Criterion for ellipticity on Heisenberg group

Abstract

We provide a semi-constructive criterion for ellipticity of the differential operator on the Heisenberg group $\mathbb {H}^1.$

1 Introduction

The criterion for ellipticity of a differential operator in Euclidean space is well known. This criterion (the invertibility of the principal symbol of the operators) is perfectly constructive.

Criteria for ellipticity of differential operators on other Lie groups are much more involved. In this paper, we provide a criterion for ellipticity on the Heisenberg group $\mathbb {H}^1$ which is almost as constructive as the one for Euclidean space. It is possible that a (heavily modified) version of this proof works for an arbitrary stratified Lie group G. For background material on this topic, see [1,2,3].

The Heisenberg group $\mathbb {H}^1$ is the subgroup in $\textrm{GL}(3,\mathbb {R})$ defined by

$$\begin{aligned} \mathbb {H}^1=\left\{ \begin{pmatrix} 1&{}x&{}t\\ 0&{}1&{}y\\ 0&{}0&{}1 \end{pmatrix},\quad x,y,t\in \mathbb {R}. \right\} . \end{aligned}$$

In other words, $\mathbb {H}^1$ is $\mathbb {R}^3$ equipped with the product

$$\begin{aligned} (x_1,y_1,t_1)\cdot (x_2,y_2,t_2)=(x_1+x_2,y_1+y_2,t_1+t_2+x_1y_2). \end{aligned}$$

The differential calculus on $\mathbb {H}^1$ consists of two left-invariant vector fields

$$\begin{aligned} X_1=\frac{\partial }{\partial x},\quad X_2=\frac{\partial }{\partial y}+M_x\frac{\partial }{\partial t}. \end{aligned}$$

In what follows, differential operators are defined on the Schwartz space $\mathcal {S}(\mathbb {H}^1)=\mathcal {S}(\mathbb {R}^3).$ Note that $X^{w}:\mathcal {S}(\mathbb {H}^1)\rightarrow \mathcal {S}(\mathbb {H}^1)$ for every word w in the alphabet with 2 letters. Here, $X^{w}$ is the word w expressed in the alphabet $\{X_1,X_2\}$ and viewed as a differential operator. Hence, $\mathcal {S}(\mathbb {H}^1)$ serves as a natural domain for differential operators. Let $C^\infty _b(\mathbb {H}^1)$ denote the algebra of smooth functions f on $\mathbb {H}^1$, such that $X^wf$ is bounded for every word w.

Definition 1.1

A differential operator on $\mathbb {H}^1$ of order m is the mapping $P:\mathcal {S}(\mathbb {H}^1)\rightarrow \mathcal {S}(\mathbb {H}^1)$ of the shape

$$\begin{aligned} P=\sum _{\textrm{len}(w)\le m}M_{a_{w}}X^{w}, \end{aligned}$$

where the sum is taken over all words of length at most m and where each $M_{a_{w}}$ is a multiplication operator with $a_{w}\in C^{\infty }_b(\mathbb {H}^1).$

Clearly, a differential operator P extends to a mapping $P:\mathcal {S}'(\mathbb {H}^1)\rightarrow \mathcal {S}'(\mathbb {H}^1).$ The Lebesgue measure on $\mathbb {R}^3$ is a bivariant Haar measure for $\mathbb {H}^1.$ We write $L_2(\mathbb {H}^1)$ for the $L_2$-space with this measure.

Definition 1.2

A differential operator P of order m on $\mathbb {H}^1$ is elliptic if, for every $f\in L_2(\mathbb {H}^1)$ with $Pf\in L_2(\mathbb {H}^1)$, we have $(1-\Delta )^{\frac{m}{2}}f\in L_2(\mathbb {H}^1)$ and

$$\begin{aligned} \Vert Pf\Vert _{L_2(\mathbb {H}^1)}+\Vert f\Vert _{L_2(\mathbb {H}^1)}\ge c_P \Vert (1-\Delta )^{\frac{m}{2}}f\Vert _{L_2(\mathbb {H}^1)}. \end{aligned}$$

Here, $\Delta =X_1^2+X_2^2$ and $c_P$ is a strictly positive constant which only depends on P.

The paper [5] provides the following criterion (strictly speaking, only the sufficiency is established in [5]; however, the necessity is easy) for ellipticity of the differential operator P on a stratified Lie group. We state it here only for the Heisenberg group $\mathbb {H}^1.$ This condition is related to the “maximal sub-ellipticity” of Helffer–Nourrigat [4, Chapter I, Definition 1.1]. Necessary and sufficient conditions for the hypoellipticity of left-invariant differential operators on the Heisenberg group were first discovered by Rockland [6].

Theorem 1.3

Let P be a formally self-adjoint differential operator of order m on $\mathbb {H}^1.$ The operator P is elliptic if and only if

$$\begin{aligned} \Vert P_gf\Vert _{L_2(\mathbb {H}^1)}\ge c_P\Vert (-\Delta )^{\frac{m}{2}}f\Vert _{L_2(\mathbb {H}^1)},\quad f\in \mathcal {S}(\mathbb {H}^1),\quad g\in \mathbb {H}^1. \end{aligned}$$

Here, we are using the notation

$$\begin{aligned} P_g=\sum _{\textrm{len}(w)=m}a_{w}(g)X^{w},\quad g\in \mathbb {H}^1. \end{aligned}$$

The latter condition is rather hard to deal with.

Definition 1.4

A differential operator P on $\mathbb {H}^1$ of order m is called formally elliptic if there exists a constant $c_P>0$, such that

$$\begin{aligned} \left| \sum _{\textrm{len}(w)=m}a_{w}(g)s^{w}\right| \ge c_P,\quad s\in \mathbb {S}^1,\quad g\in \mathbb {H}^1. \end{aligned}$$

Here, $s\in \mathbb {S}^1$ means that $s=(s_0,s_1)\in \mathbb {R}^2$ and $s_0^2+s_1^2=1.$ The sum is taken over words in the alphabet $\{0,1\}$ and $s^{w},$ $w=w_1\cdots w_m,$ is shorthand for $s_{w_1}s_{w_2}\cdots s_{w_m}.$

Equivalently, P is formally elliptic if $|\pi (P_g)|\ge c_P\pi ((-\Delta )^{\frac{m}{2}})$ for every 1-dimensional representation $\pi$ of $\mathbb {H}^1.$

A naive guess is that formal ellipticity is equivalent to ellipticity. This is not really the case: the operator $-X_1^2-X_2^2+i[X_1,X_2]$ is formally elliptic, but not elliptic (as it has a non-trivial kernel). However, the following weaker assertion is still true.

For each w,  the bounded function $g\rightarrow a_{w}(g)$ extends to a continuous function on the Stone–Čech compactification $\beta \mathbb {H}^1$ of the topological space $(\mathbb {H}^1,\tau _{\textrm{disc}})$ (here, $\tau _{\textrm{disc}}$ is the discrete topology on $\mathbb {H}^1$). Thus, we can also define $P_g$ for every $g\in \beta \mathbb {H}^1.$

Theorem 1.5

Let P be a formally self-adjoint differential operator of order m on $\mathbb {H}^1.$ The operator P is elliptic if and only if the following conditions hold:

• 1. P is formally elliptic;

• 2. if $\xi \in \textrm{dom}((-\Delta )^{\frac{m}{2}})$ and $g\in \beta \mathbb {H}^1$ are such that $P_g\xi =0,$ then $\xi =0;$

The second condition in Theorem 1.5 is stated in terms of the Stone–Čech compactification, because we do not want to introduce a topology on the set of left-invariant differential operators. If, instead, such a topology is introduced, then the second condition is the triviality of the kernel of Q for every Q in the closure of the set $\{P_g\}_{g\in \mathbb {H}^1}.$

2 Proof of the main theorem

In this section, we work in the Hilbert space $l_2(\mathbb {Z}_+),$ with standard orthonormal basis denoted $\{e_k\}_{k\ge 0}.$ Denote $E_{j,k}$ for the matrix basis operator defined as $E_{j,k}e_n = e_j\delta _{k,n}.$

We work with the following operators:

$$\begin{aligned} ip= & {} \frac{1}{\sqrt{2}}\sum _{k\ge 0}(k+1)^{\frac{1}{2}}(E_{k,k+1}-E_{k+1,k}),\quad q=\frac{1}{\sqrt{2}}\sum _{k\ge 0}(k+1)^{\frac{1}{2}}(E_{k+1,k}+E_{k,k+1}),\\ U= & {} \sum _{k\ge 0}E_{k+1,k},\quad H=\sum _{k\ge 0}(2k+1)E_{k,k}. \end{aligned}$$

To be clear, p and q are self-adjoint unbounded operators, and $i=\sqrt{-1}.$ We also identify $l_2(\mathbb {Z}_+)$ with a subspace in $l_2(\mathbb {Z}).$ Let V denotes the right shift operator on $l_2(\mathbb {Z}).$ For $n\in \mathbb {Z}_+,$ let $E_n=\sum _{k=0}^{n-1}E_{k,k}.$

Theorem 2.1

Let $\mathbb {J}$ be a set (discrete topological space) and let $(P_j)_{j\in \mathbb {J}}$ be a bounded family of homogeneous (of order m) polynomials in 2 non-commuting variables. The following conditions are equivalent:

• 1. there exists $c>0$, such that

  $$\begin{aligned} \Vert P_j(p,q)\xi \Vert _{l_2(\mathbb {Z}_+)}\ge c\Vert H^{\frac{m}{2}}\xi \Vert _{l_2(\mathbb {Z}_+)},\quad \xi \in \textrm{dom}(H^{\frac{m}{2}}),\quad j\in \mathbb {J}; \end{aligned}$$

• 2. there exists $c>0$, such that

  $$\begin{aligned} |P_j(\Re (z),\Im (z))|\ge c,\quad z\in \mathbb {C},\quad |z|=1,\quad j\in \mathbb {J}, \end{aligned}$$

  and, for every $j\in \beta \mathbb {J},$ $P_j(p,q)\xi =0,$ $\xi \in \textrm{dom}(H^{\frac{m}{2}})$ implies $\xi =0.$

Lemma 2.2

Let P be a homogeneous (of order m) polynomial in 2 non-commuting variables. The operator

$$\begin{aligned} P(p,q)H^{-\frac{m}{2}}-P(-\Im (U),\Re (U)) \end{aligned}$$

is compact.

Proof

We prove the assertion by induction on m. The base of the induction (i.e., the cases $m=1$ and $m=2$) is an easy computation.

Suppose the assertion is true for m. Let us prove it for $m+2.$ Let P be a homogeneous (of order $m+2$) polynomial in 2 non-commuting variables. We write

$$\begin{aligned} P(p,q)=p^2P_1(p,q)+pqP_2(p,q)+qpP_3(p,q)+q^2P_4(p,q), \end{aligned}$$

where $(P_k)_{k=1}^4$ are homogeneous (of order m) polynomial in 2 non-commuting variables. We have

$$\begin{aligned}{} & {} P(p,q)H^{-\frac{m+2}{2}}=H^{-1}P(p,q)H^{-\frac{m}{2}}+[P(p,q),H^{-1}]H^{-\frac{m}{2}}\\{} & {} \quad =H^{-1}P(p,q)H^{-\frac{m}{2}}-H^{-1}[P(p,q),H]H^{-\frac{m+2}{2}}\\{} & {} \quad =H^{-1}p^2\cdot P_1(p,q)H^{-\frac{m}{2}}+H^{-1}pq\cdot P_2(p,q)H^{-\frac{m}{2}}+H^{-1}qp\cdot P_3(p,q)H^{-\frac{m}{2}}\\{} & {} \qquad +H^{-1}q^2\cdot P_4(p,q)H^{-\frac{m}{2}}-H^{-1}\cdot [P(p,q),H]H^{-\frac{m+2}{2}}. \end{aligned}$$

The operators $[P(p,q),H]H^{-\frac{m+2}{2}}$ are bounded. Hence, the last summand is compact. By the inductive assumption (i.e., for m and for 2), the operator

$$\begin{aligned}{} & {} P(p,q)H^{-\frac{m+2}{2}}\!-\!\Big ((\Im (U))^2\cdot P_1(-\!\Im (U),\Re (U))-\Im (U)\Re (U)\!\cdot \! P_2(-\Im (U),\Re (U))\\{} & {} \quad -\Re (U)\Im (U)\cdot P_3(-\Im (U),\Re (U))+(\Re (U))^2\cdot P_4(-\Im (U),\Re (U))\Big ) \end{aligned}$$

is compact. Clearly

$$\begin{aligned}{} & {} P(-\!\Im (U),\Re (U))\!=\!(\Im (U))^2\!\cdot \! P_1(-\!\Im (U),\Re (U))\!-\!\Im (U)\Re (U)\cdot P_2(-\!\Im (U),\Re (U))\\{} & {} \quad -\Re (U)\Im (U)\cdot P_3(-\Im (U),\Re (U))+(\Re (U))^2\cdot P_4(-\Im (U),\Re (U)). \end{aligned}$$

This yields the step of the induction and, hence, completes the proof. $\square$

Lemma 2.3

Let P be a homogeneous (of order m) polynomial in 2 non-commuting variables. We have

$$\begin{aligned} P(-\Im (U),\Re (U))U^n\xi =P(-\Im (V),\Re (V))V^n\xi ,\quad \xi \in l_2(\mathbb {Z}_+),\quad n\ge m. \end{aligned}$$

Lemma 2.4

Let P be a homogeneous (of order m) polynomial in 2 non-commuting variables. If

$$\begin{aligned} \Vert P(p,q)\xi \Vert _{l_2(\mathbb {Z}_+)}\ge \Vert H^{\frac{m}{2}}\xi \Vert _{l_2(\mathbb {Z}_+)},\quad \xi \in \textrm{dom}(H^{\frac{m}{2}}), \end{aligned}$$

then

$$\begin{aligned} \Vert P(-\Im (V),\Re (V))\xi \Vert _{l_2(\mathbb {Z})}\ge \Vert \xi \Vert _{l_2(\mathbb {Z})},\quad \xi \in l_2(\mathbb {Z}_+). \end{aligned}$$

Proof

The assumption means

$$\begin{aligned} \Vert P(p,q)H^{-\frac{m}{2}}\xi \Vert _{l_2(\mathbb {Z}_+)}\ge \Vert \xi \Vert _{l_2(\mathbb {Z}_+)},\quad \xi \in l_2(\mathbb {Z}_+). \end{aligned}$$

Substituting $U^n\xi$ instead of $\xi ,$ we obtain

$$\begin{aligned} \Vert P(p,q)H^{-\frac{m}{2}}U^n\xi \Vert _{l_2(\mathbb {Z}_+)}\ge \Vert \xi \Vert _{l_2(\mathbb {Z}_+)},\quad \xi \in l_2(\mathbb {Z}_+),\quad n\in \mathbb {Z}_+. \end{aligned}$$

By the triangle inequality

$$\begin{aligned}{} & {} \Vert \xi \Vert _{l_2(\mathbb {Z}_+)}\le \Vert P(-\Im (U),\Re (U))(U^n\xi )\Vert _{l_2(\mathbb {Z}_+)}+\\{} & {} \quad +\Big \Vert \Big (P(p,q)H^{-\frac{m}{2}}-P(-\Im (U),\Re (U))\Big )(U^n\xi )\Big \Vert _{l_2(\mathbb {Z}_+)},\quad n\in \mathbb {Z}_+. \end{aligned}$$

By Lemma 2.3

$$\begin{aligned}{} & {} \Vert P(-\Im (U),\Re (U))(U^n\xi )\Vert _{l_2(\mathbb {Z}_+)}\\{} & {} \quad =\Vert P(-\Im (V),\Re (V))V^n(\xi )\Vert _{l_2(\mathbb {Z})}=\Vert P(-\Im (V),\Re (V))(\xi )\Vert _{l_2(\mathbb {Z})},\quad n\ge m. \end{aligned}$$

Thus

$$\begin{aligned}{} & {} \Vert \xi \Vert _{l_2(\mathbb {Z}_+)}\le \Vert P(-\Im (V),\Re (V))\eta \Vert _{l_2(\mathbb {Z})}+\\{} & {} \quad +\Big \Vert \Big (P(p,q)H^{-\frac{m}{2}}-P(-\Im (U),\Re (U))\Big )(U^n\xi )\Big \Vert _{l_2(\mathbb {Z}_+)},\quad n\in \mathbb {Z}_+. \end{aligned}$$

By Lemma 2.2

$$\begin{aligned} \Big \Vert \Big (P(p,q)H^{-\frac{m}{2}}-P(-\Im (U),\Re (U))\Big )(U^n\eta )\Big \Vert _{l_2(\mathbb {Z}_+)}\rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

This suffices to complete the proof. $\square$

Lemma 2.5

Let P be a homogeneous (of order m) polynomial in 2 non-commuting variables. If

$$\begin{aligned} \Vert P(-\Im (V),\Re (V))\xi \Vert _{l_2(\mathbb {Z})}\ge \Vert \xi \Vert _{l_2(\mathbb {Z})},\quad \xi \in l_2(\mathbb {Z}_+), \end{aligned}$$

then

$$\begin{aligned} |P(\Im (z),\Re (z))|\ge 1,\quad z\in \mathbb {C},\quad |z|=1. \end{aligned}$$

Proof

Let $\eta \in l_2(\mathbb {Z}).$ Fix $\epsilon >0$ and choose $\xi \in l_2(\mathbb {Z}_+)$ and $n\in \mathbb {Z}_+$, such that

$$\begin{aligned} \Vert \eta -V^{-n}\xi \Vert _{l_2(\mathbb {Z})}<\epsilon . \end{aligned}$$

Set $\theta =\eta -V^{-n}\xi .$ We have

$$\begin{aligned} \Vert P(-\Im (V),\Re (V))\eta \Vert _{l_2(\mathbb {Z})}\ge & {} \Vert P(-\Im (V),\Re (V))V^{-n}\xi \Vert _{l_2(\mathbb {Z})}-\Vert P(-\Im (V),\Re (V))\theta \Vert _{l_2(\mathbb {Z})}\\\ge & {} \Vert P(-\Im (V),\Re (V))V^{-n}\xi \Vert _{l_2(\mathbb {Z})}-\epsilon \Vert P(-\Im (V),\Re (V))\Vert _{\infty }\\= & {} \Vert P(-\Im (V),\Re (V))\xi \Vert _{l_2(\mathbb {Z})}-\epsilon \Vert P(\Im (V),\Re (V))\Vert _{\infty }. \end{aligned}$$

By the assumption, we have

$$\begin{aligned} \Vert P(-\Im (V),\Re (V))\xi \Vert _{l_2(\mathbb {Z})}\ge \Vert \xi \Vert _{l_2(\mathbb {Z})}. \end{aligned}$$

Thus

$$\begin{aligned} \Vert P(-\Im (V),\Re (V))\eta \Vert _{l_2(\mathbb {Z})}\ge & {} \Vert \xi \Vert _{l_2(\mathbb {Z})}-\epsilon \Vert P(-\Im (V),\Re (V))\Vert _{\infty }\\= & {} \Vert V^{-n}\xi \Vert _{l_2(\mathbb {Z})}-\epsilon \Vert P(-\Im (V),\Re (V))\Vert _{\infty }\\\ge & {} \Vert \eta \Vert _{l_2(\mathbb {Z})}-\epsilon -\epsilon \Vert P(-\Im (V),\Re (V))\Vert _{\infty }. \end{aligned}$$

Since $\epsilon >0$ is arbitrarily small, it follows that:

$$\begin{aligned} \Vert P(-\Im (V),\Re (V))\eta \Vert _{l_2(\mathbb {Z})}\ge \Vert \eta \Vert _{l_2(\mathbb {Z})},\quad \eta \in l_2(\mathbb {Z}). \end{aligned}$$

Since V is normal and since the spectrum of V is $\{z\in \mathbb {C}:\ |z|=1\},$ the assertion immediately follows. $\square$

Lemma 2.6

Let $(P_j)_{j\in \mathbb {J}}$ be a family of homogeneous (of order m) polynomials in 2 non-commuting variables. Suppose the condition (1.) in Theorem 2.1 fails. There exists a sequence $(\eta _k)_{k\ge 0}\subset l_2(\mathbb {Z}_+)$ and a sequence $(j_k)_{k\ge 0}\subset \mathbb {J}$, such that

• 1. $\Vert \eta _k\Vert _{l_2(\mathbb {Z}_+)}=1$ for every $k\ge 0.$

• 2. $\eta _k\rightarrow \eta$ weakly in $l_2(\mathbb {Z}_+).$

• 3. $P_{j_k}(p,q)H^{-\frac{m}{2}}\eta _k\rightarrow 0$ in $l_2(\mathbb {Z}_+)$ as $k\rightarrow \infty .$

Proof

Indeed, assume the contrary and choose a sequence $(j_k)_{k\ge 0}\subset \mathbb {J}$ and $\xi _k\in \textrm{dom}(H^{\frac{m}{2}})$, such that $\Vert H^{\frac{m}{2}}\xi _k\Vert _{l_2(\mathbb {Z}_+)}=1$ and such that $\Vert P_{j_k}(p,q)\xi _k\Vert _{l_2(\mathbb {Z}_+)}\rightarrow 0.$

Set $\eta _k=H^{\frac{m}{2}}\xi _k,$ $k\ge 0.$ The sequence $(\eta _k)_{k\ge 0}$ satisfies the first and third conditions. Since the unit ball in $l_2(\mathbb {Z}_+)$ is weakly compact, it follows that, passing to a subsequence if needed, we may also satisfy the second condition. $\square$

Lemma 2.7

Let $(P_j)_{j\in \mathbb {J}}$ be a bounded family of homogeneous (of order m) polynomials in 2 non-commuting variables. We have

$$\begin{aligned} \sup _{j\in \mathbb {J}}\Big \Vert (1-E_n)\Big (P_j(p,q)H^{-\frac{m}{2}}-P_j(-\Im (U),\Re (U))\Big )\Big \Vert _{\infty }\rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

Proof

By definition, we have

$$\begin{aligned} P_j=\sum _{\textrm{len}(w)=m}a_{j,w}w(p,q). \end{aligned}$$

Boundedness of the family means that

$$\begin{aligned} \sup _{j\in \mathbb {J}}|a_{j,w}|<\infty ,\quad \textrm{len}(w)=m. \end{aligned}$$

By Lemma 2.2, the operator

$$\begin{aligned} w(p,q)H^{-\frac{m}{2}}-w(-\Im (U),\Re (U)),\quad \textrm{len}(w)=m, \end{aligned}$$

is compact. Hence

$$\begin{aligned} \Big \Vert (1-E_n)\Big (w(p,q)H^{-\frac{m}{2}}-w(-\Im (U),\Re (U))\Big )\Big \Vert _{\infty }\rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

(1.2)

By triangle inequality

$$\begin{aligned}{} & {} \sup _{j\in \mathbb {J}}\Big \Vert (1-E_n)\Big (P_i(p,q)H^{-\frac{m}{2}}-P_i(-\Im (U),\Re (U))\Big )\Big \Vert _{\infty }\\{} & {} \quad \le \sup _{j\in \mathbb {J}}\sum _{\textrm{len}(w)=m}|a_{j,w}|\cdot \Big \Vert (1-E_n)\Big (w(p,q)H^{-\frac{m}{2}}-w(-\Im (U),\Re (U))\Big )\Big \Vert _{\infty }\\{} & {} \quad \le \!\sum _{\textrm{len}(w)=m}\sup _{j\in \mathbb {J}}|a_{j,w}|\!\cdot \! \max _{\textrm{len}(w)\!=\!m}\Big \Vert (1-E_n)\Big (w(p,q)H^{-\frac{m}{2}}\!-\!w(-\Im (U),\Re (U))\Big )\Big \Vert _{\infty }. \end{aligned}$$

Thus, the assertion follows from (1.2). $\square$

Lemma 2.8

Let P be a homogeneous (of order m) polynomial in 2 non-commuting variables. For every $n\ge m,$ we have

$$\begin{aligned}{} & {} E_n\cdot P(-\Im (U),\Re (U))=E_n\cdot P(-\Im (U),\Re (U))\cdot E_{n+m}.\\{} & {} [P(p,q)H^{-\frac{m}{2}},E_n]=[P(p,q)H^{-\frac{m}{2}},E_n]\cdot (1-E_{n-m}). \end{aligned}$$

Lemma 2.9

Let $(P_j)_{j\in \mathbb {J}}$ be a bounded family of homogeneous (of order m) polynomials in 2 non-commuting variables. For every $\eta \in l_2(\mathbb {Z}_+),$ we have

$$\begin{aligned} \sup _{j\in \mathbb {J}}\Vert (1-E_n)P_j(p,q)H^{-\frac{m}{2}}\eta \Vert _{l_2(\mathbb {Z}_+)}\rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

Proof

By definition, we have

$$\begin{aligned} P_j=\sum _{\textrm{len}(w)=m}a_{j,w}w(p,q). \end{aligned}$$

Boundedness of the family means that

$$\begin{aligned} \sup _{j\in \mathbb {J}}|a_{j,w}|<\infty ,\quad \textrm{len}(w)=m. \end{aligned}$$

By the triangle inequality

$$\begin{aligned}{} & {} \sup _{j\in \mathbb {J}}\Vert (1-E_n)P_j(p,q)H^{-\frac{m}{2}}\eta \Vert _{l_2(\mathbb {Z}_+)}\\{} & {} \quad \le \sup _{j\in \mathbb {J}}\sum _{\textrm{len}(w)=m}|a_{j,w}|\Vert (1-E_n)w(p,q)H^{-\frac{m}{2}}\eta \Vert _{l_2(\mathbb {Z}_+)}\\{} & {} \quad \le \sum _{\textrm{len}(w)=m}\sup _{j\in \mathbb {J}}|a_{j,w}|\cdot \max _{\textrm{len}(w)=m}\Vert (1-E_n)w(p,q)H^{-\frac{m}{2}}\eta \Vert _{l_2(\mathbb {Z}_+)}. \end{aligned}$$

Since

$$\begin{aligned} \Vert (1-E_n)w(p,q)H^{-\frac{m}{2}}\eta \Vert _{l_2(\mathbb {Z}_+)}\rightarrow 0,\quad n\rightarrow \infty , \end{aligned}$$

for every word w with $\textrm{len}(w)=m$ and since there are finitely many ($2^m,$ to be precise) words of length m,  it follows that:

$$\begin{aligned} \max _{\textrm{len}(w)=m}\Vert (1-E_n)w(p,q)H^{-\frac{m}{2}}\eta \Vert _{l_2(\mathbb {Z}_+)}\rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

This completes the proof. $\square$

Lemma 2.10

Let $(P_j)_{j\in \mathbb {J}}$ be a bounded family of homogeneous (of order m) polynomials in 2 non-commuting variables. Suppose that

$$\begin{aligned} |P_j(\Im (z),\Re (z))|\ge c,\quad z\in \mathbb {C},\quad |z|=1,\quad j\in \mathbb {J}. \end{aligned}$$

Suppose also that condition (1.) in Theorem 2.1 fails. Let $(\eta _k)_{k\ge 0}$ and $\eta$ be as in Lemma 2.6. We have $\eta _k\rightarrow \eta$ in $l_2(\mathbb {Z}_+).$

Proof

By definition, we have

$$\begin{aligned} P_j=\sum _{\textrm{len}(w)=m}a_{j,w}w(p,q). \end{aligned}$$

Boundedness of the family means that

$$\begin{aligned} \sup _{j\in \mathbb {J}}|a_{j,w}|<\infty ,\quad \textrm{len}(w)=m. \end{aligned}$$

We may assume without loss of generality that

$$\begin{aligned} \sum _{\textrm{len}(w)=m}\sup _{j\in \mathbb {J}}|a_{j,w}|\le 1. \end{aligned}$$

It follows from the preceding paragraph that:

$$\begin{aligned} \sup _{j\in \mathbb {J}}\Vert P_j(-\Im (U),\Re (U))\Vert _{\infty }\le 1. \end{aligned}$$

(2.2)

Fix $\epsilon >0$ and choose, using Lemmas 2.7 and 2.9, $n(\epsilon )\ge 2m$, such that

$$\begin{aligned} \sup _{j\in \mathbb {J}}\Big \Vert (1-E_{n(\epsilon )})\Big (P_j(p,q)H^{-\frac{m}{2}}-P_j(-\Im (U),\Re (U))\Big )\Big \Vert _{\infty }<\epsilon , \end{aligned}$$

(3.2)

$$\begin{aligned}&\displaystyle \Vert (1-E_{n(\epsilon )-2m})\eta \Vert _{l_2(\mathbb {Z}_+)}<\epsilon , \end{aligned}$$

(4.2)

$$\begin{aligned}&\displaystyle \sup _{j\in \mathbb {J}}\Vert (1-E_{n(\epsilon )})P_j(p,q)H^{-\frac{m}{2}}\eta \Vert _{l_2(\mathbb {Z}_+)}<\epsilon . \end{aligned}$$

(5.2)

Using the third and second conditions in Lemma 2.6, we can choose $k(\epsilon )$, such that

$$\begin{aligned}{} & {} \Vert P_{j_k}(p,q)H^{-\frac{m}{2}}\eta _k\Vert _{l_2(\mathbb {Z}_+)}<\epsilon ,\quad k\ge k(\epsilon ), \end{aligned}$$

(6.2)

$$\begin{aligned}{} & {} \Vert E_{n(\epsilon )+m}(\eta _k-\eta )\Vert _{l_2(\mathbb {Z}_+)}<\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

(7.2)

It follows from (6.2) that:

$$\begin{aligned} \Vert (1-E_{n(\epsilon )})P_{j_k}(p,q)H^{-\frac{m}{2}}\eta _k\Vert _{l_2(\mathbb {Z}_+)}<\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

Using (5.2) and the triangle inequality, we write

$$\begin{aligned} \Vert (1-E_{n(\epsilon )})P_{j_k}(p,q)H^{-\frac{m}{2}}(\eta _k-\eta )\Vert _{l_2(\mathbb {Z}_+)}<2\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

Using (3.2), we write

$$\begin{aligned} \Vert (1-E_{n(\epsilon )})P_{j_k}(-\Im (U),\Re (U))(\eta _k-\eta )\Vert _{l_2(\mathbb {Z}_+)}<3\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

It follows from Lemma 2.8 that:

$$\begin{aligned} E_{n(\epsilon )}\cdot P_{j_k}(-\Im (U),\Re (U))=E_{n(\epsilon )}\cdot P_{j_k}(-\Im (U),\Re (U))\cdot E_{n(\epsilon )+m}. \end{aligned}$$

Thus

$$\begin{aligned}{} & {} \Vert E_{n(\epsilon )}P_{j_k}(-\Im (U),\Re (U))(\eta _k-\eta )\Vert _{l_2(\mathbb {Z}_+)}\\{} & {} \quad \le \Vert P_{j_k}(-\Im (U),\Re (U))\Vert _{\infty }\Vert E_{n(\epsilon )+m}(\eta _k-\eta )\Vert _{l_2(\mathbb {R})}\\{} & {} \quad {\mathop {<}\limits ^{(7)}}\epsilon \Vert P_{j_k}(-\Im (U),\Re (U))\Vert _{\infty }{\mathop {\le }\limits ^{(2)}}\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

Hence

$$\begin{aligned} \Vert P_{j_k}(-\Im (U),\Re (U))(\eta _k-\eta )\Vert _{l_2(\mathbb {Z}_+)}<4\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

Again, using (7.2), we obtain

$$\begin{aligned} \Vert P_{j_k}(-\Im (U),\Re (U))(1-E_{n(\epsilon )})(\eta _k-\eta )\Vert _{l_2(\mathbb {Z}_+)}<5\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

Taking into account that $n(\epsilon )\ge m$ and using Lemma 2.3, we write

$$\begin{aligned} \Vert P_{j_k}(-\Im (V),\Re (V))(1-E_{n(\epsilon )})(\eta _k-\eta )\Vert _{l_2(\mathbb {Z})}<5\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

By the assumption, we have

$$\begin{aligned} \Vert P_j(-\Im (V),\Re (V))\xi \Vert _{l_2(\mathbb {Z})}\ge c\Vert \xi \Vert _{l_2(\mathbb {Z})},\quad \xi \in l_2(\mathbb {Z}). \end{aligned}$$

Thus

$$\begin{aligned} c\Vert (1-E_{n(\epsilon )})(\eta _k-\eta )\Vert _{l_2(\mathbb {Z})}<5\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

It follows now from (7.2):

$$\begin{aligned} \Vert \eta _k-\eta \Vert _{l_2(\mathbb {Z})}\le (5c^{-1}+1)\epsilon ,\quad k\ge k(\epsilon ). \end{aligned}$$

Since $\epsilon >0$ can be chosen arbitrarily small, the assertion follows. $\square$

Proof of Theorem 2.1

If the condition (1.) holds, then

$$\begin{aligned} \Vert P_j(p,q)\xi \Vert _{l_2(\mathbb {Z}_+)}\ge c\Vert H^{\frac{m}{2}}\xi \Vert _{l_2(\mathbb {Z}_+)},\quad \xi \in \textrm{dom}(H^{\frac{m}{2}}),\quad j\in \mathbb {J}. \end{aligned}$$

Hence, exactly the same estimate holds for $j\in \beta \mathbb {J}.$ In particular, if $P_j(p,q)\xi =0,$ $\xi \in \textrm{dom}(H^{\frac{m}{2}}),$ $j\in \beta \mathbb {J},$ then $H^{\frac{m}{2}}\xi =0$ and, therefore, $\xi =0.$ Necessity of the condition (2.) follows now from Lemma 2.5.

Suppose now that the condition (1.) fails and that

$$\begin{aligned} |P_j(\Im (z),\Re (z))|\ge c,\quad z\in \mathbb {C},\quad |z|=1,\quad j\in \mathbb {J}. \end{aligned}$$

Let $(\eta _k)_{k\ge 0}$ and $\eta$ be as in Lemma 2.6. By Lemma 2.10, we have $\eta _k\rightarrow \eta$ in $l_2(\mathbb {Z}_+).$ Since $\Vert \eta _k\Vert _{l_2(\mathbb {Z}_+)}=1$ for every $k\ge 0,$ it follows that $\Vert \eta \Vert _{l_2(\mathbb {Z}_+)}=1.$ The third condition in Lemma 2.6 asserts that $P_{j_k}(p,q)H^{-\frac{m}{2}}\eta _k\rightarrow 0$ in $l_2(\mathbb {Z}_+)$ as $k\rightarrow \infty .$ Since the family $\{P_j\}_{j\in \mathbb {J}}$ is bounded, it follows that $P_{j_k}(p,q)H^{-\frac{m}{2}}\eta \rightarrow 0$ in $l_2(\mathbb {Z}_+)$ as $k\rightarrow \infty .$

Set $\xi =H^{-\frac{m}{2}}\eta \in \textrm{dom}(H^{\frac{m}{2}}).$ We have $\xi \ne 0$ and $P_{j_k}(p,q)\xi \rightarrow 0$ in $l_2(\mathbb {Z}_+)$ as $k\rightarrow \infty .$ Passing to a subsequence if needed, we may assume without loss of generality that $j_k\rightarrow j\in \beta \mathbb {J}.$ Thus, $P_j\xi =0$ and, hence, the condition (2.) fails. $\square$

3 Proof of Theorem 1.5

Consider the position and momentum operators q and p on $L_2(\mathbb {R}).$ Let $\{\psi _k\}_{k\ge 0}$ be the Hermite basis in $L_2(\mathbb {R}).$ Recall that

$$\begin{aligned} ip\psi _k=\frac{1}{\sqrt{2}}(k^{\frac{1}{2}}\psi _{k-1}-(k+1)^{\frac{1}{2}}\psi _{k+1}),\quad q\psi _k=\frac{1}{\sqrt{2}}(k^{\frac{1}{2}}\psi _{k-1}+(k+1)^{\frac{1}{2}}\psi _{k+1}) \end{aligned}$$

for every $k\in \mathbb {Z}_+.$ In what follows, we identify $\xi \in L_2(\mathbb {R})$ with the sequence $\{\langle \xi ,\psi _k\rangle \}_{k\in \mathbb {Z}_+}\in l_2(\mathbb {Z}_+).$ In this way, we identify the Hilbert spaces $L_2(\mathbb {R})$ and $l_2(\mathbb {Z}_+)$ and, hence, we fall exactly into the setting of the preceding section. Consequently, Theorem 2.1 applies for these q and p.

Proof of Theorem 1.5

By continuity, the ellipticity condition holds for the closure of $\mathcal {S}(\mathbb {H}^1)$ in the graph norm of $(-\Delta )^{\frac{m}{2}}.$ Hence, the condition

$$\begin{aligned} \Vert P_gf\Vert _{L_2(\mathbb {H}^1)}\ge c_P\Vert (-\Delta )^{\frac{m}{2}}f\Vert _{L_2(\mathbb {H}^1)},\quad f\in \mathcal {S}(\mathbb {H}^1),\quad g\in \mathbb {H}^1 \end{aligned}$$

is equivalent to the condition

$$\begin{aligned} \Vert P_gf\Vert _{L_2(\mathbb {H}^1)}\ge c_P\Vert (-\Delta )^{\frac{m}{2}}f\Vert _{L_2(\mathbb {H}^1)},\quad f\in \textrm{dom}((-\Delta )^{\frac{m}{2}}),\quad g\in \mathbb {H}^1. \end{aligned}$$

We want to apply Theorem 2.1 with $\mathbb {J}=\mathbb {H}^1\times \{-1,1\}$ and with

$$\begin{aligned} P_{(g,\pm 1)}=\sum _{\textrm{len}(w)=m}a_{w}(g)w(\pm p,q),\quad g\in \mathbb {H}^1. \end{aligned}$$

Recall the Plancherel decomposition

$$\begin{aligned} L_2(\mathbb {H}^1)=\int _{\mathbb {R}\backslash \{0\}}^{\oplus }L_2(\mathbb {R})\textrm{d}\nu (s), \end{aligned}$$

where $\nu$ is a Plancherel measure (in fact, $\textrm{d}\nu (s)=sds,$ up to a constant but the precise formula is not very important)

$$\begin{aligned} X_1=i\int _{\mathbb {R}\backslash \{0\}}^{\oplus }\textrm{sgn}(s)|s|^{\frac{1}{2}}p\textrm{d}\nu (s),\quad X_2=i\int _{\mathbb {R}\backslash \{0\}}^{\oplus }|s|^{\frac{1}{2}}q\textrm{d}\nu (s). \end{aligned}$$

Thus

$$\begin{aligned} (-\Delta )^{\frac{m}{2}}= & {} \int _{\mathbb {R}\backslash \{0\}}^{\oplus }|s|^mH^{\frac{m}{2}}\textrm{d}\nu (s),\\ P_g= & {} \int _{\mathbb {R}\backslash \{0\}}^{\oplus }|s|^m(P_{(g,1)}\chi _{(0,\infty )}(s)+P_{(g,-1)}\chi _{(-\infty ,0)}(s))\textrm{d}\nu (s). \end{aligned}$$

We have

$$\begin{aligned} \textrm{dom}((-\Delta )^{\frac{m}{2}})=\Big \{\int _{\mathbb {R}\backslash \{0\}}^{\oplus }f_s\textrm{d}\nu (s):\ \int _{\mathbb {R}\backslash \{0\}}\Vert H^{\frac{m}{2}}f_s\Vert _{L_2(\mathbb {R})}^2|s|^{2m}\textrm{d}\nu (s)<\infty \Big \}. \end{aligned}$$

Hence, the ellipticity condition

$$\begin{aligned} \Vert P_gf\Vert _{L_2(\mathbb {H}^1)}\ge c_P\Vert (-\Delta )^{\frac{m}{2}}f\Vert _{L_2(\mathbb {H}^1)},\quad f\in \textrm{dom}((-\Delta )^{\frac{m}{2}}),\quad g\in \mathbb {H}^1 \end{aligned}$$

can be equivalently rewritten as

$$\begin{aligned} \int _{\mathbb {R}\backslash \{0\}}|s|^{2m}\Vert P_{(g,\textrm{sgn}(s))}f_s\Vert _{L_2(\mathbb {R})}^2\textrm{d}\nu (s)\ge c_P^2\int _{\mathbb {R}\backslash \{0\}}\Vert H^{\frac{m}{2}}f_s\Vert _{L_2(\mathbb {R})}^2|s|^{2m}\textrm{d}\nu (s) \end{aligned}$$

whenever the right-hand side is finite.

Fix $\xi \in \textrm{dom}(H^{\frac{m}{2}})$ and set

$$\begin{aligned} f_s= {\left\{ \begin{array}{ll} \xi ,&{} s\in (0,1)\\ 0,&{} s\notin (0,1) \end{array}\right. } \text{ or } \text{ alternatively } f_s= {\left\{ \begin{array}{ll} \xi ,&{} s\in (-1,0)\\ 0,&{} s\notin (-1,0). \end{array}\right. } \end{aligned}$$

The ellipticity condition yields

$$\begin{aligned} \Vert P_{(g,\pm 1)}\xi \Vert _{L_2(\mathbb {R})}\ge c_P\Vert H^{\frac{m}{2}}\xi \Vert _{L_2(\mathbb {R})},\quad \xi \in \textrm{dom}(H^{\frac{m}{2}}). \end{aligned}$$

(8.2)

Conversely, the condition (8.2) clearly yields the ellipticity condition.

By Theorem 2.1, the condition (8.2) is equivalent to formal ellipticity of P (this is exactly the condition (1.) in Theorem (1.5) and the condition that $P_{(g,\pm 1)}(p,q)\xi =0,$ $\xi \in \textrm{dom}((-\Delta )^{\frac{m}{2}}),$ $g\in \beta \mathbb {H}^1,$ implies $\xi =0.$ Again, using the Plancherel decomposition, we see that the latter condition is equivalent to the condition (2.) in Theorem 1.5.

Hence, the ellipticity condition is equivalent to the condition (8.2) which is, in turn, equivalent to the conditions (1.) and (2.) in Theorem 1.5. $\square$