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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