}\left\langle
s^{-1}2^{-2k_{\max}}\right\rangle^{-1}2^{-2k_{\max}}P_{k}\mathcal{O}{\bf
N}(\partial_{t}A^{df}_{k_{1}},\partial_{t}A^{df}_{k_{2}})$ (10.15)

where $\mathcal{O}$ is disposable.

Since we assume the conditions in Proposition 4.1 part (2) we have
$I\partial_{t}A^{df}\in\varepsilon X^{0,\frac{3}{4}+}$. Additionally, from
Lemma 2.4

$\chi e^{s\Delta}P_{k}\partial_{t}A^{df}\in\varepsilon\left\langle
2^{k}/N\right\rangle^{1-\sigma}X^{0,\frac{3}{4}+}$

We apply Prop. 10.2 to bound the contribution of (10.5) to the LHS of (10.1)
by

$(N^{2}s)^{1-\sigma}2^{(-\frac{1}{2}-)k_{\max}}\left\langle
s2^{2k}\right\rangle^{-10}\left\langle
s^{-1}2^{-2k_{\max}}\right\rangle^{-1}\left\langle\frac{2^{k}}{N}\right\rangle^{1-\sigma}\left\langle\frac{2^{k_{1}}}{N}\right\rangle^{1-\sigma}\left\langle\frac{2^{k_{2}}}{N}\right\rangle^{1-\sigma}\varepsilon^{3}$

After summation in $k_{1},k_{2},k$ the result is $\lesssim$ RHS of (10.1). ∎

We next estimate the terms (10.7) - (10.13), beginning by decomposing them
into

$(N^{2}s)^{1-\sigma}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}(P_{k}\mathbf{W}(\partial_{t}A^{(1)}_{k_{1}},\nabla_{x}\partial_{t}A^{(2)}_{k_{2}},s),\chi
e^{s\Delta}\partial_{t}A_{k}^{(3)})\,\mathrm{d}x\,\mathrm{d}t$ (10.16)

####  10.1.2. The terms (10.10)-(10.13) with low-high inputs in $w$

Here we consider the contributions of (10.16) when $2^{k}\simeq 2^{k_{\max}}$
and at most one of $A^{(1)},A^{(2)},A^{(3)}$ is $A^{df}$. By Lemma 9.2 and
translation-invariance, without loss of generality it suffices to bound

$2^{-k}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\partial_{t}A^{(1)}_{k_{1}}\cdot\partial_{t}A^{(2)}_{k_{2}}\cdot
P_{k}(\chi
e^{s\Delta}\partial_{t}A_{k}^{(3)})\,\mathrm{d}x\,\mathrm{d}t\times\left\langle
s2^{2k}\right\rangle^{-10}\left\langle
s^{-1}2^{-2k}\right\rangle^{-1}(N^{2}s)^{1-\sigma}$ (10.17)

when $2^{k_{1}}\lesssim 2^{k_{2}}\simeq 2^{k}$. We bound this term as
$L^{2}L^{\infty}\times L^{\infty}L^{2}\times L^{\infty}L^{2}$, the low
frequency in $L^{2}L^{\infty}$ using (4.12), (9.18), (9.16), (9.17). After
summing in $k_{1}$, and then in $k$ using the $\left\langle
s2^{2k}\right\rangle^{-10}\left\langle s^{-1}2^{-2k}\right\rangle^{-1}$
factors one obtains a bound of
$s^{\frac{1}{2}-}(Ns^{\frac{1}{2}})^{-3(1-\sigma)}.$

####  10.1.3. The terms (10.10)-(10.13) with high-high inputs in $w$

Here we consider the contributions of (10.16) when $2^{k}\ll 2^{k_{1}}\simeq
2^{k_{2}}$ and at most one of $A^{(1)},A^{(2)},A^{(3)}$ is $A^{df}$. By Lemma
9.2, without loss of generality it suffices to bound

$2^{-{k_{2}}}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}P_{k}(\partial_{t}A^{(1)}_{k_{1}}\cdot\partial_{t}A^{(2)}_{k_{2}})\chi
e^{s\Delta}\partial_{t}A_{k}^{(3)}\,\mathrm{d}x\,\mathrm{d}t\times\left\langle
s2^{2k}\right\rangle^{-10}\left\langle
s^{-1}2^{-2{k_{2}}}\right\rangle^{-1}(N^{2}s)^{1-\sigma}$ (10.18)

We use the norms of $L^{\infty}L^{2},L^{2}L^{2}$ and $L^{2}L^{\infty}$ as
follows: a high frequency $\partial_{t}A^{cf}$ is always placed in
$L^{2}L^{2}$ using (4.13), the term $\partial_{t}A^{df}$ (if present) is
placed in $L^{2}L^{\infty}$ using (9.18) (otherwise place the low frequency
$\partial_{t}A^{cf}$ in $L^{2}L^{\infty}$ using (4.12)) and the remaining
$\partial_{t}A^{cf}$ term is placed in $L^{\infty}L^{2}$ by (9.16). After
summation we obtain either $s^{\frac{5}{8}-}(Ns^{\frac{1}{2}})^{-4(1-\sigma)}$
or $s^{\frac{3}{8}-}(Ns^{\frac{1}{2}})^{-(1-\sigma)}$.

####  10.1.4. The terms (10.7)-(10.9) with low-high inputs in $w$

Now consider the contributions of (10.16) when $2^{k}\simeq 2^{k_{\max}}$ and
two of $A^{(1)},A^{(2)},A^{(3)}$ are $A^{df}$ and one is $A^{cf}$. Again it
suffices to bound (10.17) when $2^{k_{1}}\lesssim 2^{k_{2}}\simeq 2^{k}$. We
place the $\partial_{t}A^{cf}$ term in $L^{2}L^{2}$ using (4.13). In the cases
of (10.7) and (10.9) when $\partial_{t}A^{df}$ is low frequency, place this
term in $L^{2}L^{\infty}$ using (9.18) and the other term in
$L^{\infty}L^{2}$. In the case of (10.8) place the product of the two high
frequency df terms in $L^{2}L^{2}$ using (2.31). After summing in $k_{1}$, and
then in $k$ using the $\left\langle s2^{2k}\right\rangle^{-10}\left\langle
s^{-1}2^{-2k}\right\rangle^{-1}$ factors one obtains a bound of
$s^{\frac{3}{8}-}(Ns^{\frac{1}{2}})^{-2(1-\sigma)}$ or $s^{\frac{1}{4}-}$.

####  10.1.5. The terms (10.7)-(10.9) with high-high inputs in $w$

Here we consider the contributions of (10.16) when $2^{k}\ll 2^{k_{1}}\simeq
2^{k_{2}}$ and two of $A^{(1)},A^{(2)},A^{(3)}$ are $A^{df}$ and one is
$A^{cf}$. Again it suffices to bound (10.18) and we place the
$\partial_{t}A^{cf}$ term in $L^{2}L^{2}$ using (4.13). In the case of (10.7)
we place the product of the two high frequency df terms in $L^{2}L^{2}$ using
(2.31). In the cases of (10.8), (10.9), place the low frequency
$\partial_{t}A^{df}$ in $L^{2}L^{\infty}$ using (9.18) and the high frequency
$\partial_{t}A^{df}$ in $L^{\infty}L^{2}$. In all cases, after summations, we
obtain a bound slightly better than
$s^{\frac{1}{4}-}(Ns^{\frac{1}{2}})^{-(1-\sigma)}$. ∎

We continue with the commutator-type terms (10.20) and (10.6), but first let
us make some preparations. Recall the commutator identity [51, Lemma 2]:

$[P_{k},\chi]f=2^{-k}L(\nabla\chi,f)$ (10.19)

where $L$ is a disposable operator. It’s possible to pass $I$ bounds from $f$
to $L(\nabla\chi,f)$:

######  Lemma 10.4.

Let $\chi^{\prime}$ be a bump function and $L$ a disposable operator. One has

$\|P_{k}L(\chi^{\prime},f)\|_{L^{2}_{x}}\lesssim\left\langle\frac{2^{k}}{N}\right\rangle^{1-\sigma}\|If\|_{L^{2}_{x}}$

###### Proof.

By translation-invariance and the integrable kernel, it suffices to prove this
for $P_{k}(\chi^{\prime}f)$ which we separate as
$P_{k}(\chi^{\prime}f_{<k+c})$ and $P_{k}(\chi^{\prime}_{>k+c-3}f_{>k+c})$.
Normalize $\|If\|_{L^{2}_{x}}=1$. Then

$\displaystyle\|P_{k}(\chi^{\prime}f)\|_{L^{2}_{x}}$
$\displaystyle\lesssim\|f_{<k+c}\|_{L^{2}_{x}}+\sum_{k_{2}=k_{1}+O(1)\geq
k+c}\|\chi^{\prime}_{k_{1}}\|_{L^{\infty}_{x}}\|f_{k_{2}}\|_{L^{2}_{x}}$
$\displaystyle\lesssim\left\langle
2^{k}/N\right\rangle^{1-\sigma}+\sum_{k_{2}\geq k+c}2^{-5k_{2}}\left\langle
2^{k_{2}}/N\right\rangle^{1-\sigma}\lesssim\left\langle
2^{k}/N\right\rangle^{1-\sigma}.$

∎

######  Lemma 10.5.

Let $f_{k}$ be a function localized at frequencies
$\left\langle\xi\right\rangle\simeq 2^{k}$, $\chi$ be a bump function and let
$m(\xi)$ be a homogeneous multiplier of degree zero. Then

$\|P_{k}[m(D),\chi]f_{k}\|_{L^{2}_{x}}\lesssim 2^{-k}\|f_{k}\|_{L^{2}_{x}}.$

###### Proof.

Note that

$P_{k}[m(D),\chi]f_{k}=P_{k}[m(D),\chi_{<k-3}]f_{k}+P_{k}[m(D),P_{[k-3,k+3]}\chi]f_{k}$

For the latter term, bounds the two terms of this commutator separately as

$\|P_{k}[m(D),P_{[k-3,k+3]}\chi]f_{k}\|_{L^{2}_{x}}\lesssim\|P_{[k-3,k+3]}\chi\|_{L^{\infty}_{x}}\|f_{k}\|_{L^{2}_{x}}\leq
2^{-10k}\|f_{k}\|_{L^{2}_{x}}.$

For the former term, denoting $Tf_{k}=[m(D),\chi_{<k-3}]f_{k}$,

$\displaystyle\widehat{Tf_{k}}(\xi)$
$\displaystyle=\int[m(\xi)-m(\xi-\eta)]\hat{\chi}_{<k-3}(\eta)\hat{f_{k}}(\xi-\eta)\,\mathrm{d}\eta$
$\displaystyle=\int_{0}^{1}\int_{\eta\ll 2^{k}}\nabla
m(\xi-t\eta)\eta\hat{\chi}_{<k-3}(\eta)\hat{f_{k}}(\xi-\eta)\,\mathrm{d}\eta\,\mathrm{d}t$

Apply $P_{k}$ and take the $L^{2}_{\xi}$ norm, use Minkowski’s inequality,
take absolute values, bound
$\left|\nabla m(\xi-t\eta)\right|\lesssim 2^{-k}$ and use the integrability of
$\left|\eta\right|\hat{\chi}_{<k-3}(\eta)$. ∎

####  10.1.6. Proof of the first commutator bound:

$(N^{2}s)^{1-\sigma}\sum_{k}\left|\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}(P_{k}w^{(2)\ell}(s),\tilde{P}_{k}([\chi,P_{k}]e^{s\Delta}\partial_{t}A_{\ell}))\,\mathrm{d}x\,\mathrm{d}t\right|\lesssim\frac{s^{\frac{1}{4}-}}{(Ns^{\frac{1}{2}})^{(1-\sigma)}}\varepsilon^{2}$
(10.20)

We bound $P_{k}w^{(2)}(s)$ in $L^{2}_{t}L^{2}_{x}$ using (9.11). The term
$\tilde{P}_{k}([\chi,P_{k}]e^{s\Delta}\partial_{t}A_{\ell})$ is viewed as

$2^{-k}\tilde{P}_{k}L(\nabla\chi,e^{s\Delta}\partial_{t}A)$

using (10.19) which we bound using Lemma 10.4 as

$2^{-k}\|\tilde{P}_{k}L(\nabla\chi,e^{s\Delta}\partial_{t}A)\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\varepsilon
2^{-k}\left\langle 2^{k}/N\right\rangle^{1-\sigma}.$

After summing in $k$ we bound the product by the RHS of (10.20).

####  10.1.7. The second commutator bound (10.6)

We prove that

$(N^{2}s)^{1-\sigma}\sum_{k}\left|\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}(P_{k}{\bf
P}^{\perp}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{df},s),\tilde{P}_{k}(\chi
e^{s\Delta}\partial_{t}A^{df}_{k}))\,\mathrm{d}x\,\mathrm{d}t\right|$

is $\lesssim$ the RHS of (10.20). Since ${\bf P}^{\perp}\partial_{t}A^{df}=0$
one can write

${\bf P}^{\perp}\chi e^{s\Delta}\partial_{t}A^{df}_{k}=[{\bf
P}^{\perp},\chi]e^{s\Delta}\partial_{t}A^{df}_{k}$

By Lemma 10.5

$\|\tilde{P}_{k}[{\bf
P}^{\perp},\chi]e^{s\Delta}\partial_{t}A^{df}_{k}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\varepsilon
2^{-k}\left\langle 2^{k}/N\right\rangle^{1-\sigma}$

By Lemma 9.11 and Remark 9.4 we obtain an $L^{2}_{t}L^{2}_{x}$ bound for
$P_{k}w^{(2)}(\partial_{t}A^{df},\partial_{t}A^{df},s)$. The numerology is the
same as in the proof of (10.20). ∎

###  10.2. Proof of Proposition 10.2

Integrating by parts if needed, one may assume without loss of generality that
$k_{1}=\min(k_{1},k_{2},k_{3})$. One may also assume that the Fourier
transforms on $\mathbb{R}\times\mathbb{R}^{3}$ of the inputs are real and non-
negative, as one then estimates

$\int_{\mathbb{R}\times\mathbb{R}^{3}}1_{[t_{0},t_{1}]}(t){\bf
N}(f_{k_{1}},g_{k_{2}})h_{k_{3}}\,\mathrm{d}x\,\mathrm{d}t$

using Plancharel’s identity by

$\int\frac{\left|\xi_{1}\wedge\xi_{2}\right|}{\left|\xi_{1}\right|\left\langle\tau_{1}+\tau_{2}+\tau_{3}\right\rangle}\hat{f}_{k_{1}}(\tau_{1},\xi_{1})\hat{g}_{k_{2}}(\tau_{2},\xi_{2})\hat{h}_{k_{3}}(\tau_{3},\xi_{3})$

since
$\left|\hat{1}_{[t_{0},t_{1}]}(\tau)\right|\lesssim\left\langle\tau\right\rangle^{-1}$.
The integral is taken over the subset of

$(\tau_{1},\xi_{1},\tau_{2},\xi_{2},\tau_{3},\xi_{3})\in(\mathbb{R}\times\mathbb{R}^{3})^{3}$

with $\xi_{1}+\xi_{2}+\xi_{3}=0$. Denote the modulations by
$\lambda_{i}\vcentcolon=\left\langle\left|\xi\right|_{i}-\left|\tau\right|_{i}\right\rangle$.

Denoting $a_{\alpha}(t)$ the Fourier transform of
$\left\langle\tau\right\rangle^{-\alpha}$, we recall

$a_{\alpha}(t)\in L^{p}_{t},\qquad\forall\ p<(1-\alpha)^{-1},\ \alpha\in(0,1]$
(10.21)

from [22, Lemma 15.2]. We also recall

$\left|\xi_{1}\wedge\xi_{2}\right|\lesssim
2^{\frac{k_{1}}{2}}2^{\frac{k_{2}}{2}}2^{\frac{k_{3}}{2}}(\left\langle\tau_{1}+\tau_{2}+\tau_{3}\right\rangle+\lambda_{1}+\lambda_{2}+\lambda_{3})^{\frac{1}{2}}$
(10.22)

from [28, Prop. 1] or [22, Lemma 15.1].

Then it suffices to consider three cases and bound (10.23), (10.24), (10.25)
as follows.

(1) Bound

$2^{k_{2}-\frac{k_{1}}{2}}\int\frac{1}{\left\langle\tau_{1}+\tau_{2}+\tau_{3}\right\rangle^{\frac{1}{2}}}\hat{f}_{k_{1}}(\tau_{1},\xi_{1})\hat{g}_{k_{2}}(\tau_{2},\xi_{2})\hat{h}_{k_{3}}(\tau_{3},\xi_{3})$
(10.23)

by the RHS of (10.2). Undoing the Fourier transform one proves

$2^{k_{2}-\frac{k_{1}}{2}}\left|\int_{\mathbb{R}\times\mathbb{R}^{3}}a_{\frac{1}{2}}(t)f_{k_{1}}g_{k_{2}}h_{k_{3}}\,\mathrm{d}x\,\mathrm{d}t\right|\lesssim
RHS\eqref{CoreTrilEst}$

which is bounded as $L^{2-}_{t}L^{\infty}_{x}\times
L^{2+}_{t}L^{\infty}_{x}\times L^{\infty}_{t}L^{2}_{x}\times
L^{\infty}_{t}L^{2}_{x}$ using (10.21) and Strichartz estimates (2.30).

(2) When $\lambda_{1}$ dominates in (10.22), by dyadic decomposition and
Cauchy-Schwarz, it suffices to integrate on the subregion
$\lambda_{2},\lambda_{3}\lesssim 2^{j}\simeq\lambda_{1}$ and prove

$\displaystyle
2^{-\frac{k_{1}}{2}}\int\frac{1}{\left\langle\tau_{1}+\tau_{2}+\tau_{3}\right\rangle}\hat{u}_{k_{1},j}(\tau_{1},\xi_{1})$
$\displaystyle\hat{g}_{k_{2}}(\tau_{2},\xi_{2})\hat{h}_{k_{3}}(\tau_{3},\xi_{3})\lesssim$
(10.24) $\displaystyle 2^{(\frac{1}{2}+)k_{2}}$
$\displaystyle\|u_{k_{1},j}\|_{X^{0,0+}}\|g_{k_{2}}\|_{X^{0,\frac{1}{2}+}}\|h_{k_{3}}\|_{X^{0,\frac{1}{2}+}}$

where we denote $\hat{u}_{k_{1},j}=\lambda_{1}^{\frac{1}{2}}\hat{f}_{k_{1}}$.
It suffices to prove

$\left|\int_{\mathbb{R}\times\mathbb{R}^{3}}a_{1}(t)u_{k_{1},j}D^{-(\frac{1}{2}-)}(g_{k_{2}}h_{k_{3}})\,\mathrm{d}x\,\mathrm{d}t\right|\lesssim
RHS\eqref{Est2}$

Use (10.21) to place $a_{1}(t)$ in $L^{\infty-}_{t}L^{\infty}_{x}$, the
embedding (2.27) for $u_{k_{1},j}$ and the bilinear Strichartz estimate (2.31)
for the product.

(3) In the remaining case, we can assume without loss of generality that
$\lambda_{2}$ dominates and integrate on the subregion
$\lambda_{1},\lambda_{3}\lesssim 2^{j}\simeq\lambda_{2}$, proving

$\displaystyle\int\frac{1}{\left\langle\tau_{1}+\tau_{2}+\tau_{3}\right\rangle}\hat{f}_{k_{1}}(\tau_{1},\xi_{1})\hat{v}_{k_{2},j}(\tau_{2},\xi_{2})$
$\displaystyle\hat{h}_{k_{3}}(\tau_{3},\xi_{3})\lesssim$ (10.25)
$\displaystyle 2^{\frac{k_{1}}{2}}2^{(\frac{1}{2}+)k_{2}}$
$\displaystyle\|f_{k_{1}}\|_{X^{0,\frac{1}{2}+}}\|\hat{v}_{k_{2},j}\|_{L^{2}L^{2}}\|h_{k_{3}}\|_{X^{0,\frac{1}{2}+}}$

where we denote $\hat{v}_{k_{2},j}=\lambda_{2}^{\frac{1}{2}}\hat{g}_{k_{2}}$.
Undoing the Fourier transform this is reduced to

$\left|\int_{\mathbb{R}\times\mathbb{R}^{3}}a_{1}(t)f_{k_{1}}v_{k_{2},j}h_{k_{3}}\,\mathrm{d}x\,\mathrm{d}t\right|\lesssim
RHS\eqref{Est3}$

which is bounded as $L^{\infty-}_{t}L^{\infty}_{x}\times
L^{4}_{t}L^{4}_{x}\times L^{2}_{t}L^{2}_{x}\times L^{4+}_{t}L^{4}_{x}$ using
(10.21) and Strichartz estimates (2.30).

##  Appendix A Modified energy for Maxwell-Klein-Gordon

In this appendix we formulate the gauge invariant modified energy for the
Maxwell-Klein-Gordon equation, we show that the same favorable balance of
derivatives as for Yang-Mills is present when it is differentiated in time
(see Remark 1.14) and we sketch how this can be used to increase the range of
regularities for which global well-posedness is known to hold, from
$\sigma>\frac{\sqrt{3}}{2}$ in [22] to $\sigma>\frac{5}{6}$.

The Yang-Mills equations in the case of the abelian group $G=SO(2)=U(1)$,
$\mathfrak{g}=\mathfrak{so}(2)$ correspond to Maxwell’s equations, where
$A_{\alpha}$ can be viewed as a real one-form
$A_{\alpha}:\mathbb{R}^{1+3}\to\mathbb{R}$. Coupling with a complex field
$\phi:\mathbb{R}^{1+3}\to\mathbb{C}$ one obtains the (massless) Maxwell-Klein-
Gordon system [26], [36], [22], as the Euler-Lagrange equations for the
Lagrangian

$\frac{1}{2}\int_{\mathbb{R}^{1+3}}D_{\alpha}\phi\overline{D^{\alpha}\phi}+\frac{1}{2}F_{\alpha\beta}F^{\alpha\beta}\,\mathrm{d}x\,\mathrm{d}t$

Here one denotes the covariant derivative $D_{\alpha}$ and the curvature
$F_{\alpha\beta}$ by

$D_{\alpha}\phi\vcentcolon=(\partial_{\alpha}+iA_{\alpha})\phi,\qquad
F_{\alpha\beta}\vcentcolon=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}.$

The system becomes

$\displaystyle\partial^{\beta}F_{\alpha\beta}$
$\displaystyle=\mathrm{Im}(\phi\overline{D_{\alpha}\phi})$ (MKG)
$\displaystyle\Box_{A}\phi$ $\displaystyle=0.$

where we denote $\Box_{A}\vcentcolon=D^{\alpha}D_{\alpha}$. The gauge
invariance takes the form

$\phi\mapsto e^{i\chi}\phi,\qquad A_{\alpha}\mapsto
A_{\alpha}-\partial_{\alpha}\chi,\qquad\chi:\mathbb{R}^{1+3}\to\mathbb{R}$

while the energy is

$H[A,\phi](t)\vcentcolon=\frac{1}{2}\int_{\mathbb{R}^{3}}\sum_{\alpha<\beta}\left|F_{\alpha\beta}\right|^{2}+\sum_{\alpha}\left|D_{\alpha}\phi\right|^{2}\,\mathrm{d}x.$
(A.1)

Initial data sets consist of
$(\underline{A}_{i},\underline{E}_{i},\phi_{0},\phi_{1})=(A_{i},F_{0i},\phi,D_{t}\phi)(t=0)$
satisfying the constraint equation
$\partial^{i}\underline{E}_{i}=\mathrm{Im}(\phi_{0}\overline{\phi_{1}})$.

In [22] Keel, Roy and Tao prove global well-posedness in the (global) Coulomb
gauge $\partial^{i}A_{i}=0$. In this gauge, $A_{0}$ is determined from the
initial data.

######  Theorem A.1 ([22]).

Let $\sigma>\frac{\sqrt{3}}{2}$. The Maxwell-Klein-Gordon equation in the
Coulomb gauge is globally well-posed for initial data in $H^{\sigma}\times
H^{\sigma-1}\times H^{\sigma}\times H^{\sigma-1}$.

After reducing to global smooth solutions, rescaling and choosing $N$ such
that $H[IA,I\phi](0)\leq 1$, their proof proceeds by proving the almost
conservation law

$H[I\Phi](t)=H[I\Phi](t_{0})+O(N^{\frac{1}{2}-\sigma+})$ (A.2)

under the assumption $H[I\Phi](t_{0})\leq 2$, where one denotes
$\Phi=(A_{0},A_{x},\phi)$.

The spacetime control on small intervals is $\nabla_{t,x}(IA_{x},I\phi)\in
X^{0,\sigma-}+L^{\infty}L^{3}$ where the latter part consists of low
frequencies.

For arbitrary fields [22][Lemma 11.1] writes the differentiated Hamiltonian as

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}H[\Phi](t)=-\mathrm{Re}\int_{\mathbb{R}^{3}}D_{t}\phi\overline{\Box_{A}\phi}+F_{\mu
0}(\partial^{\alpha}F_{\alpha}^{\
\mu}+\mathrm{Im}(\phi\overline{D^{\mu}\phi})\,\mathrm{d}x$ (A.3)

The dominant part of $H[I\Phi](t_{1})-H[I\Phi](t_{0})$ is proved in [22,
(106),(124)] to be (1.38).

We now formulate the invariant modified energy for (MKG).

The gradient flow for the functional

$\frac{1}{2}\int_{\mathbb{R}^{3}}\sum_{i<j}\left|F_{ij}\right|^{2}+\sum_{i}\left|D_{i}\phi\right|^{2}\,\mathrm{d}x.$

gives rise to the equations $\partial_{s}A_{i}=\partial^{\ell}F_{\ell
i}+\mathrm{Im}(\phi\overline{D_{i}\phi})$ and
$\partial_{s}\phi=D^{\ell}D_{\ell}\phi$. Adding temporal and parabolic
connections $A_{0},A_{s}$ and writing the system in a gauge-invariant way one
obtains a Maxwell-Klein-Gordon Heat Flow

$\displaystyle
F_{s\alpha}=\partial^{\ell}F_{\ell\alpha}+\mathrm{Im}(\phi\overline{D_{\alpha}\phi})$
(MKG-HF) $\displaystyle(D_{s}-\Delta_{A})\phi=0.$

The extent of failure of $A_{\alpha}(s),\phi(s)$ to satisfy (MKG) for $s>0$ is
measured by the tension fields

$v\vcentcolon=\Box_{A}\phi,\qquad
w_{\beta}\vcentcolon=\partial^{\alpha}F_{\alpha\beta}+\mathrm{Im}(\phi\overline{D_{\beta}\phi}).$

Up to lower order terms, the main components of the parabolic equations
satisfied by the tension fields can be written as

$\displaystyle(\partial_{s}-\Delta){\bf P}_{j}w$ $\displaystyle=-2{\bf
P}_{j}\mathrm{Im}(\partial_{t}\phi\overline{\partial_{x}\partial_{t}\phi})+\dots$
(A.4) $\displaystyle(D_{s}-\Delta_{A})v$ $\displaystyle=4i\partial_{t}{\bf
P}^{j}A\partial_{j}\partial_{t}\phi+\dots$

The dots consist of trilinear terms or terms with higher regularity, such as
those involving $A^{cf}$ or $A_{0}$. Assuming deTurck’s gauge
$A_{s}=\partial^{\ell}A_{\ell}$, expanding and decomposing (A.4) similarly to
Section 9 one obtains the leading terms

$\displaystyle{\bf P}_{j}w(s)$ $\displaystyle=-2{\bf
P}_{j}\mathrm{Im}\mathbf{W}(\partial_{t}\phi,\overline{\partial_{x}\partial_{t}\phi})(s)+\dots$
(A.5) $\displaystyle v(s)$ $\displaystyle=4i\mathbf{W}(\partial_{t}{\bf
P}^{j}A,\partial_{j}\partial_{t}\phi)(s)+\dots$

where $\mathbf{W}$ is defined by (9.10) and the inputs are taken at $s=0$.

For a regular global solution of (MKG) we define the modified energy similarly
to (1.6). Denote the energies (A.1) defined by
$A_{\alpha}(t,\cdot,s),\phi(t,\cdot,s)$ by $H(t,s)$. Then, for $s_{0}=N^{-2}$,
let

$\mathcal{IH}(t)\vcentcolon=\sup_{s\in[0,s_{0}]}(N^{2}s)^{1-\sigma}H(t,s)+\int_{0}^{s_{0}}(N^{2}s)^{1-\sigma}H(t,s)\frac{\,\mathrm{d}s}{s}$
(A.6)

The goal is to have $\left|\mathcal{IH}(t)-\mathcal{IH}(t_{0})\right|\ll
N^{-\frac{1}{2}+}$ like in Proposition 1.7 as this can be used to obtain
global well-posedness for (MKG) as in Section 1.4, with the name numerology.
The $N^{-\frac{1}{2}+}$ bound compared to (A.2) reduces the minimum initial
data regularity from $\sigma>\frac{\sqrt{3}}{2}$ to $\sigma>\frac{5}{6}$.

Regarding the choices of gauge there are several options and we avoid making a
definitive choice. One may set the temporal gauge $A_{0}=0$. Working with
local gauges and extensions like we did for Yang-Mills is probably not
necessary, but can be a natural choice in view of potential extensions to
Yang-Mills-Higgs (which generalizes both (YM) and (MKG)). This also avoids
some low frequency technicalities. On the other hand, after fixing the
interval $[t_{0},t_{1}]$, one could set the global Coulomb gauge at
$s=s_{0},t=t_{0}$ and obtain an $H^{1}+N^{1-\sigma}H^{\sigma}$ bound at $s=0$
from the caloric condition like in Section 7, but globally instead of locally.

The key spacetime bound remains $\nabla_{t,x}(IA_{x}^{df},I\phi)\in
X^{0,\frac{3}{4}+}$, perhaps with a low frequency component.

To estimate the variation of the modified energy one writes similarly to
(1.21), using (A.3)

$H(t_{1},s)-H(t_{0},s)=-\mathrm{Re}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}D_{t}\phi\overline{v}+F_{j0}w_{j}\,\mathrm{d}x\,\mathrm{d}t$
(A.7)

As (10.3), (10.4) show for quadrilinear terms and (10.7)-(10.13) show for
terms involving $A^{cf}$, in the case of temporal gauge + DeTurck gauge, all
these terms can be fairly easily bounded using spacetime Lebesgue norms. In
the case of the Coulomb gauge, sections 12 and 14 in [22] show that such terms
are not problematic.

Up to error terms, (A.7) is written, from (A.5) and the identities (2.7), as

$\mathrm{Re}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}4i\partial_{t}\phi\mathbf{W}{\bf
N}(\partial_{t}A^{df},\overline{\partial_{t}\phi})(s)+2\partial_{t}A^{df}\mathrm{Im}\mathbf{W}{\bf
N}(\partial_{t}\phi,\overline{\partial_{t}\phi})(s)\,\mathrm{d}x\,\mathrm{d}t+\dots$
(A.8)

Schematically, these terms are exactly like (10.5) together with (10.15) and
are estimated using Proposition 10.2.

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