2.1 Graded Lie groups and their homogeneous structure

A Lie group $G$ is said to be graded if it is connected and simply connected, and its Lie algebra $\mathfrak {g}$ is endowed with a vector space decomposition $\mathfrak {g}=\oplus _{k =1}^{\infty } \mathfrak {g}_k$ (where all but finitely many of the $\mathfrak {g}_{k}$'s are $\{0\}$) such that $[\mathfrak {g}_{k},\, \mathfrak {g}_{k'}]\subset \mathfrak {g}_{k + k'}$ for all $k,\, k' \in \mathbb {N}$. Such a group is necessarily nilpotent, and the exponential map $\exp : \mathfrak {g} \rightarrow G$ is a diffeomorphism. Examples of graded Lie groups include the Euclidean space $\mathbb {R}^{n}$, the Heisenberg group $\mathbb {H}^{n}$ and, more generally, all stratified Lie groups.

We choose and fix a basis $\{X_1,\, \cdots,\, X_n\}$ of $\mathfrak {g}$, so that it is adapted to the gradation, i.e., $\{X_1,\, \cdots,\, X_{n_1}\}$ (possibly $\emptyset$) is a basis of $\mathfrak {g}_1$, $\{X_{n_1 +1},\, \cdots,\, X_{n_1 + n_2}\}$ (possibly $\emptyset$) is a basis of $\mathfrak {g}_2$, and so on. Via the map

(2.1)\begin{align} \mathbb{R}^{n} \ni (x_1, \cdots, x_n) \mapsto \exp(x_1 X_1 + \cdots + x_n X_n) \equiv x\in G, \end{align}

each point $(x_1,\, \cdots,\, x_n) \in \mathbb {R}^{n}$ is identified with the point $x \in G$. This map takes the Lebesgue measure on $\mathbb {R}^{n}$ to a bi-invariant Haar measure $\mu$ on $G$. We denote the group identity of $G$ by $e$.

The Lie algebra $\mathfrak {g}$ is equipped with a natural family of dilations $\{\delta _r\}_{r>0}$ which are linear mappings from $\mathfrak {g}$ to $\mathfrak {g}$ determined by

\[ \delta_r X= r^{k} X \quad \mbox{ for } X \in \mathfrak{g}_k \]

For each $j \in \{1,\, \cdots,\, n\}$, let $v_j$ be the unique positive integer such that $X_j \in \mathfrak {g}_{v_j}$. Then we have $\delta _r X_j = r^{v_j} X_j$, $j =1,\, \cdots,\, n$. The associated group dilation is given by

\[ \delta_r x = (r^{v_1}x_1, \cdots, r^{v_n} x_n), \]

for $x = (x_1,\, \cdots,\, x_n) \in G$ and $r>0$. The integers $v_1,\, \cdots,\, v_n$ are referred to as weights of the dilations $\{\delta _t\}_{t>0}$, and the positive integer

\[ Q:= \sum_{k=1}^{\infty} k (\dim \mathfrak{g}_k )=\sum_{j=1}^{n} v_j \]

is called the homogeneous dimension of $G$.

A homogeneous quasi-norm on G is a continuous function $x\rightarrow |x|$ from $G$ to $[0,\, \infty )$ which vanishes only at $e$ and satisfies that $|x^{-1}| = |x|$ and $|\delta _r x | = r|x|$ for all $x \in G$ and $r > 0$. An example of homogeneous quasi-norm on $G$ is given by

(2.2)\begin{equation} |x|_\kappa = \left(\sum_{j=1}^{n} x_j^{2\kappa /v_j} \right)^{1/(2\kappa)}, \end{equation}

where $\kappa$ is the smallest common multiple to the weights $v_1,\, \cdots,\, v_n$. Any two homogeneous quasi-norms on $G$ are equivalent (see [Reference Folland and Stein14]). Henceforth we fix a homogenous quasi-norm $|\cdot |$ on $G$. It satisfies a quasi-triangle inequality: there exists a constant $\gamma \geq 1$ such that

(2.3)\begin{equation} |xy| \leq \gamma (|x| + |y|) \end{equation}

for all $x,\,y \in G$.

There is an analogue of polar coordinates on homogeneous groups with the homogeneous dimension $Q$ replacing the topological dimension $n$, see [Reference Folland and Stein14]:

\[ \forall f \in L^{1} (G) \quad \int_G f(x)d\mu(x) = \int_0^{\infty} \int_{\mathfrak{S}} f(\delta_r y)r^{Q-1}d\sigma (y)dr, \]

where $d\sigma$ is a (unique) positive Borel measure on the unit sphere $\mathfrak {S} := \{x \in G: |x| =1\}$. This implies that for $0< r < R < \infty$ and $\theta \in \mathbb {R}$,

(2.4)\begin{equation} \int_{r \leq |x| \leq R} |x|^{\theta -Q}d\mu(x) = \begin{cases} C \theta^{{-}1} (R^{\theta} - r^{\theta}) & \mbox{if } \theta \neq 0, \\ C \log(R /r) & \mbox{if } \theta =0. \end{cases} \end{equation}

Consequently, if $\theta >0$ then $|\cdot |^{\theta -Q}$ is integrable near the group identity $e$, and if $\theta <0$ then $|\cdot |^{\theta - Q}$ is integrable near $\infty$.

Since $G$ has been identified with $\mathbb {R}^{n}$ via the map given in (2.1), functions on $G$ can be viewed as functions on $\mathbb {R}^{n}$, and vise versa. This leads naturally to the notions of test function classes $\mathcal {D}(G)$, $\mathcal {S}(G)$ and the distribution spaces $\mathcal {D}'(G)$, $\mathcal {S}'(G)$. For example, a function $f$ is said to be in the Schwartz class $\mathcal {S}(G)$ if $f \circ \exp$ is a Schwartz function on $\mathbb {R}^{n}$. The coordinate function $G \ni x =(x_1,\, \cdots,\, x_n) \mapsto x_1 \in \mathbb {R}$ is denoted by $x_1$. For a multi-index $\alpha =(\alpha _1,\, \cdots,\, \alpha _n) \in \mathbb {N}_0^{n}$, we define $x^{\alpha } = x_1^{\alpha _1} \cdots x_n^{\alpha _n}$, as a function on $G$. Similarly, we set $X^{\alpha } = X_1^{\alpha _1} \cdots X_n^{\alpha _n}$ in the universal enveloping Lie algebra $\mathfrak {U}(\mathfrak {g})$ of $\mathfrak {g}$. We shall follow the usual custom of identifying each vector of $\mathfrak {g}$ with a left-invariant vector field on $G$ and, more generally, we identify the universal enveloping Lie algebra of $\mathfrak {g}$ with the left-invariant differential operators. In what follows we keep the same notation for the vectors and the corresponding operators. By the Poincaré–Birkhoff–Witt theorem, the $X^{\alpha }$'s form a basis for the algebra of the left-invariant differential operators on $G$.

In a canonical way the dilations $\{\delta _r\}_{r>0}$ lead to the notions of homogeneity for functions and operators. For instance the degree of homogeneity of the function $x^{\alpha }$ and the differential operator $X^{\alpha }$ is

\[ [\alpha] : = \sum_{j =1}^{n} v_j \alpha_j. \]

A function $P: G \rightarrow \mathbb {C}$ is called a polynomial, if it is of the form

\[ P(x) =\sum_{\alpha \in \mathbb{N}_0^{n}} c_\alpha x^{\alpha} \]

where all but finitely many of the complex coefficients $c_\alpha$ vanish. The homogeneous degree of the polynomial $P$ is defined as $\max \{[\alpha ]: c_\alpha \neq 0 \}$. For $M \in \mathbb {N}_0$, we set

\[ \mathcal{P}_M := \{\mbox{all polynomials on }G \mbox{ with homogeneous degree } \leq M\}.\hat{} \]

We denote by $\widetilde {X}_1,\, \cdots,\, \widetilde {X}_n$ the corresponding basis for right-invariant vector fields, that is,

\[ \widetilde{X}_j f (x) = \frac{d}{dt} f \big(\exp(tX_j) x\big)\big|_{t=0}, \quad j =1, \cdots, n. \]

Also, for $\alpha \in \mathbb {N}_0^{n}$, we set $\widetilde {X}^{\alpha } = \widetilde {X}_1^{\alpha _1} \cdots \widetilde {X}_n^{\alpha _n}$.

If $f$ and $g$ are measurable functions on $G$, then their convolution is defined by

\[ f \ast g (x) = \int_G f(y)g(y^{{-}1}x)d\mu(y) = \int_G f(xy^{{-}1})g(y)d\mu(y), \]

provided that the integrals converge. For any multi-index $\alpha \in \mathbb {N}_0^{n}$ and sufficiently good functions $f$ and $g$, we have (see [Reference Folland and Stein14, chapter 1])

(2.5)\begin{equation} X^{\alpha} (f \ast g) = f \ast (X^{\alpha} g), \quad \widetilde{X}^{\alpha} (f \ast g) = (\widetilde{X}^{\alpha} f) \ast g, \quad (X^{\alpha} f) \ast g = f \ast (\widetilde{X}^{\alpha} g). \end{equation}

2.2 Fourier analysis on graded Lie groups

The general theory of representation of Lie groups may be found in [Reference Dixmier and Jellett9]. Here we also refer to [Reference Fischer and Ruzhansky12] for a description which is more adapted to our particular context.

A representation $\pi$ of a Lie group $G$ on a Hilbert space $\mathcal {H}_{\pi } \neq \{0\}$ is a homomorphism from $G$ into the group of bounded linear operators on $\mathcal {H}_{\pi }$ with bounded inverse. More precisely,

• • for every $x \in G$, the linear mapping $\pi (x) : \mathcal {H}_{\pi } \rightarrow \mathcal {H}_{\pi }$ is bounded and has bounded inverse;

• • for every $x,\, y \in G$, we have $\pi (xy )=\pi (x) \pi (y)$.

A representation $\pi$ of $G$ is called irreducible if it has no closed invariant subspaces. $\pi$ is called unitary if $\pi (x)$ is unitary for every $x \in G$, and is called strongly continuous if the mapping $\pi : G \rightarrow \mathscr {L}(\mathcal {H_{\pi }})$ is continuous with respect to the strong operator topology in $\mathscr {L}(\mathcal {H}_{\pi })$. Two representations $\pi _1$ and $\pi _2$ are said to be equivalent if there exists a bounded linear mapping $A: \mathcal {H}_{\pi _1} \rightarrow \mathcal {H}_{\pi _2}$ between their representation spaces with a bounded inverse such that the relation $A \pi _1 (x) = \pi _2 (x)A$ holds for all $x \in G$. In this case we write $\pi _1 \sim \pi _2$, and denote their equivalence class by $[\pi _1] = [\pi _2]$. The set of all equivalence classes of strongly continuous irreducible unitary representations of $G$ is called the unitary dual of $G$ and is denoted by $\widehat {G}$. In what follows, we will identify one representation $\pi$ with its equivalent class $[\pi ]$.

For a unitary representation of $G$, the corresponding infinitesimal representation which acts on the universal enveloping algebra $\mathfrak {U}(\mathfrak {g})$ of the Lie algebra $\mathfrak {g}$ is still denoted by $\pi$. This is characterized by its action on $\mathfrak {g}$:

\[ \pi (X) = \partial_{t =0} \pi (e^{tX}), \quad X \in \mathfrak{g}. \]

The infinitesimal action acts on the space $\mathcal {H}_{\pi }^{\infty }$ of smooth vectors, that is, the space of vectors $v \in \mathcal {H}_{\pi }$ such that the function $G \ni x \mapsto \pi (x) v \in \mathcal {H}_{\pi }$ is of class $C^{\infty }$.

The Fourier coefficients or group Fourier transform of a function $f \in L^{1} (G)$ at $\pi \in \widehat {G}$ is defined by

\[ \mathcal{F}_G f (\pi) \equiv \widehat{f}(\pi) \equiv \pi (f) := \int_G f(x) \pi(x)^{{\ast}} d\mu(x). \]

It is readily seen that

\[ \|\widehat{f}(\pi)\|_{\mathscr{L}(\mathcal{H}_\pi)} \leq \|f\|_{L^{1} (G)}. \]

For $f,\, g \in L^{1} (G)$, we also have

\[ \widehat{f \ast g} (\pi) = \widehat{g}(\pi) \widehat{f}(\pi). \]

There exists a unique positive Borel measure $\widehat {\mu }$ on $\widehat {G}$, called the Plancherel measure, such that for any continuous function $f$ on $G$ with compact support, one has

\[ \int_G |f(x)|^{2} d\mu(x) = \int_{\widehat{G}} \|\mathcal{F}_G f (\pi)\|_{HS(\mathcal{H}_{\pi})}^{2} d\widehat{\mu}(\pi), \]

where $\|\cdot \|_{HS(\mathcal {H}_{\pi })}$ denotes the Hilbert–Schmidt norm on the space $HS(\mathcal {H}_\pi ) \sim \mathcal {H}_{\pi } \otimes \mathcal {H}_{\pi }^{\ast }$ of Hilbert–Schmidt operators on the Hilbert space $\mathcal {H}_{\pi }$. Since $L^{1}(G) \cap L^{2}(G)$ is dense in $L^{2}(G)$, the Fourier transform $\mathcal {F}_G$ extends to a unitary operator from $L^{2}(G)$ onto $L^{2} (\widehat {G})$.

By the general theory on locally compact unimodular groups of type I (see e.g. [Reference Dixmier and Jellett9]), if $T$ is an $L^{2}$-bounded operator on $G$ which commutes with left-translations, then there exists a field of bounded operators $\widehat {T}(\pi )$ such that for all $f \in L^{2} (G)$,

\[ \mathcal{F}_G (Tf)(\pi) = \widehat{T} (\pi) \widehat{f}(\pi) \quad \text{a.e. } \pi \in \widehat{G}. \]

Moreover, we have

\[ \|T\|_{\mathscr{L}(L^{2}(G))} = \sup_{\pi \in \widehat{G}} \|\widehat{T}(\pi)\|_{\mathscr{L}(\mathcal{H}_\pi)}, \]

where the supremum here is understood as the essential supremum with respect to the Plancherel measure $\mu$. Conversely, given any $\sigma =\{\sigma (\pi ),\, \pi \in \widehat {G}\} \in L^{\infty } (\widehat {G})$, there is a corresponding operator $T_\sigma$ given by

\[ \mathcal{F}_G (T_\sigma f) (\pi) = \sigma(\pi) \widehat{f}(\pi), \quad f \in L^{2} (G). \]

By the Plancherel theorem, $T_\sigma$ is bounded on $L^{2}(G)$ with $\|T_\sigma \|_{\mathscr {L}(L^{2}(G))} =\|\sigma \|_{L^{\infty } (\widehat {G})}$.

If $\pi$ is a unitary irreducible representation of $G$ and $r >0$, we define $r \cdot \pi$ to be the unitary irreducible representation such that

\[ r \cdot \pi (x) = \pi (\delta_r x), \quad x \in G. \]

2.3 Rockland operators

Let $G$ be a graded Lie group. A left-invariant differential operator $\mathcal {R}$ on $G$ is called a Rockland operator if it is homogeneous of positive degree and for each unitary irreducible non-trivial representation $\pi$ of $G$, the operator $\pi (\mathcal {R})$ is injective on $\mathcal {H}_{\pi }^{\infty }$. Rockland operators may be defined on any homogeneous group, however it turns out that the existence of a Rockland operator on a homogeneous group implies that (the Lie algebra of) the group admits a gradation. This is the reason why we and the authors in [Reference Fischer and Ruzhansky11] consider the setting of graded Lie groups. On any graded Lie group $G$, the operator

\[ \sum_{j =1}^{n} ({-}1)^{\frac{\nu_0}{v_j}} c_j X_j^{2 \frac{\nu_0}{v_j}} \]

with $c_j >0$ is a Rockland operator of homogeneous degree 2$\nu _0$ if $\nu _0$ is any common multiple of $v_1,\, \cdots,\, v_n$.

We will mainly consider positive Rockland operators. A Rockland operator $\mathcal {R}$ is said to be positive, if

\[ \int_G \mathcal{R}f (x)\overline{f(x)}d\mu(x) \geq 0 \]

for all $f \in \mathcal {S}(G)$. If a Rockland operator $\mathcal {R}$ is positive then $\mathcal {R}$ and $\pi (\mathcal {R})$ admit self-adjoint extensions on $L^{2} (G)$ and $\mathcal {H}_{\pi }$, respectively. We use the same notation for their self-adjoint extensions. By the spectral theory, we have

\[ \mathcal{R} = \int_0^{\infty} \lambda dE_{\mathcal{R}}(\lambda)\quad \mbox{and} \quad \pi(\mathcal{R}) = \int_0^{\infty} \lambda dE_{\pi(\mathcal{R})}(\lambda), \]

where $E_{\mathcal {R}}(\lambda )$ (resp. $E_{\pi (\mathcal {R})} (\lambda )$) is the resolution of the identity associated to $\mathcal {R}$ (resp. $\pi (\mathcal {R})$).

For any bounded Borel function $\varphi$ on $[0,\,\infty )$, the operator

\[ \varphi(\mathcal{R}) = \int_{0}^{\infty} \varphi(\lambda)dE_{\mathcal{R}}(\lambda) \]

is bounded on $L^{2}(G)$, and commutes with left translations. Thus, by the Schwartz kernel theorem, there exists a distribution $K_{\varphi (\mathcal {R})} \in \mathcal {S}'(G)$ such that

\[ \varphi(\mathcal{R})f =f \ast K_{\varphi(\mathcal{R})}, \quad \forall f \in \mathcal{S}(G). \]

Note that the point $\lambda =0$ may be neglected in the spectral resolution, since the projection measure of $\{0\}$ is zero (see [Reference Hulanicki, Jenkins and Ludwig17] or [Reference Fischer and Ruzhansky12, remark 4.2.8]). Consequently we should regard $\varphi$ as a function on $(0,\, \infty )$ rather than on $[0,\,\infty )$.

We now recall Hulanicki's theorem, which will play an important role in the proof of our main result.

Theorem 2.1 Hulanicki [Reference Hulanicki16]

Let $G$ be a graded Lie groups and let $\mathcal {R}$ be a positive Rockland operator on $G$. For any $M_1 \in \mathbb {N}$ and $M_2 \geq 0,$ there exist $C =C(M_1,\,M_2) >0$ and $k = k(M_1,\, M_2),\, k'=k'(M_1,\,M_2) \in \mathbb {N}_0$ such that, for any $\varphi \in C^{k} (0,\, \infty ),$ the convolution kernel $K_{\varphi (\mathcal {R})}$ of $\varphi (\mathcal {R})$ satisfies

\[ \sum_{[\alpha] \leq M_1} \int_G |X^{\alpha} K_{\varphi(\mathcal{R})} (x)|(1+|x|_\kappa)^{M_2}d\mu(x) \leq C \sup_{\substack{\lambda\in (0,\infty) \\ \ell \in \{0,1,\cdots, k\}\\ \ell' \in \{0,1,\cdots, k'\}}} (1+\lambda)^{\ell'}\left|\frac{d^{\ell}}{d\lambda^{\ell}}\varphi(\lambda) \right|. \]

The same result with the right-invariant vector fields $\widetilde {X}_j$'s instead of the left-invariant vector fields $X_j$'s also holds.

2.4 Difference operators

The Mihlin condition (1.1) is formulated in terms of the derivatives with respect to the Fourier variable $\xi$. However, for a field $\sigma =\{\sigma (\pi ),\, \pi \in \widehat {G}\}$ of operators, there is no direct way to define an analogue of derivatives with respect to the Fourier variable $\pi$. To generalize the symbolic conditions to the setting of graded Lie groups, Fischer and Ruzhansky [Reference Fischer and Ruzhansky12] introduced the so-called difference operators, whose definition we now recall.

For $a,\, b \in \mathbb {R}$, we denote by $\mathscr {L}_L (L_a^{2}(G),\, L_b^{2}(G))$ the subspace of all $T \in \mathscr {L}(L^{2}_a(G),\, L^{2}_b(G))$ which are left-invariant. Here $L^{2}_a(G)$ is the Bessel potential space (fractional Sobolev space) defined in [Reference Fischer and Ruzhansky13]. Define

\begin{align*} \mathcal{K}_{a,b} (G):= \Big\{K \in \mathcal{S}'(G): \ & \mbox{the operator } \mathcal{S}(G) \ni f \mapsto f \ast K \mbox{ extends to} \\ & \mbox{a bounded operator from } L_a^{2}(G) \mbox{ to } L_b^{2}(G)) \Big\} \end{align*}

and define $L_{a,b}^{\infty }(\widehat {G})$ to be the space of all fields $\sigma = \{\sigma (\pi ),\, \pi \in \widehat {G}\}$ such that

\[ \|\sigma\|_{L^{\infty}_{a,b}(\widehat{G})} := \sup_{\pi \in \widehat{G}} \|\pi(I +\mathcal{R})^{\frac{b}{\nu}} \sigma(\pi)\pi(I +\mathcal{R})^{-\frac{a}{\nu}} \|_{\mathscr{L}(\mathcal{H}_{\pi})} <\infty. \]

From [Reference Fischer and Ruzhansky12, proposition 5.1.24] we see that, if $\sigma \in L^{\infty }_{a,b}(\widehat {G})$ then the Fourier multiplier operator $T_{\sigma }$ corresponding to $\sigma$ belongs to $\mathscr {L}_L (L_a^{2} (G),\, L_b^{2}(G))$ with

\[ \|T_{\sigma}\|_{\mathscr{L}(L_a^{2} (G), L_b^{2} (G))} = \|\sigma\|_{L^{\infty}_{a,b}(\widehat{G})}. \]

Conversely, if $T \in \mathscr {L}_L (L_a^{2}(G),\, L_b^{2}(G))$, then there exists a unique $\sigma \in L^{\infty }_{a,b}(\widehat {G})$ such that

\[ \mathcal{F}_G (T f)(\pi) = \sigma(\pi) \widehat{f}(\pi), \quad f \in L^{2}(G). \]

In this case, denoting by $K \in \mathcal {K}_{a,b}(G)$ the convolution kernel of $T$, we define

\[ \mathcal{F}_G K = \sigma \quad \mbox{and} \quad \mathcal{F}_G^{{-}1}\sigma = K. \]

This extends the definition of Fourier transform to the space $\mathcal {K}_{a,b} (G)$. See [Reference Fischer and Ruzhansky12, definition 5.1.25].

For $\alpha \in \mathbb {N}_0^{n}$ and $\sigma =\{\sigma (\pi ),\, \pi \in \widehat {G}\}\in L^{\infty }_{a,b}(\widehat {G})$, the difference operator $\Delta ^{\alpha }$ acting on $\sigma$ is defined according to the formula (see [Reference Fischer and Ruzhansky12, definition 5.2.1])

\[ \Delta^{\alpha} \sigma (\pi) = \mathcal{F}_G (q_\alpha \mathcal{F}_G^{{-}1}\sigma)(\pi) \quad \mbox{for a.e. } \pi \in \widehat{G}, \]

where $q_\alpha (x) =x^{\alpha }$. Analogously to the derivatives in the Euclidean setting, the operator $\Delta ^{\alpha }$ satisfies the Lebnitz rule [Reference Fischer and Ruzhansky12, section 5.2.2]:

(2.6)\begin{equation} \Delta^{\alpha} (\sigma \tau) =\sum_{\alpha_1 +\alpha_2 =\alpha} C_{\alpha_1, \alpha_2} \Delta^{\alpha_1} (\sigma) \Delta^{\alpha_2} (\tau), \quad \sigma, \tau \in L^{\infty}_{a,b}(\widehat{G}). \end{equation}

3.1 Definition of Hardy spaces on $G$

We first introduce some maximal functions. Given a function $\Phi \in \mathcal {S}(G)$, we defined the nontangential maximal function $M_\Phi f$ and the radial maximal function $M_\Phi ^{0} f$ of $f \in \mathcal {S}'(G)$ by

\[ M_\Phi f (x) : = \sup_{|x^{{-}1}y| < t} |f \ast \Phi_t (y)| \quad \mbox{and} \quad M_\Phi^{0} f (x): = \sup_{t >0 } |f \ast \Phi_t (x)|, \]

respectively, where $\Phi _t (x):= t^{-Q} \Phi (\delta _{t^{-1}}x)$. We then define the grand maximal function $M_{(N)}f$ for each $N\in \mathbb {N}$ by

\[ M_{(N)} f(x) := \sup_{\Phi \in \mathcal{S}(G), \|\Phi\|_{(N)} \leq 1} M_\Phi f(x), \]

where

\[ \|\Phi\|_{(N)}: = \sup_{|\alpha| \leq N, x \in G} (1 + |x|_\kappa)^{(N+1)(Q+1)} |\widetilde{X}^{\alpha} \Phi (x)|. \]

Moreover, given $N \in \mathbb {N}$, we define the grand maximal function $M_{(N)}f$ of $f \in \mathcal {S}'(G)$ by

Definition 3.1 For $0< p<\infty$, the Hardy space $H^{p}(G)$ is defined as

\[ H^{p} (G): = \big\{f \in \mathcal{S}'(G) : M_{(N_p)} f <\infty\big\}, \]

where

\[ N_p := \min\{[\alpha]: \alpha \in \mathbb{N}_0^{n} \mbox{ with } [\alpha] > Q (1/p-1)\}. \]

The quasi-norm on $H^{p}(G)$ is defined by

\[ \|f\|_{H^{p} (G)}: = \|M_{(N_p)} f\|_{L^{p} (G)}. \]

The Hardy spaces $H^{p} (G)$, initially defined via grand maximal function, can be characterized by radial maximal function and nontangential maximal function equivalently. To recall these maximal characterizations, we need the notion of commutative approximate identities introduced in [Reference Folland and Stein14]. A commutative approximate identity on $G$ is a function $\Phi \in \mathcal {S}(G)$ such that $\int _{G} \Phi (x)dx =1$ and $\Phi _s \ast \Phi _t =\Phi _t \ast \Phi _s$ for all $s,\, t >0$. On a graded Lie group it is easy to construct a commutative approximate identity. Indeed, if $\mathcal {R}$ is a positive Rockland operator on $G$, and $\varphi \in \mathcal {S}(\mathbb {R})$ such that $\varphi (0) =1$, then the convolution kernel of the operator $\varphi (\mathcal {R})$ is a commutative approximate identity.

Proposition 3.2 [Reference Folland and Stein14, corollary 4.17]

Suppose $0< p <\infty$ and $\Phi$ is a commutative approximate identity. Then for $f \in \mathcal {S}'(G),$ the following are equivalent:

1. (i) $M_{\Phi }^{0} f \in L^{p}(G);$

2. (ii) $M_{\Phi } f \in L^{p}(G);$

3. (iii) $M_{(N_p)} f \in L^{p}(G)$.

Moreover, we have

\[ \|M_{\Phi}^{0} f\|_{L^{p}(G)} \sim \|M_{\Phi}f\|_{L^{p}(G)} \sim \|M_{(N_p)}f\|_{L^{p} (G)} \]

with the implicit constants depending only on $\Phi$ and $p$.

3.2 Atomic decomposition

Atomic decomposition is a very useful tool for the study of boundedness of operators on Hardy spaces. Analogously to the Euclidean case, Hardy spaces on graded Lie groups also admit an atomic decomposition, which we now recall. See [Reference Folland and Stein14] for more details.

A triplet $(p,\,q,\,M)$ is said to be admissible, if $0< p \leq 1 \leq q \leq \infty$, $p \neq q$ and $M \in \mathbb {N}_0$ with $M \geq \max \big \{[\alpha ]: \alpha \in \mathbb {N}_0^{n} \mbox { with } [\alpha ] \leq Q(1/p-1)\big \}$.

The atomic decomposition of $H^{p} (G)$ can be stated as follows.

Proposition 3.5 [Reference Folland and Stein14]

Let $(p,\,q,\,M)$ be an admissible triplet. Then there is a constant $c_1>0$ such that for all any $(p,\,q,\,M)$-atom $a,$ one has

\[ \|a\|_{H^{p} (G)} \leq c_1. \]

Conversely, given any $f \in H^{p} (G)$, there exist a sequence $\{a_j\}_{j=1}^{\infty }$ of $(p,\,q,\,M)$-atoms and a sequence $\{\lambda _j\}_{j=1}^{\infty }$ of complex numbers such that $f = \sum _{j=1}^{\infty }\lambda _j a_j$ with convergence in $\mathcal {S}'(G)$ and

\[ \left(\sum_{j=1}^{\infty}|\lambda_j|^{p}\right)^{1/p} \leq c_2 \|f\|_{H^{p} (G)}, \]

where $c_2$ is a constant independent of $f$.