## Lecture 23. Davie’s estimate (2)

We now turn to the proof of Davie’s estimate. We follow the approach by Friz-Victoir who smartly use interpolations by geodesics in Carnot groups.

Theorem: Let $\gamma > p \ge 1$. Assume that $V_1, \cdots, V_d$ are $(\gamma-1)$-Lipschitz vector fields in $\mathbb{R}^n$. Let $x \in C^{1-var}([0,T], \mathbb{R}^d)$. Let $y$ be the solution of the equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.$
There exists a constant $C$ depending only on $p$ and $\gamma$ such that for every $0 \le s < t \le T$,
$\| y \|_{p-var, [s,t]} \le C \left(\| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[s,t]} +\| V \|^p_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^p_{p-var,[s,t]} \right),$
where $S_{[p]} (x)$ is the lift of $x$ in $\mathbb{G}_{[p]}(\mathbb{R}^d)$.

Proof: For $s \le t$, we denote by $x^{s,t}$ a path in $C^{1-var}([s,t],\mathbb{R}^d)$ such that $S_{[\gamma]}( x^{s,t})(s)=S_{[\gamma]}( x)(s)$, $S_{[\gamma]}( x^{s,t})(t)=S_{[\gamma]}( x )(t)$ and $S_{[\gamma]}( x^{s,t})(u)$, $s\le u \le t$, is a geodesic for the Carnot-Caratheodory distance. We consider then $y^{s,t}$ to be the solution of the equation
$y^{s,t} (u)=y(s)+\sum_{i=1}^d \int_s^u V_i(y^{s,t} (v)) dx^i(v), \quad s \le u \le t.$
We can readily observe that from the continuity of Lyons’ lift:
$\| x^{s,t} \|_{1-var, [s,t]} = d( S_{[\gamma]}( x)(s), S_{[\gamma]}( x)(t)) \le \| S_{[\gamma]} (x) \|_{p-var,[s,t]} \le K \| S_{[p]} (x) \|_{p-var,[s,t]} .$
Let us now denote
$\Gamma_{s,t} =(y(t) -y(s)) -( y^{s,t} (t) -y^{s,t} (s)).$
For fixed $s \le t \le u$, we have then:
$\Gamma_{s,u}-\Gamma_{s,t}-\Gamma_{t,u}=(y^{s,u}(s)-y^{s,u}(u))-( y^{s,t}(s)-y^{s,t}(t))-(y^{t,u}(t)-y^{t,u}(u)).$
To estimate this quantity, we consider the path $y^{s,t,u}(v)$, $s \le v \le u$, that solves the ordinary differential equation driven by the concatenation of $x^{s,t}$ and $x^{t,u}$. We first estimate $y^{s,t,u}(u)-y^{s,u}(u)$ by observing that $y^{s,t,u}(u)$ and $y^{s,u}(u)$ have the same Taylor expansion up to order $[\gamma]$. Thus by using the lemma of the previous lecture and the triangle inequality, we easily get that:
$\| y^{s,t,u}(u)-y^{s,u}(u) \| \le C_1 \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \left( \int_s^t \| dx^{s,t} (r)\| + \int_t^u \| dx^{t,u} (r)\| \right)^\gamma$
$\le C_2 \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^\gamma_{p-var,[s,u]} .$
We then estimate $(y^{s,t,u}(u) - y^{s,t,u}(s)) + ( y^{s,t}(s)-y^{s,t}(t))+(y^{t,u}(t)-y^{t,u}(u))$ by observing that $y^{s,t,u}(s)=y^{s,t}(s)$, $y^{s,t,u}(t)=y^{s,t}(t)$. Thus,
$(y^{s,t,u}(u) - y^{s,t,u}(s)) + ( y^{s,t}(s)-y^{s,t}(t))+(y^{t,u}(t)-y^{t,u}(u))$
$= (y^{s,t,u} (u)- y^{s,t,u} (t))-( y^{t,u}(u) - y^{t,u}(t) )$
This last term is estimated by using basic continuity estimates with respect to the initial condition which gives
$\| (y^{s,t,u} (u)- y^{s,t,u} (t))-( y^{t,u}(u) - y^{t,u}(t) )\|$
$\le \|y^{s,t,u} (t) -y^{t,u}(t) \| \| V \|_{\text{Lip}^{\gamma-1}} \int_t^u \| dx^{t,u}(r) \| \exp \left( \| V \|_{\text{Lip}^{\gamma-1}} \int_t^u \| dx^{t,u}(r)\|\right)$
$\le C_3 \| \Gamma_{s,t} \| \| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[t,u]} \exp \left( C_3 \| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[t,u]} \right)$
We conclude
$\| \Gamma_{s,u}-\Gamma_{s,t}-\Gamma_{t,u}\| \le C_2 \omega(s,u)^{\gamma/p} +C_3 \| \Gamma_{s,t} \| \omega(t,u)^{1/p} \exp \left(C_3 \omega(t,u)^{1/p} \right),$
where
$\omega(s,t)= \left( \| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[s,t]}\right)^p.$
The basic inequality $1+x e^x \le e^{2x}$ combined with the triangle inequality gives:
$\| \Gamma_{s,u} \| \le \| \Gamma_{t,u} \|+\|\Gamma_{s,t}\| \exp\left(2C_3 \omega(s,u)^{1/p} \right)+ C_2 \omega(s,u)^{\gamma/p}$
$\le \left( \| \Gamma_{t,u} \|+\|\Gamma_{s,t}\| + C_2 \omega(s,u)^{\gamma/p} \right) \exp\left(2C_3 \omega(s,u)^{1/p} \right)$.

We are now in position to apply the lemma of the previous lecture (we let the reader check that the assumptions are satisfied). We deduce then
$\| \Gamma_{s,t} \| \le C_4 \omega(s,t)^{\gamma /p} \exp\left(C_4 \omega(s,t)^{1 /p} \right).$
We now keep in mind that
$\Gamma_{s,t} =(y(t) -y(s)) -( y^{s,t} (t) -y^{s,t} (s)),$
and $y^{s,t} (t) -y^{s,t} (s)$ can be estimated by using basic estimates on differential equations:
$\| y^{s,t} (t) -y^{s,t} (s) \| \le C_5 \| V \|_{\text{Lip}^{\gamma-1}} \int_s^t \| dx^{s,t}(u) \|$
$\le C_6 \omega(s,t)^{1/p}.$
From the triangle inequality, we conclude then:
$\| y(s)-y(t) \| \le C_6 \omega(s,t)^{1/p}+ C_4 \omega(s,t)^{\gamma /p} \exp\left(C_4 \omega(s,t)^{1 /p} \right),$
In particular we have for $s,t$ such that $\omega(s,t)\le 1$,
$\| y(s)-y(t) \| \le C_7 \omega(s,t)^{1/p}.$
This easily gives the required estimate (see Proposition 5.10 in the book by Friz-Victoir) $\square$

We can remark that the proof actually also provided the following estimate which is interesting in itself:

Proposition: Let $\gamma > p \ge 1$. Assume that $V_1, \cdots, V_d$ are $(\gamma-1)$-Lipschitz vector fields in $\mathbb{R}^n$. Let $x \in C^{1-var}([0,T], \mathbb{R}^d)$. Let $y$ be the solution of the equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.$
There exists a constant $C$ depending only on $p$ and $\gamma$ such that for every $0 \le s < t \le T$,
$\left\| y(t)-y(s)-\sum_{k=1}^{[\gamma]} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k} \right\|$
$\le C \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^\gamma_{p-var,[s,t]}$.

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