The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
Show the existence of a constant $c_1 > 0$ such that for all $(p,q,r) \in B(1)$, we have $$c_1^{-1} \leq p^2 + q^2 + |r| \leq c_1$$

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
(a) Show that for all $(p,q,r) \in \mathbf{R}^3 \setminus \{(0,0,0)\}$, there exists a unique $\lambda > 0$ such that: $$(\lambda p, \lambda q, \lambda^2 r) \in B(1).$$
(b) Deduce that for every point $A \in \mathbf{H}$, there exists a positive real $T(A)$ and parameters $(\theta, \varphi)$ (also depending on $A$) such that $A$ is the endpoint of the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi}) \in E(T(A))$.
(c) Show the existence of a constant $c_2 > 0$ such that for all $(p,q,r) \in \mathbf{R}^3$, $$c_2^{-1}\sqrt{p^2 + q^2 + |r|} \leq T\left(\exp\left(M_{p,q,r}\right)\right) \leq c_2\sqrt{p^2 + q^2 + |r|}$$

We assume that $m = 0$, that is $$\mathcal{M} = \left(\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left(G(x)\right) \wedge G^{\prime}(x)$$ and that moreover there exists $\lambda > 0$ such that $$G(0) = (0, 0, 2\lambda), \quad G^{\prime}(0) = (1, 0, 0)$$
Show that $G \in C^{\infty}(\mathbb{R}, \mathbb{R}^{3})$.

We denote $L$ the operator that associates to a function $f : \mathbb { R } \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 2 }$, the function $Lf$ defined by $$\forall x \in \mathbb { R } , \quad L f ( x ) = \frac { 1 } { 2 } f ^ { \prime \prime } ( x ) - x f ^ { \prime } ( x )$$ We recall that the measure $\mu$ is defined by $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$.
3a. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be of class $\mathscr { C } ^ { 2 }$. Show that $L f = \frac { 1 } { 2 \mu } \left( \mu f ^ { \prime } \right) ^ { \prime }$.
3b. Let $h _ { 1 } , h _ { 2 }$ be two functions in $\mathscr { C } _ { b } ^ { 2 }$. Show that $$\int h _ { 1 } ( x ) \left( L h _ { 2 } \right) ( x ) \mu ( x ) d x = - \frac { 1 } { 2 } \int h _ { 1 } ^ { \prime } ( x ) h _ { 2 } ^ { \prime } ( x ) \mu ( x ) d x$$ after having justified the existence of each term of the formula.

We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $Lf(x) = \frac{1}{2} f''(x) - x f'(x)$. We assume that $f \in \mathscr { C } _ { b } ^ { 2 }$.
5a. Show that, on $\mathbb { R } ^ { 2 } , \Phi _ { f }$ is of class $\mathscr { C } ^ { 1 }$ and $\partial _ { x x } \Phi _ { f }$ is well defined, continuous and bounded.
5b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Find a relation between $\partial _ { x } \Phi _ { f } ( t , x )$ and $\Phi _ { f ^ { \prime } } ( t , x )$.
5c. Show that for all $( t , x ) \in \mathbb { R } ^ { 2 }$, we have $\partial _ { t } \Phi _ { f } ( t , x ) \cos t = L \Phi _ { f } ( t , x ) \sin t$.
5d. Show that for all $t \in \mathbb { R }$, we have $\int \Phi _ { f } ( t , x ) \mu ( x ) d x = \int f ( x ) \mu ( x ) d x$.

Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. We assume throughout this question that $f \in \mathscr { C } _ { b } ^ { 2 }$ and that there exists $\delta > 0$ such that $$\forall x \in \mathbb { R } , \quad f ( x ) \geqslant \delta .$$ We denote $g = \left( f ^ { \prime } \right) ^ { 2 } / f$.
7a. Show that $J$ is then of class $\mathscr { C } ^ { 1 }$ on $\mathbb { R }$ and that $$\forall t \in \mathbb { R } , \quad J ^ { \prime } ( t ) \cos t = - \frac { \sin t } { 2 } \int \frac { \left( \partial _ { x } \Phi _ { f } ( t , x ) \right) ^ { 2 } } { \Phi _ { f } ( t , x ) } \mu ( x ) d x$$
7b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Show that $$\Phi _ { f ^ { \prime } } ( t , x ) ^ { 2 } \leqslant \Phi _ { f } ( t , x ) \Phi _ { g } ( t , x )$$
7c. Conclude that $$\int h ( f ( x ) ) \mu ( x ) d x - h \left( \int f ( y ) \mu ( y ) d y \right) \leqslant \frac { 1 } { 4 } \int g ( x ) \mu ( x ) d x$$

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$ where $P^*(X) = P(-X)$.
Show that $T$ and $M$ satisfy hypotheses (H1), (H2), (H3) and (H4).

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. The Fourier transform is defined as $\mathcal{F}(f) : \left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & \xi \mapsto \int_{-\infty}^{+\infty} f(x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x \end{aligned}\right.$. Show that $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that $f$ tends to zero at $+\infty$ and at $-\infty$.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that, for any real $\xi$, $\mathcal{F}\left(f^{\prime}\right)(\xi) = 2\mathrm{i}\pi\xi \mathcal{F}(f)(\xi)$.

Let $f$ be a function satisfying the diffusion equation, the three domination conditions, and the boundary condition $\lim_{t\to 0^+} f(t,x) = g_\sigma(x)$. Justify that, for any real $t > 0$ and any real $\xi$, the function $x \mapsto f(t, x) \exp(-2\mathrm{i}\pi \xi x)$ is integrable on $\mathbb{R}$.

We define $\hat{f}$ on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ by: $\forall(t, \xi) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \hat{f}(t, \xi) = \int_{-\infty}^{+\infty} f(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$. Show that, for any real number $\xi$, $\lim_{t \rightarrow 0^{+}} \hat{f}(t, \xi) = \widehat{g_{\sigma}}(\xi)$. One may use any sequence $\left(t_{n}\right)_{n \in \mathbb{N}}$ of strictly positive reals converging to zero.

Show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = \int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$.

Give a non-constant function belonging to $\mathcal{H}(U)$. Is the product of two harmonic functions a harmonic function?

We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$. Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations $$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Show that $f$ belongs to $\mathcal{H}\left(\mathbb{R}^2 \setminus \{(0,0)\}\right)$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$

We consider two functions of class $\mathcal{C}^2$, $u: \mathbb{R}^{*+} \to \mathbb{R}$ and $v: \mathbb{R} \to \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables. Show that, if $f$ is not identically zero, then $v$ is $2\pi$-periodic.

We consider two functions of class $\mathcal{C}^2$, $u: \mathbb{R}^{*+} \to \mathbb{R}$ and $v: \mathbb{R} \to \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ and $v$ is a solution of the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$

We assume here that $\lambda = 0$. Deduce, in the case $\lambda = 0$, the harmonic functions with separable polar variables.

Give a non-constant function belonging to $\mathcal{H}(U)$. Is the product of two harmonic functions a harmonic function?

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations $$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Show that $f$ belongs to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is not identically zero, then $v$ is $2\pi$-periodic.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ and $v$ is a solution of the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Deduce from this, in the case $\lambda = 0$, the harmonic functions with separable polar variables.