1 Introduction

Definition 1.1 .

A bounded function $u$ is a mild bounded ancient solution if and only if there exists a pressure $p$ such that $p=p^{1}+p^{2}$ , where the even extension of $p^{1}$ to the whole $\mathbb{R}^{3}$ with respect to $x_{3}$ is $L_{\infty}(-\infty,0;BMO(\mathbb{R}^{3}))$ -function,

\triangle p^{1}=-{\rm divdiv}\,u\otimes u

(1.14)

in $Q^{+}_{-}$ with $p^{1}_{,3}(x^{\prime},0,t)=0$ and $p^{2}(\cdot,t)$ is a harmonic function in $\mathbb{R}^{3}_{+}$ whose gradient satisfies the estimate

|\nabla p^{2}(x,t)|\leq c\ln(2+1/{x_{3}})

(1.15)

for all $(x,t)\in Q_{-}^{+}$ and has the property

\sup\limits_{x^{\prime}\in\mathbb{R}^{2}}|\nabla p^{2}(x,t)|\to 0

(1.16)

as $x_{3}\to\infty$ ; $u$ and $p$ satisfy ( 1.12 ) and

\int\limits_{Q^{+}_{-}}\Big{(}u\cdot(\partial_{t}\varphi+\Delta\varphi)+u% \otimes u:\nabla\varphi+p{\rm div}\,\varphi\Big{)}dxdt=0

(1.17)

for any $\varphi\in C^{\infty}_{0}(Q_{-})$ with $\varphi(x^{\prime},0,t)=0$ for any $x^{\prime}\in\mathbb{R}^{2}$ and for any $t<0$ .

Definition 2.2 .

An H-type algebra is a finite-dimensional real Lie algebra $(\mathfrak{g},[\cdot,\cdot])$ which can be endowed with an inner product $\left\langle\cdot,\cdot\right\rangle$ such that

[\mathfrak{z}^{\bot},\mathfrak{z}^{\bot}]=\mathfrak{z},

where $\mathfrak{z}$ is the center of $\mathfrak{g}$ . Moreover, for any fixed $z\in\mathfrak{z}$ , the map $J_{z}:\mathfrak{z}^{\bot}\longrightarrow\mathfrak{z}^{\bot}$ defined by

\left\langle J_{z}(v),w\right\rangle=\left\langle z,[v,w]\right\rangle\qquad% \forall\,w\in\mathfrak{z}^{\bot}

is an orthogonal map whenever $\left\langle z,z\right\rangle=1$ . We say that a simply connected Lie group is an H-type group if its Lie algebra is an H-type algebra.

Définition 2.2 .

Un élément $z\in\hbox{IK}$ est $\alpha$ -stable avec $\alpha\neq 0$ , si

a^{1/\alpha}z+b^{1/\alpha}z=(a+b)^{1/\alpha}z

pour tous $a,b>0$ .

Definition 2.7 ( $\alpha$ -virtual value) .

For $\alpha\geq 0$ , the $\alpha$ -virtual value of $v$ is

\varphi^{\alpha}(v)=v-\alpha\lambda(v).

Definition 3.3 (Absolute continuity and singularity of measures) .

Let $(\Omega,\Sigma)$ be a measurable space, $\mu:\Sigma\rightarrow\mathsf{X}$ a vector measure, and $\nu:\Sigma\rightarrow[0,\infty]$ be a (real-valued, positive) measure. We say that $\mu$ is absolutely continuous w.r.t. to $\nu$ , and write $\mu\ll\nu$ , if

\forall A\in\Sigma:\Big{[}\ \nu(A)=0\quad\Longrightarrow\quad\mu(A)=0\ \Big{]}\,.

(3.3)

Moreover, we say that two real-valued, positive measures $\mu$ and $\nu$ are singular, and write $\mu\perp\nu$ , if there exist $B_{1},B_{2}\in\Sigma$ with $B_{1}\cup B_{2}=\Omega$ and $B_{1}\cap B_{2}=\emptyset$ such that

\forall A\in\Sigma:\quad\quad\mu(A)=\mu\left(A\cap B_{1}\right)\quad\text{and}% \quad\nu(A)=\nu\left(A\cap B_{2}\right)

(3.4)

Definition 5.22 (The embedding of $X^{\prime}$ into $X$ )

With the same notation and assumptions as in Definition 5.20 , the map $\iota:X^{\prime}\to X$ is defined by

\iota(x^{\prime})=x,\textrm{ provided $\pi(x)=x^{\prime}$ and ${\rm b}(x)=0$},

Definition 1

Assume the following commutator condition

[b,a]=ba-ab=u

and define

H=\frac{1}{2}(ab+ba)

where $u$ is a phase times the unit operator.

Definition 2.16 .

( [ 19 ] ) A left PostLie algebra is a triple $(A,\circ,[\;,\;])$ consisting of a vector space $A$ , a product $\circ$ and a skew-symmetric product $[\;,\;]$ satisfying the relations:

(25)		$\displaystyle[[x,y],z]+[[z,x],y]+[[y,z],x]=0,$
(26)		$\displaystyle(x\circ y)\circ z-x\circ(y\circ z)-(y\circ x)\circ z+y\circ(x% \circ z)-[x,y]\circ z=0,$
(27)		$\displaystyle z\circ[x,y]-[z\circ x,y]-[x,z\circ y]=0.$

Definition 4.7 .

An element $r$ of a ring $R$ is said regular or simplifiable for the multiplicative law of $R$ (or multiplicatively regular or multiplicatively simplifiable) if for any couple $(x,y)\in R^{2}$ , we have:

rx=ry\,\Rightarrow\,x=y

and

xr=yr\,\Rightarrow\,x=y

Definition 3.1 .

We say that $\varphi:\partial U\to\partial\tilde{U}$ is conformally fit if there are Riemann maps $g:({\mathbb{D}},0)\to(U,c)$ and $\tilde{g}:({\mathbb{D}},0)\to(\tilde{U},\tilde{c})$ for which the induced homeomorphism $\varphi^{\circ}=[\tilde{g}]^{-1}\circ[\varphi]\circ[g]:{\mathbb{T}}\to{\mathbb% {T}}$ defined by ( 6 ) is the identity map. By Lemma 2.1 , this is equivalent to the condition

\tilde{g}=\varphi\circ g\quad\text{on the set of landing angles for}\ g.

Definition 2.5 .

In any dialgebra $D$ the dicommutator is the bilinear operation

\langle x,y\rangle=x\dashv y-y\vdash x.

In what follows, we will write $D^{-}$ to denote $(D,\langle-,-\rangle)$ , i.e., the underlying vector space of $D$ with the dicommutator.

Definition 1 (See loos69sp1 )

A symmetric space is a manifold $M$ with a differentiable symmetric product $\cdot$ obeying the following conditions:

(i)

$x\cdot x=x,$
(ii)

$x\cdot(x\cdot y)=y,$
(iii)

$x\cdot(y\cdot z)=(x\cdot y)\cdot(x\cdot z),$

and moreover

(iv)

every $x$ has a neighbourhood $U$ such that for all $y$ in $U$ $x\cdot y=y$ implies $y=x$ .

Definition 3.3 .

[ 27 , p.24] A trace form on a Jordan algebra $J$ is a symmetric bilinear form $\gamma$ such that

\gamma(u\bullet v,w)=\gamma(u,v\bullet w)

(7)

for all $u,v,w\in J$ . Equivalently, the operator $L_{v}$ is self-adjoint with respect to $\gamma$ for all $v\in J$ .

Definition 3.12 .

[ 24 , p.52] Let $J$ be a Jordan algebra. The unital hull $\hat{J}$ of $J$ is the vector space $\mathbb{R}\times J$ equipped with a multiplication $\hat{\bullet}$ defined by

(\alpha,u)\hat{\bullet}(\beta,v)=(\alpha\beta,\alpha v+\beta u+u\bullet v).

(10)

Definition 2.4 .

Let $(\Sigma,\xi)$ denote a closed connected coorientable contact manifold, and fix a contact form $\alpha\in\Omega^{1}(\Sigma)$ generating $\xi$ . Fix $\varphi\in\mbox{Cont}_{0}(\Sigma,\xi)$ . We can write $\varphi^{*}\alpha=\rho\alpha$ for a smooth positive function $\rho$ on $\Sigma$ . A translated point of $\varphi$ is a point $x\in$ with the property that there exists $\eta\in\mathbb{R}$ such that

\varphi(x)=\theta^{\eta}(x),\ \ \ \mbox{and}\ \ \ \rho(x)=1.

(2.11)

We call $\eta$ the time-shift of $x$ . Note that if the leaf $\{\theta^{t}(x)\}_{t\in\mathbb{R}}$ is closed (which is always the case when $\alpha$ is periodic) then the time-shift is not unique. Indeed, if the leaf $\{\theta^{t}(x)\}_{t\in\mathbb{R}}$ has period $T>0$ then $\varphi(x)=\theta^{\eta+\nu T}(x)$ for all $\nu\in\mathbb{Z}$ . In this case one needs to keep track of the $\eta$ . Finally we say a point $x\in\Sigma$ is an iterated translated point of $\varphi$ if it is a translated point for some iteration $\varphi^{\nu}$ .

Definition 2.8 .

The co-adjoint operators $d^{*}$ : Given $M$ to be a $2$ -dimensional oriented manifold equipped with a Riemannian metric $g(\cdot,\cdot)$ . For each $p=1,2$ , the coadjoint operator $d^{*}:C^{\infty}(\wedge^{p}T^{*}M)\rightarrow C^{\infty}(\wedge^{p-1}T^{*}M)$ sending the space of $p$ -forms on $M$ into the space of $p-1$ -forms on $M$ is defined through the following relation.

(2.20)

d^{*}=(-1)^{2(p+1)+1}*d*=-*d*,

where the symbol $*$ stands for the Hodge-Star operators as defined in Definition 2.5 .