1 Introduction and Statement of the Results

1.1 Grushin Operator

On \({\mathbb {R}}^{n+1}={\mathbb {R}}^n\times {\mathbb {R}}\) we consider the Grushin operator

$$\begin{aligned} {\mathcal {L}}= - \sum _{j=1}^n \left( \frac{\partial ^2}{\partial {x'_{j}}^{2}} + (x'_j)^2\frac{\partial ^2}{\partial {x''\,}^2}\right) =-\Delta _{x'}-|x'|^2\frac{\partial ^2}{\partial {x''\,}^2}, \end{aligned}$$

where \({\mathbb {R}}^n\times {\mathbb {R}}\ni {\mathbf {x}}=(x',x'')\), \(x'=(x'_1,x'_2,\ldots ,x'_n)\in {\mathbb {R}}^n\), \(x''\in {\mathbb {R}}\). The operator \({\mathcal {L}}\) is homogeneous of degree 2 with respect to the dilations

$$\begin{aligned} \delta _t{\mathbf {x}}=(tx',t^2x''), \end{aligned}$$

that is, \({\mathcal {L}}(f\circ \delta _t)({\mathbf {x}})= t^2 ({\mathcal {LF}})(\delta _t{\mathbf {x}})\). It is well known \(\mathcal {L}\) is a hypoelliptic operator. It is related to the Heisenberg group \(\mathbb {H}_n\). Actually the Grushin operator \({\mathcal {L}}\) is the image of a sub-Laplacian L associated to \(\mathbb {H}_n\) under a representation \(\pi \) acting on functions on \(\mathbb {R}^{n+1}.\) In fact we make use of this relation to prove some crucial estimates on some kernels related to \({\mathcal {L}}\).

The control distance on \({\mathbb {R}}^{n+1}\) associated with \({\mathcal {L}}\) is defined by

$$\begin{aligned} d({\mathbf {x}},{\mathbf {y}})=\sup _{\psi \in {\mathcal {D}}}| \psi ({\mathbf {x}})-\psi ({\mathbf {y}})|, \end{aligned}$$

where \({\mathcal {D}}=\{\psi \in W^{1,\infty }({\mathbb {R}}^{n+1}):\sum _{j=1}^n( |\partial _{x'_j} \psi |^2+|x_j'\partial _{x''}\psi |^2)\le 1\}\). It is homogeneous, that is,

$$\begin{aligned} d(\delta _s{\mathbf {x}},\delta _s{\mathbf {y}})= sd({\mathbf {x}},{\mathbf {y}}) \end{aligned}$$

and behaves like:

$$\begin{aligned} d({\mathbf {x}},{\mathbf {y}})\sim |x'-y'|+ {\left\{ \begin{array}{ll} \frac{|x''-y''|}{|x'|+|y'|} \ \ \ &{} \text {if } \ |x''-y''|^{1\slash 2}\le |x'|+|y'|,\\ |x''-y''|^{1\slash 2} \ \ \ &{} \text {if } \ |x''-y''|^{1\slash 2} > |x'|+|y'|,\\ \end{array}\right. } \end{aligned}$$
(1.1)

see, e.g., [18, 20] for details. Clearly,

$$\begin{aligned} |f({\mathbf {x}})-f({\mathbf {y}})|\le d({\mathbf {x}},{\mathbf {y}})\sum _{j=1}^n\left( \Vert \partial _{x'_j} f\Vert _{\infty }+\Vert x_j'\partial _{x''}f\Vert _{\infty }\right) . \end{aligned}$$
(1.2)

Let \(B({\mathbf {x}}, r)=\{ {\mathbf {y}}\in {\mathbf {X}}: d({\mathbf {x}},{\mathbf {y}})<r\}\) denote the ball with center \({\mathbf {x}}\) and radius \(r>0\) in the metric \(d({\mathbf {x}},{\mathbf {y}})\) and \(|B({\mathbf {x}},r)|\) be its Lebesgue measure volume. Then

$$\begin{aligned} |B({\mathbf {x}},r)|\sim r^{n+1}\max \{r, |x'|\}\sim r^{n+1}(r+|x'|) \end{aligned}$$
(1.3)

and, consequently,

$$\begin{aligned} \left( \frac{R}{r}\right) ^{n+1}\lesssim \frac{|B({\mathbf {x}}, R)|}{|B({\mathbf {x}},r)|}\lesssim \left( \frac{R}{r}\right) ^{n+2}, \ \ R\ge r>0. \end{aligned}$$
(1.4)

The homogeneity of the distance d implies

$$\begin{aligned} |B(\delta _s {\mathbf {x}}, sr)|=s^{n+2} |B( {\mathbf {x}}, r)|. \end{aligned}$$
(1.5)

The space \({\mathbf {X}}={\mathbb {R}}^n\times {\mathbb {R}}\) equipped with the Lebesgue measure \(d{\mathbf {x}}\) and the distance \(d({\mathbf {x}},{\mathbf {y}})\) is the space of homogeneous type in the sense of Coifman–Weiss [7]. It is well known (see e.g., [20]) that \(-{\mathcal {L}}\) generates a semigroup of self-adjoint linear operators \(e^{-t{\mathcal {L}}}\) on \(L^2({\mathbf {X}}) \) which has the form

$$\begin{aligned} e^{-t{\mathcal {L}}}f({\mathbf {x}})=\int _{{\mathbf {X}}} H_t({\mathbf {x}},{\mathbf {y}})f({\mathbf {y}})d{\mathbf {y}}, \end{aligned}$$

where the heat kernel \(H_t({\mathbf {x}},{\mathbf {y}})\) satisfies the Gaussian upper bound estimates (see (2.13)).

1.2 Hardy Space \(H^1_{\mathcal L}\)

Let \({\mathcal {M}}_{{\mathcal {L}}}f(x)=\sup _{t>0} |e^{-t{\mathcal {L}}}f(x)|\) be the maximal function associated with the semigroup \(e^{-t{\mathcal {L}}}\). The upper Gaussian estimates (2.13) imply that \({\mathcal {M}}_{{\mathcal {L}}}\) is bounded on \(L^p({\mathbf {X}})\) for \(1<p\le \infty \) and of weak-type (1,1). We define the Hardy space

$$\begin{aligned} H^1_{{\mathcal {L}}}=\big \{ f\in L^1({\mathbf {X}}): {\mathcal M}_{\mathcal L} f\in L^1({\mathbf {X}})\big \}, \end{aligned}$$
$$\begin{aligned} \Vert f\Vert _{H^1_{{\mathcal {L}}}}= \Vert {\mathcal {M}}_{{\mathcal {L}}} f\Vert _{L^1({\mathbf {X}})}. \end{aligned}$$

Now we define atoms associated to the homogeneous space \({\mathbf {X}}.\)

1.3 Atoms

Fix \(1<q\le \infty \). A function a is called a (1, q)-atom for the Hardy space \(H^1({\mathbf {X}})\) if there is a ball \(B=B({\mathbf {x}},r)=\{ {\mathbf {y}}: \ d({\mathbf {x}},{\mathbf {y}})<r\}\) such that

\(\text {supp}\, a\subset B\),

\(\Vert a\Vert _{L^q}\le |B|^{\frac{1}{q}-1}\) (\(\Vert a\Vert _{L^\infty }\le |B|^{-1}\) if \(q=\infty \)),

\(\int a({\mathbf {y}})\, d{\mathbf {y}}=0\).

The atomic norm is given by

$$\begin{aligned} \Vert f\Vert _{H^1_{\mathrm{atom},\, q}({\mathbf {X}}, d)}=\sum |\lambda _j|, \end{aligned}$$

where the infimum is taken over all decompositions \(f=\sum \lambda _j a_j\), \(\lambda _j\in {\mathbb {C}}\), \(a_j\) are (1, q)-atoms for \(H^1({\mathbf {X}})\).

We are now in a position to state our first result.

Theorem 1.1

For every \(q\in (1,\infty ]\) the space \(H^1_{{\mathcal {L}}}\) admits atomic decomposition and the norms \(\Vert f\Vert _{H^1_{\mathcal {L}}}\) and \(\Vert f\Vert _{H^1_{\mathrm{atom},\, q}({\mathbf {X}})}\) are equivalent.

1.4 Riesz Transforms

The system of Riesz transforms \({\mathcal {R}}_j\), \(j=1,2,\ldots ,2n\), associated with \({\mathcal {L}}\) is defined by

$$\begin{aligned} {\mathcal {R}}_j = \partial _{x_j'} {\mathcal {L}}^{-1\slash 2}, \ \ \ {\mathcal {R}}_{n+j}=x_j'\partial _{x''} {\mathcal {L}}^{-1\slash 2}, \ \ j=1,2,\ldots ,n. \end{aligned}$$

This formal definition has a precise meaning and the operators \({\mathcal {R}}_j\) are Calderón–Zygmund operators on \({\mathbf {X}}\). Moreover, \( {\mathcal {R}}_j\) are well-defined in the sense of distributions on \(L^1({\mathbf {X}})\) (see Sect. 4). Our second main result is the following theorem.

Theorem 1.2

An \(L^1({\mathbf {X}})\) function F belongs to \(H^1_{{\mathcal {L}}}\) if and only if \({\mathcal {R}}_jF\in L^1({\mathbf {X}})\) for \(j=1,2,\ldots ,2n\). Moreover, there is a constant \(C>0\) such that

$$\begin{aligned} C^{-1} \Vert F\Vert _{H^1_{{\mathcal {L}}}}\le \Vert F\Vert _{L^1({\mathbf {X}})}+\sum _{j=1}^{2n} \Vert {\mathcal {R}}_j F\Vert _{L^1({\mathbf {X}})}\le C \Vert F\Vert _{H^1_{{\mathcal {L}}}}. \end{aligned}$$
(1.6)

The theory of the classical real Hardy spaces on \({\mathbb {R}}^n\) has its origin in studying holomorphic function of one variable. The reader is referred to the very original works: Stein and Weiss [23], Burkholder et al. [2], Fefferman and Stein [11], and Coifman [6]. The spaces are natural extensions of \(L^p\) spaces and many operators occurring in harmonic analysis, like convolution singular integral operators, are bounded on them. The theory was then extended to the spaces of homogeneous type (see [8, 17, 25]). More information about the classical real \(H^p\) spaces with their characterizations and historical remarks can be also found in [22]. In [13] the authors provide a very general approach to the theory of \(H^1\) spaces for semigroups of linear operators satisfying Davies–Gaffney estimates and in particular Gaussian bounds. Let us point out, that in the context of semigroups, the classical Hardy spaces can be thought as those associated with the Laplace operator on \({\mathbb {R}}^n\).

In the monograph [12] Folland and Stein study \(H^p\) spaces on homogeneous nilpotent Lie groups proving equivalence of their definitions by means of maximal functions, square functions, and atoms. Important contributions to the theory of Hardy spaces on homogeneous groups are their characterizations by Riesz transforms proved in Christ and Geller [5]. To this end the authors of [5] extended Uchiyama’s theorem (see [26]) about Fefferman–Stein decomposition of BMO functions on \({\mathbb {R}}^n\) to homogeneous nilpotent Lie groups. Let us emphasise that our proof of the Riesz transforms characterization of Hardy space \(H^1_{{\mathcal {L}}}\) associated with the Grushin operator (see Theorem 1.2) takes an inspiration from [5]. We make use of the relation between \(\mathcal {L}\) and L via the already mentioned representation \(\pi \) and transfer the methods of [5] into the space of homogeneous type \({\mathbf {X}}\).

Let us also remark that our proof of Theorem 1.1 is based on Uchiyama results [25] about characterization of Hardy spaces on spaces of homogeneous type by maximal functions and atomic decompositions.

The Grushin operator \({\mathcal {L}}\) we consider here is a special example of operators of the form \({\mathbf {L}}=-\Delta _{x'}-|x'|^2\Delta _{x''}\), \((x',x'')\in {\mathbb {R}}^{n}\times {\mathbb {R}}^{m}\). It seems likely the methods we present here combined with [5] and relation of \({\mathbf {L}}\) with the Heisenberg–Reiter groups (see, e.g., [18]) will allow to develop the theory of Hardy spaces for \({\mathbf {L}}\).

1.5 Organization of the Paper

In Sect. 2 we describe relation of the Grushin operator and the sub-Laplacian on the Heisenberg group via a unitary representation and derive estimates on the heat kernel of \(\exp (-t{\mathcal {L}})\). Section 3 is devoted to proving Theorem 1.1. In Sect. 4 we study properties of kernels which are obtained as images by the representation of some singular integral kernels on \({\mathbb {H}}_n\). The crucial theorem about decompositions of compactly supported BMO functions by means of singular integrals is stated in Sect. 5 and its proof is completed in Appendixes 1 and 2. The proof of the Riesz transforms characterization of the Hardy space \(H^1_{{\mathcal {L}}}\) is presented in Sect. 6.

2 Relation with the Heisenberg Group

In this section we describe relation between the Grushin operator \({\mathcal {L}}\) and the sub-Laplacian L on the Heisenberg group \({\mathbb {H}}_n\). As we will see \({\mathcal {L}}\) occurs as an image of L in a special unitary representation \(\pi \) of \({\mathbb {H}}_n\) (see [15, 19]). We start this section by recalling basic facts from the analysis on the Heisenberg group.

2.1 Heisenberg Group

The Heisenberg group \({\mathbb {H}}_n\) is a Lie group with the underlying manifold \({\mathbb {R}}^{2n+1}={\mathbb {R}}^n\times {\mathbb {R}}^n\times {\mathbb {R}}\) and the group multiplication

$$\begin{aligned} (x,y,t)(u,v,s)=\left( x+u,y+v,t+s+\frac{1}{2}(y\cdot u-x\cdot v)\right) , \end{aligned}$$

where \(x\cdot y\) is the standard inner product in \({\mathbb {R}}^n\). We shall also denote the elements of the Heisneberg group by \(\varvec{x}=(x,y,t)\). Then \(\varvec{x}^{-1}=-\varvec{x}=(-x,-y,-t)\). The Lebesgue measure \(d\varvec{x}\) on \({\mathbb {R}}^{2n+1}\) turns out to be the bi-invariant Haar measure on \({\mathbb {H}}_n\). Clearly, \({\mathbb {H}}_n\) is a homogeneous nilpotent Lie group with dilations \(\delta _s(x,y,t)=(sx,sy,s^2t)\). We fix a homogeneous norm on \({\mathbb {H}}_n\) to be so called Koranyi norm given by

$$\begin{aligned} |\varvec{x}|=|(x,y,t)|=\big ( (|x|^2+|y|^2)^2+16t^2\big )^{1\slash 4}. \end{aligned}$$
(2.1)

The function \({\mathbb {H}}_n\ni \varvec{x} \mapsto |\varvec{x}|\in {\mathbb {R}}_+\cup \{0\}\) is smooth away from the origin, homogeneous of degree one, that is, \(|\delta _s \varvec{x}|=s|\varvec{x} |\), and symmetric \((|\varvec{x}|=|\varvec{-}\varvec{x}|)\). Moreover, \(|\varvec{x}\varvec{y}|\le |\varvec{x}|+|\varvec{y}|\). Clearly, \(|(x,y,t)|\sim |x|+|y|+ |t|^{1\slash 2}\). The homogeneous dimension of \({\mathbb {H}}_n\) is denoted by D and in our case \(D=2n+2\).

We choose the standard basis of the left-invariant vector fields:

$$\begin{aligned} X_j=\partial _{x_j}+\frac{1}{2}y_j\partial _t, \ \ X_{n+j}=\partial _{y_j}-\frac{1}{2}x_j\partial _t,\ \ j=1,2,\ldots ,n, \ \ X_{2n+1}=\partial _t, \end{aligned}$$

and the corresponding right-invariant vector fields:

$$\begin{aligned} Y_{j}=\partial _{x_j}-\frac{1}{2}y_{j}\partial _t, \ \ Y_{n+j}=\partial _{y_j}+\frac{1}{2}x_{j}\partial _t, \ \ j=1,2,\ldots ,n, \ \ Y_{2n+1}=\partial _t. \end{aligned}$$

Obviously, for \(j=1,2,\ldots ,n\) we have

$$\begin{aligned} X_j=Y_{j}+w_{j} Y_{2n+1}, \ \ X_{n+j}=Y_{n+j}+w_{n+j}Y_{2n+1}, \ \ X_{2n+1}=Y_{2n+1}, \end{aligned}$$
(2.2)

where \(w_j(\varvec{x})=w_j(x,y,t)=y_j\), \(w_{n+j}(\varvec{x})=w_{n+j}(x,y,t)=-x_j\).

We apply the usual notation for higher order derivatives (see [12]). If \(I=(i_1,i_2,\ldots ,i_{2n+1})\in ({\mathbb {N}}\cup \{0\})^{2n+1}\) is a multi-index, we set

$$\begin{aligned} X^I=X_1^{i_1}X_2^{i_2} \ldots X_{2n+1}^{i_{2n+1}}, \ \ \ Y^I=Y_1^{i_1}Y_2^{i_2} \ldots Y_{2n+1}^{i_{2n+1}}, \end{aligned}$$
$$\begin{aligned} d(I)=i_1+i_2+\cdots +i_{2n}+ 2i_{2n+1} \ \ \text {is the homogeneous degree of } \ I. \end{aligned}$$

Let \(L=-\sum _{k=1}^{2n} X_k^2\) denote the left-invariant sub-Laplacian on \({\mathbb {H}}_n\). It is well-known (see e.g., [27]) that the corresponding heat semigroup \(e^{-sL}\) is given by the convolution \(e^{-sL} f(x,y,t)=f*h_s(x,y,t)\) with a heat kernel \(h_s(x,y,t)=h_s(-x,-y,-t)\) which satisfies

$$\begin{aligned} h_s(x,y,t)=s^{-D\slash 2} h_1(\delta _{s^{-1\slash 2}}(x,y,t)), \end{aligned}$$
(2.3)
$$\begin{aligned} s^{-D\slash 2}e^{-C|(x,y,t)|^2\slash s}\lesssim h_s(x,y,t)\lesssim s^{-D\slash 2}e^{-c|(x,y,t)|^2\slash s}, \end{aligned}$$
(2.4)
$$\begin{aligned} |X^IY^Jh_s(x,y,t)|\le C_{I,J} s^{-(D+d(I)+d(J))\slash 2}e^{-c|(x,y,t)|^2\slash s}. \end{aligned}$$
(2.5)

2.2 Unitary Representation

We define the unitary representation of \({\mathbb {H}}_n\) on \(L^2({\mathbf {X}})\) by

$$\begin{aligned} \pi _{(x,y,t)}f({\mathbf {x}})=\pi _{(x,y,t)} f(x', x'')=f\left( x' +y, x''+t+\frac{1}{2}x\cdot y+x \cdot x'\right) \end{aligned}$$

(cf. Meyer [19]). It is easy to see that

$$\begin{aligned} \pi (X_j) f(x',x'')=\pi (Y_j) f(x',x'')=x'_j\partial _{x''} f(x',x''), \ \ j=1,2,\ldots ,n, \end{aligned}$$
$$\begin{aligned} \pi (X_{n+j})f(x',x'')=\pi (Y_{n+j})f(x',x'')=\partial _{x'_j}f(x',x''), \ \ j=1,2,\ldots ,n, \end{aligned}$$
$$\begin{aligned} \pi (X_{2n+1})f(x',x'')=\pi (Y_{2n+1})f(x',x'')=\partial _{x''}f(x',x''). \end{aligned}$$

Hence, \(\pi (L)={\mathcal {L}}\).

For a function \(F\in L^1({\mathbb {H}}_n)\) we set

$$\begin{aligned} \pi (F) f(x',x'')=\int _{{\mathbb {H}}_n} F(x,y,t)\pi _{(x,y,t)} f(x',x'')=\int _{{\mathbf {X}}} \pi (F)({\mathbf {x}},{\mathbf {y}}) f({\mathbf {y}})\, d{\mathbf {y}}, \end{aligned}$$

where

$$\begin{aligned} \pi (F)({\mathbf {x}},{\mathbf {y}})= \int _{{\mathbb {R}}^n} F\left( z,y'-x',y''-x'' -\frac{1}{2}z\cdot \big (y'+x'\big )\right) \, dz. \end{aligned}$$
(2.6)

Clearly, if \(F\in L^1({\mathbb {H}}_n)\), then

$$\begin{aligned} \int _{{\mathbf {X}}}\pi (F)({\mathbf {x}},{\mathbf {y}})\, d{\mathbf {x}}= \int _{{\mathbf {X}}}\pi (F)({\mathbf {x}},{\mathbf {y}})\, d{\mathbf {y}}=\int _{{\mathbb {H}}_n}F(x,y,t)\, dx\, dy\, dt, \end{aligned}$$
(2.7)
$$\begin{aligned} \pi (F_s)({\mathbf {x}},{\mathbf {y}})=s^{-(n+2)\slash 2} \pi (F)\big (\delta _{s^{-1\slash 2}}{\mathbf {x}},\delta _{s^{-1\slash 2}}{\mathbf {y}})\big ), \end{aligned}$$
(2.8)

where here and subsequently \(F_s(x,y,t)=s^{-D\slash 2}F(\delta _{s^{-1\slash 2}}(x,y,t))\).

Further, for suitable functions F on \({\mathbb {H}}_n\) one has

$$\begin{aligned} \pi (X_k)_{{\mathbf {x}}}\pi (F)({\mathbf {x}},{\mathbf {y}})=-\pi (Y_k F)({\mathbf {x}},{\mathbf {y}}), \ \ \ \pi (X_k)_{{\mathbf {y}}}\pi (F)({\mathbf {x}},{\mathbf {y}})=\pi (X_k F)({\mathbf {x}},{\mathbf {y}}). \end{aligned}$$
(2.9)

Lemma 2.1

There is a constant \(C_1>0\) such that if \(F\in L^1({\mathbb {H}}_n)\), \(\mathrm{supp}\, F\subset B_{{\mathbb {H}}_n}((0,0,0), R)\subset {\mathbb {H}}_n\), then

$$\begin{aligned} \pi (F)({\mathbf {x}},{\mathbf {y}})=0 \ \ \ \text {for} \ d({\mathbf {x}},{\mathbf {y}})>C_1R. \end{aligned}$$

Here \(B_{{\mathbb {H}}_n}((0,0,0), R)=\{(x,y,t)\in {\mathbb {H}}_n: |(x,y,t)|<R\}\) and |(x, y, t)| is the homogeneous norm in \({\mathbb {H}}_n\).

Proof

In the proof we will frequently use, without any comment, the formulas (2.1), (2.6), and (1.1). Assume that \(d({\mathbf {x}},{\mathbf {y}})>C_1R\) with \(C_1\) being large. If \(|x'-y'|>R\), then \(\pi (F)({\mathbf {x}},{\mathbf {y}})=0\). Thus for the remaining part of the proof we assume that \(|x'-y'|\le R\). We shall consider two cases.

Case 1: \(|x''-y''|^{1\slash 2}\le |x'|+|y'|\). Then \(|x''-y''|>CR(|x'|+|y'|)\), where C is large if \(C_1\) is chosen to be large. Consequently, \(|x''-y''|>(CR)^2\). Hence, for \(|z|<R\) we have

$$\begin{aligned} \big |x''-y''-\frac{1}{2}z\cdot (y'+x')\big |\ge & {} |x''-y''|-R(|y'|+|x'|)\\\ge & {} |x''-y''|\left( 1-\frac{1}{C}\right) \ge (CR)^2\left( 1-\frac{1}{C}\right) . \end{aligned}$$

Thus \(\pi (F)({\mathbf {x}},{\mathbf {y}})=0\) if \(C_1\) is large and, consequently, so is C.

Case 2: \(|x''-y''|^{1\slash 2}>|x'|+|y'|\). Then \(|x''-y''|^{1\slash 2}>CR\) and, again C is large if \(C_1\) is chosen to be large. For \(|z|<R\) we have \(|\frac{1}{2}z\cdot (y'+x')|<R(|x'|+|y'|)\le R|x''-y''|^{1\slash 2}\). Therefore,

$$\begin{aligned} \big |x''-y''-\frac{1}{2}z\cdot (y'+x')\big |\ge & {} |x''-y''|-R|x''-y''|^{1\slash 2}\nonumber \\ {}= & {} |x''-y''|^{1\slash 2}(|x''-y''|^{1\slash 2}-R)\nonumber \\\ge & {} CR(CR-R)=C(C-1)R^2, \end{aligned}$$
(2.10)

which implies \(\pi (F)({\mathbf {x}},{\mathbf {y}})=0\) if \(C_1\) is large enough. \(\square \)

Lemma 2.2

There is a constant \(C_2>0\) such that if F is a bounded function on \({\mathbb {H}}_n\), \(\mathrm{supp}\, F\subset B((0,0,0), R)\subset {\mathbb {H}}_n\), then

$$\begin{aligned} |\pi (F)({\mathbf {x}},{\mathbf {y}})|\le C_2R^D|B({\mathbf {x}}, R)|^{-1} \Vert F\Vert _{L^\infty ({\mathbb {H}}_n)}. \end{aligned}$$

Proof

It suffices to prove the lemma for F being the characteristic function of the ball \(B_{{\mathbb {H}}_n}((0,0,0),R)\) for every \(R>0\). Then, by (2.6),

$$\begin{aligned} |\pi (F)({\mathbf {x}},{\mathbf {y}})|\le & {} \int _{{\mathbb {R}}^n} \chi _{[-R,R]^n}(z) \chi _{[-R,R]^n}(y'-x') \chi _{[-R^2,R^2]}\nonumber \\&\times \, \left( y''-x''-\frac{1}{2}z\cdot (x'+y')\right) \, dz. \end{aligned}$$
(2.11)

Assume that \(\pi (F)({\mathbf {x}},{\mathbf {y}})>0\). We consider two cases.

Case 1: \(R>|x'|\slash C\), where \(C>0\) is a large constant. Then, by (1.3), \(|B({\mathbf {x}}, R)|\sim R^{n+2}\) and, consequently, \(|\pi (F)({\mathbf {x}},{\mathbf {y}}) |\le 2C R^n\sim R^D|B({\mathbf {x}}, R)|^{-1}\).

Case 2: \(R\le |x'|\slash C\). Notice that \(|x'+y'|\sim |x'|+|y'|\sim |x'|\), since \(|x'-y'|<R\) and \(C>0\) is large. Hence, by (2.11),

$$\begin{aligned} |\pi (F)({\mathbf {x}},{\mathbf {y}})|&\lesssim R^{n-1} \frac{R^{2}}{|x'+y'|}\sim \frac{R^{n+1}}{|x'|} \sim \frac{R^D}{|B({\mathbf {x}},R)|}. \end{aligned}$$

\(\square \)

2.3 Heat kernel for \({\mathcal {L}}\)

The kernels of the semigroups \(e^{-s{\mathcal {L}}}\) and \(e^{-sL}\) are related by

$$\begin{aligned} H_s({\mathbf {x}},{\mathbf {y}})= \pi (h_s) ({\mathbf {x}},{\mathbf {y}}). \end{aligned}$$
(2.12)

Let us also note that thanks to the homogeneity of \({\mathcal {L}}\) one has

$$\begin{aligned} H_s({\mathbf {x}},{\mathbf {y}})=s^{-(n+2)\slash 2} H_1\Big (\delta _{s^{-1\slash 2}}{{\mathbf {x}}},\delta _{s^{-1\slash 2}}{{\mathbf {y}}}\Big ). \end{aligned}$$

Proposition 2.3

(Gaussian bounds for \(H_s\)) There are constants \(c,C>0\) such that

$$\begin{aligned} H_s({\mathbf {x}},{\mathbf {y}})\le \frac{C}{|B({\mathbf {x}}, \sqrt{s})|}e^{-cd({\mathbf {x}},{\mathbf {y}})^2\slash s}. \end{aligned}$$
(2.13)

Proof

The proposition is well-known. For the convenience of the reader we present a short proof based on estimates of the heat kernel for the sub-Laplacian L on the Heisenberg group combined with Lemmas 2.1 and 2.2. To this end from (2.4) we have

$$\begin{aligned} 0\le h_s(\varvec{x})&\lesssim s^{-D\slash 2} \sum _{k=1}^\infty e^{-\alpha k^2} \chi _{B_{{\mathbb {H}}_n}(0, k)}(\delta _{s^{-1\slash 2} }\varvec{x})\nonumber \\&= s^{-D\slash 2} \sum _{k=1}^\infty e^{-\alpha k^2} \chi _{B_{{\mathbb {H}}_n}(0,\sqrt{s} k)}(\varvec{x}). \end{aligned}$$
(2.14)

Using (2.12), Lemmas 2.1 and 2.2, we obtain

$$\begin{aligned} 0\le H_s({\mathbf {x}},{\mathbf {y}})\lesssim s^{-D\slash 2} \sum _{k=1}^\infty (\sqrt{s}k)^{D}|B({\mathbf {x}}, \sqrt{s}k)|^{-1}e^{-\alpha k^2} \chi _{B({\mathbf {x}}, C_1\sqrt{s}k)}({\mathbf {y}}) \end{aligned}$$
(2.15)

Applying (1.4), we get

$$\begin{aligned} 0\le H_s({\mathbf {x}},{\mathbf {y}})&\lesssim \sum _{k=1}^\infty k^{D}\frac{B({\mathbf {x}},\sqrt{s})|}{|B({\mathbf {x}}, \sqrt{s}k)|} |B({\mathbf {x}}, \sqrt{s})|^{-1}e^{-\alpha k^2} \chi _{B({\mathbf {x}}, C_1\sqrt{s}k)}({\mathbf {y}})\nonumber \\&\lesssim \sum _{k=1}^\infty k^{D-n-1} |B({\mathbf {x}}, \sqrt{s})|^{-1}e^{-\alpha k^2} \chi _{B({\mathbf {x}}, C_1\sqrt{s}k)}({\mathbf {y}})\nonumber \\&\lesssim |B({\mathbf {x}}, \sqrt{s})|^{-1} e^{-c d({\mathbf {x}},{\mathbf {y}})^2\slash s}. \end{aligned}$$
(2.16)

\(\square \)

Lemma 2.4

There is a constant \(C>0\) such that

$$\begin{aligned} |H_s({\mathbf {x}},{\mathbf {y}})-H_s({\mathbf {z}},{\mathbf {y}})|\le \frac{C}{|B({\mathbf {y}}, \sqrt{s})|} \frac{d({\mathbf {x}},{\mathbf {z}})}{\sqrt{s}}. \end{aligned}$$
(2.17)

Proof

Fix \({\mathbf {y}}\in {\mathbf {X}}\) and \(s>0\) and set \(F({\mathbf {x}})=H_s({\mathbf {x}},{\mathbf {y}})\). Now, using (2.5) and the same arguments we have used in the proof of (2.13), we obtain

$$\begin{aligned} \sum _{j=1}^n|\partial _{x_j'}F({\mathbf {x}})|^2+|x_j'\partial _{x''}F({\mathbf {x}})|^2\le \frac{C}{s|B({\mathbf {y}},\sqrt{s})|^2}. \end{aligned}$$

Finally, by the definition of the distance d, we have

$$\begin{aligned} |F({\mathbf {x}})-F({\mathbf {z}})|\le \frac{Cd({\mathbf {x}},{\mathbf {z}})}{\sqrt{s}|B({\mathbf {y}},\sqrt{s})|}. \end{aligned}$$
(2.18)

\(\square \)

Corollary 2.5

(Hölder-type estimates for \(H_t\)) For \(0<\gamma <1\) there are constants \(C,c_0>0\) such that

$$\begin{aligned} |H_s({\mathbf {x}},{\mathbf {y}})-H_s({\mathbf {x}},{\mathbf {z}})|\le \frac{C}{|B({\mathbf {x}}, \sqrt{s})|} \left( \frac{d({\mathbf {y}},{\mathbf {z}})}{\sqrt{s}}\right) ^\gamma . \end{aligned}$$
(2.19)

with the improvement

$$\begin{aligned}&|H_s({\mathbf {x}},{\mathbf {y}})-H_s({\mathbf {x}},{\mathbf {z}})|\le \frac{C}{|B({\mathbf {x}}, \sqrt{s})|} \left( \frac{d({\mathbf {y}},{\mathbf {z}})}{\sqrt{s}}\right) ^\gamma e^{-c_0d({\mathbf {x}},{\mathbf {y}})^2\slash s}\nonumber \\&\quad \text {if} \ \ d({\mathbf {y}},{\mathbf {z}})\le d({\mathbf {x}},{\mathbf {y}})\slash 2. \end{aligned}$$
(2.20)

Lemma 2.6

(On diagonal lower bound of \(H_s\)) There is a constant \(C>0\) such that

$$\begin{aligned} H_s({\mathbf {x}},{\mathbf {x}})\ge C|B({\mathbf {x}},\sqrt{s})|^{-1} \ \ \text {for every } {\mathbf {x}}\in {\mathbf {X}}. \end{aligned}$$
(2.21)

Proof

By the homogeneity it suffices to prove the estimate for \(s=1\). To this end

$$\begin{aligned} H_1({\mathbf {x}},{\mathbf {x}}) \gtrsim \int _{{\mathbb {R}}^n} e^{-C|z|^2} e^{-C |z\cdot x'|}\, dz\gtrsim \frac{1}{1+|x'|}\sim |B({\mathbf {x}}, 1)|^{-1}. \end{aligned}$$
(2.22)

\(\square \)

3 Proof of Theorem 1.1

Proof

To prove the theorem we use Uchiyama’s results [25]. For this purpose we set

$$\begin{aligned} \rho ({\mathbf {x}},{\mathbf {y}})= \inf |B|, \end{aligned}$$

where the infimum is taken over all closed balls B containing \({\mathbf {x}}\) and \({\mathbf {y}}\). Then \( \rho \) is a quasi-distance such that \( \rho ({\mathbf {x}}, {\mathbf {y}})\asymp |B({\mathbf {x}},d({\mathbf {x}},{\mathbf {y}}))|\) for all \({\mathbf {x}}, {\mathbf {y}}\!\in \!{\mathbf {X}} \) and

$$\begin{aligned} \,|{B_\rho }({\mathbf {x}},r)|\asymp r \qquad \text {for every } \,{\mathbf {x}}\!\in \!{\mathbf {X}} \ \text { and }\, r\!>\!0, \end{aligned}$$
(3.1)

where \(\,{B_\rho }({\mathbf {x}},r)\) denotes the closed quasi-ball with center \( {\mathbf {x}} \) and radius r (see, e.g. [1, Lemma 6.4] for the proof).

Define the new kernel \(K_r(\mathbf {x},\mathbf {y})\) by

$$\begin{aligned} K_r(\mathbf {x},\mathbf {y}) ={H}_{t}(\mathbf {x},\mathbf {y})\,, \end{aligned}$$
(3.2)

where  \(r=|B(\mathbf {x},\sqrt{t})|\). The kernel \(K_r(\mathbf {x},\mathbf {y})\) satisfies the following assumptions of Uchiyama’s theorem, which are stated in conditions (3.3)–(3.5) below. \(\square \)

  • The on-diagonal lower estimate:

    $$\begin{aligned} K_r(\mathbf {x},\mathbf {x})\ge \tfrac{1}{Ar}. \end{aligned}$$
    (3.3)

  • Upper estimate: for every \(\delta \!>\!0\),

    $$\begin{aligned} K_r(\mathbf {x},\mathbf {y})\le \tfrac{A}{r}\,\bigl (1+\tfrac{\rho (\mathbf {x},\mathbf {y})}{r}\bigr )^{-1-\delta }. \end{aligned}$$
    (3.4)

  • Hölder estimate: there exist \(C_3\!>\!0\), \(\delta \!>\!0\), such that

    $$\begin{aligned} \bigl | K_r(\mathbf {x},\mathbf {y}) -K_r(\mathbf {x},\mathbf {z})\bigr |\le & {} \tfrac{A}{r}\bigl ( 1+\tfrac{\rho (\mathbf {x},\mathbf {y})}{r} \bigr )^{-1-2\delta } \bigl (\tfrac{\rho (\mathbf {y},\mathbf {z})}{r} \bigr )^{\delta } \text {if}\;\rho (\mathbf {y},\mathbf {z})\nonumber \\\le & {} C_3\max \,\{ r,\rho (\mathbf {x},\mathbf {y})\}. \end{aligned}$$
    (3.5)

The estimates (3.3)–(3.5) are consequences of (1.4), (2.21), (2.13), and Corollary 2.5 (see, e.g., [1, Appendix 3]).

Now we define the Hardy spaces \(H^1_{\text {max},K_r}\) as the set of all \(L^1({\mathbb {R}}^{n+1})\)-functions f such that \(\Vert f^{(+)}\Vert _{L^1({\mathbf {X}})}<\infty \), where \(f^{(+)}=\sup _{r>0} |\int K_r({\mathbf {x}},{\mathbf {y}}) f({\mathbf {y}})\, d{\mathbf {y}}|\).

The atomic Hardy space \(H^1_{\text {atom},\, \infty }({\mathbf {X}}, \rho )\) is defined in the standard way. A function a is called an atom for \(H^1_{\text {atom},\, \infty }({\mathbf {X}}, \rho )\), if there is a ball \(B_\rho ({\mathbf {x}}_0,r)\) such that \(\text {supp}\, a\subset B_\rho ({\mathbf {x}}_0, r)\), \(\Vert a\Vert _{L^\infty }\le |B_\rho ({\mathbf {x}}_0, r)|^{-1}\sim r^{-1}\), \(\int a=0\). Now a function f is an element of \(H^1_{\text {atom},\, \infty }({\mathbf {X}}, \rho )\) if \(f({\mathbf {x}})=\sum _{k} \lambda _k a_k({\mathbf {x}})\), where \(a_k({\mathbf {x}})\) are atoms for \(H^1_{\text {atom},\, \infty }({\mathbf {X}}, \rho )\) and \(\lambda _k\in {\mathbb {C}}\) with \(\sum _{k}|\lambda _k|<\infty \). For such f we set \(\Vert f\Vert _{H^1_{\text {atom},\, \infty }({\mathbf {X}}, \rho )}=\inf \sum _{k}|\lambda _k|\), where infimum is taken over all such representations.

We are now in a position to state the following theorem of Uchiyama about atomic and maximal characterizations of Hardy spaces on a space of homogeneous type.

Theorem 3.1

[25, Corollary 1’] Assume that \(\rho ({\mathbf {x}},{\mathbf {y}})\) and \(K_r({\mathbf {x}},{\mathbf {y}})\) satisfy (3.1) and (3.3)–(3.5). Then the spaces \(H^1_{\text {max},K_r}\) and \(H^1_{\text {atom},\, \infty }({\mathbf {X}}, \rho )\) coincide and the norms \(\Vert f^{(+)}\Vert _{L^1({\mathbf {X}})}\) and \(\Vert f\Vert _{H^1_{\text {atom},\, \infty }({\mathbf {X}}, \rho )}\) are equivalent.

It is easy to prove that there exists a constant \(c\ge 1\) such that if \(r= |B(\mathbf {x},\sqrt{t})|\), then

$$\begin{aligned} B(\mathbf {x},\sqrt{t}) \subset {B_\rho }(\mathbf {x},r) \subset B(\mathbf {x},c\sqrt{t}). \end{aligned}$$
(3.6)

The above inclusions imply that the atomic Hardy spaces for \(d({\mathbf {x}},{\mathbf {y}})\) and \(\rho ({\mathbf {x}},{\mathbf {y}})\) coincide. Moreover, the maximal functions for the kernels \(K_r\) and \(H_t\) are equal. Hence, Theorem 1.1 follows from Theorem 3.1.

4 Farther Properties of \(\pi \)

4.1 Homogeneous Kernels

A tempered distribution S on \({\mathbb {H}}_n\) is said to be a regular kernel of order \(r\in {\mathbb {R}}\) if S coincides with a \(C^\infty \) function \(m(\varvec{x})\) away from the origin and satisfies

$$\begin{aligned} \langle S, f\circ \delta _s\rangle = s^r\langle S,f\rangle \end{aligned}$$

for any \(f\in {\mathcal {S}}({\mathbb {H}})\). Any regular kernel of order r gives rise to the convolution operator

$$\begin{aligned} f\mapsto f*\check{S}(\varvec{x})=\langle S,f_{\varvec{x}}\rangle , \ \ f_{\varvec{x}}(\varvec{y})=f(\varvec{x}\varvec{y}), \end{aligned}$$

which will be denoted by the same symbol S.

Any tempered distribution S on \({\mathbb {H}}_n\) which is a regular kernel of order 0 is of the form

$$\begin{aligned} \langle S, f\rangle= & {} c_1 f(0) + \lim _{\varepsilon \rightarrow 0} \int _\varepsilon ^\infty \int _{\Sigma } m (\bar{\varvec{x}}) f(\delta _r \bar{ \varvec{x}})\, d \sigma (\bar{\varvec{x}}) \frac{dr}{r}\nonumber \\= & {} c_1f(0)+\lim _{\varepsilon \rightarrow 0}\int _{|\varvec{x}|>\varepsilon } m(\varvec{x})f(\varvec{x})\, d\varvec{x}, \end{aligned}$$
(4.1)

where m is a \(C^\infty \) function away from the origin, \(m(\delta _s \varvec{x})=s^{-D} m(\varvec{x})\), \(\int _{\Sigma } m(\bar{\varvec{x}}) \, d\sigma (\bar{\varvec{x}})=0\) (see [3, Lemma 2.4]). Here \(\Sigma =\{ \bar{\varvec{x}}\in {\mathbb {H}}_n: |\bar{\varvec{x}}|=1\}\) is the unit sphere in \({\mathbb {H}}_n\) and \(d\sigma (\bar{\varvec{x}})\) is the Radon measure on \(\Sigma \) such that \(\int _{{\mathbb {H}}_n} f(\varvec{x})d\varvec{x} = \int _0^\infty \int _{\Sigma } f(\delta _s \bar{\varvec{x}}) s^{D-1}d\sigma (\bar{\varvec{x}})\, ds\) (see [12, Proposition 1.5]).

Let \(\varphi \) be a \(C^\infty \) function on \({\mathbb {H}}_n\) such that \(0\le \varphi \le 1\), \(\varphi (\varvec{x})=\varphi (\varvec{y})\) whenever \(|\varvec{x}|=|\varvec{y}|\), \(\text {supp}\, \varphi \subset \{\varvec{x}\in {\mathbb {H}}_n: \frac{1}{2}\le |\varvec{x}|\le 2\}\), \(\sum _{j=-\infty }^\infty \varphi (\delta _{2^{j}}\varvec{x})=1\) for \(\varvec{x}\ne 0\). Then any regular kernel S of order 0 can be written as

$$\begin{aligned} \langle S,f\rangle =c_1f(0)+\sum _{j=-\infty }^\infty \int _{{\mathbb {H}}_n} m_j(\varvec{x})f(\varvec{x})\, d\varvec{x}, \end{aligned}$$

where \(m_j(\varvec{x})= \varphi (\delta _{2^j}\varvec{x}) m(\varvec{x})\). Clearly,

$$\begin{aligned}&|X^IY^Jm_j(\varvec{x})|\le C2^{j(D+d(I)+d(J))} \Vert X^IY^Jm_0\Vert _{L^\infty },\nonumber \\&\int _{{\mathbb {H}}_n} m_j(\varvec{x})\, d\varvec{x}=0, \ \ \int _{{\mathbb {H}}_n} X^Jm_j(\varvec{x})\, d\varvec{x}=0, \int _{{\mathbb {H}}_n} Y^Jm_j(\varvec{x})\, d\varvec{x}=0. \end{aligned}$$
(4.2)

Let \(m_j^*(\varvec{x})=\overline{m_j(\varvec{x}^{-1})}=\overline{m_j(-\varvec{x})}\), \(\check{m}_j(\varvec{x})=m_j(-\varvec{x})\). It is not difficult to check that there are constants \(C,c>0\) such that

$$\begin{aligned} \Vert (m_j^**m_k)\check{\ }\Vert _{L^1({\mathbb {H}}_n)}+ \Vert (m_j*m_k^*)\check{\ }\Vert _{L^1({\mathbb {H}}_n)}\le C2^{-c|j-k|}\Vert m\varphi \Vert _{C^1}. \end{aligned}$$

Thus, by the Cotlar–Stein lemma, \(Sf=f*\check{S}=c_1f+\sum _{j=-\infty }^\infty f*\check{m}_j\) defines a bounded operator on \(L^2({\mathbb {H}}_n)\) and, then on \(L^p({\mathbb {H}}_n)\), \(1<p<\infty \), since S is a Calderón–Zygmund operator on \({\mathbb {H}}_n\). Moreover, \(\Vert S f\Vert _{L^p({\mathbb {H}}_n)}\le (C_p \Vert m\varphi \Vert _{C^1}+c_1)\Vert f\Vert _{L^p({\mathbb {H}}_n)}\). The space of convolution operators with regular kernels of order 0 is an algebra with involution. Clearly,

$$\begin{aligned} \int _{{\mathbb {H}}_n} Sf\cdot \overline{g}=\int _{{\mathbb {H}}_n} f\cdot \overline{ S^*g}, \ \ \ \text {where}\ \langle S^*, f\rangle = \bar{c}_1f(0)+ \text {pv}\int _{{\mathbb {H}}_n} m^*(\varvec{x})f(\varvec{x})\, d\varvec{x}. \end{aligned}$$

Set \(\pi (S)f= c_1 f+\sum _{j=-\infty }^\infty \pi (m_j) f \). Then using the Coifman–Weiss transference principle [9] we have \(\Vert \pi (S)\Vert _{L^2({\mathbf {X}})\rightarrow L^2( {\mathbf {X}})}\le C\Vert S\Vert _{L^2({\mathbb {H}}_n)\rightarrow L^2({\mathbb {H}}_n)}\). Moreover, from (2.7), (2.9), (4.2), Lemmas 2.1 and 2.2 we conclude

$$\begin{aligned} \int _{{\mathbf {X}}} \pi (m_j)({\mathbf {x}},{\mathbf {y}})\, d{\mathbf {x}}= \int _{{\mathbf {X}}} \pi (m_j)({\mathbf {x}},{\mathbf {y}})\, d{\mathbf {y}}=0, \end{aligned}$$
(4.3)
$$\begin{aligned} \pi (m_j)({\mathbf {x}},{\mathbf {y}})=0 \ \ \text { for } d({\mathbf {x}},{\mathbf {y}})>2C_12^{-j}, \end{aligned}$$
(4.4)
$$\begin{aligned} |\pi (X^I)_{{\mathbf {x}}}\pi (X^J)_{{\mathbf {y}}} \pi (m_j)({\mathbf {x}},{\mathbf {y}})|\le C \Vert X^JY^I m_0\Vert _{L^\infty } 2^{(d(I)+d(J))j}|B({\mathbf {x}},2^{-j})|^{-1}, \end{aligned}$$
(4.5)
$$\begin{aligned} \pi (m_j^*)({\mathbf {x}},{\mathbf {y}})=\overline{\pi (m_j)({\mathbf {y}},{\mathbf {x}})}. \end{aligned}$$
(4.6)

Consequently, \(\pi (S)\) is a Calderón–Zygmund operator with the associated kernel

$$\begin{aligned} k({\mathbf {x}},{\mathbf {y}})=\sum _{j=-\infty }^\infty \pi (m_j)({\mathbf {x}},{\mathbf {y}}) \end{aligned}$$

which satisfies

$$\begin{aligned} \pi (S) f({\mathbf {x}})= \int k({\mathbf {x}},{\mathbf {y}}) f({\mathbf {y}})\, d{\mathbf {y}} \ \ \ \text {for } \ {\mathbf {x}}\notin \text {supp} f, \end{aligned}$$
(4.7)
$$\begin{aligned} |\pi (X^I)_{{\mathbf {x}}}\pi (X^J)_{{\mathbf {y}}} k({\mathbf {x}},{\mathbf {y}})|\le C_{I,J} \Vert X^JY^I m_0\Vert _{L^\infty } |B({\mathbf {x}},d({\mathbf {x}},{\mathbf {y}}))|^{-1} d({\mathbf {x}},{\mathbf {y}})^{-d(I)-d(J)},\nonumber \\ \end{aligned}$$
(4.8)
$$\begin{aligned} k(\delta _t {\mathbf {x}},\delta _t {\mathbf {y}})= t^{-(n+2)}k( {\mathbf {x}}, {\mathbf {y}}). \end{aligned}$$

Lemma 4.1

There is a constant \(C>0\) such that for any regular kernel S of order 0 on \({\mathbb {H}}_n\) and for every function \(f\in C^1_c(B({\mathbf {x}}_0, r_0))\) we have

$$\begin{aligned} |\pi (S)f({\mathbf {x}})|\le (2c_1+C\Vert m_0\Vert _{L^\infty }) r_0\sum _{k=1}^{2n} \Vert \pi (X_k)f\Vert _{L^\infty }. \end{aligned}$$
(4.9)

Proof

Note that (1.2) and (4.5) imply

$$\begin{aligned} \Vert f\Vert _{L^\infty }\le 2 r_0\sum _{k=1}^{2n}\Vert \pi (X_k)f\Vert _{L^\infty } \end{aligned}$$
(4.10)

and

$$\begin{aligned} |\pi (m_j)({\mathbf {x}},{\mathbf {y}})|\le C|B({\mathbf {x}}, 2^{-j})|^{-1} \Vert m_0\Vert _{L^\infty }. \end{aligned}$$
(4.11)

Let \(j_0\) be such that \(2^{-j_0}<r_0\le 2^{-j_0+1}\). Clearly, by (4.3) and (4.4), we get

$$\begin{aligned}&\Big | \sum _{j=-\infty }^\infty \int \pi (m_j)({\mathbf {x}},{\mathbf {y}})f({\mathbf {y}})\, d{\mathbf {y}} \Big |\\&\quad \le \sum _{j<j_0}\int _{B({\mathbf {x}}_0,r_0)\cap B({\mathbf {x}}, C_12^{-j+1})} |\pi (m_j)({\mathbf {x}},{\mathbf {y}})| |f({\mathbf {y}})|\, d{\mathbf {y}}\\&\qquad +\sum _{j \ge j_0}\int _{B({\mathbf {x}}, C_12^{-j+1})} |\pi (m_j)({\mathbf {x}},{\mathbf {y}})||f({\mathbf {y}})-f({\mathbf {x}})|\, d{\mathbf {y}}\\&\quad = J_1+J_2. \end{aligned}$$

Observe that if \(B({\mathbf {x}}_0,r_0)\cap B({\mathbf {x}}, C_12^{-j+1})\ne \emptyset \) with \(j<j_0\), then \(|B({\mathbf {x}}, 2^{-j})|\sim |B({\mathbf {x}}_0, 2^{-j})|\). Hence, applying (4.10), (4.11) and (1.4) we obtain

$$\begin{aligned} J_1\le & {} \sum _{j<j_0}\Vert m_0\Vert _{L^\infty } |B({\mathbf {x}}, 2^{-j})|^{-1} 2r_0 \left( \sum _{k=1}^{2n}\Vert \pi (X_k)f\Vert _{L^\infty }\right) |B({\mathbf {x}}_0, r_0)|\\\le & {} Cr_0\Vert m_0\Vert _{L^\infty } \left( \sum _{k=1}^{2n}\Vert \pi (X_k)f\Vert _{L^\infty } \right) \sum _{j<j_0}\left( \frac{r_0}{2^{-j}}\right) ^{n+1}\\\le & {} Cr_0\Vert m_0\Vert _{L^\infty } \sum _{k=1}^{2n}\Vert \pi (X_k)f\Vert _{L^\infty } . \end{aligned}$$

To estimate \(J_2\) we use (1.2) together with (4.11) and get

$$\begin{aligned} J_2\le & {} C \sum _{j\ge j_0 } \Vert m_0\Vert _{L^\infty } \left( \sum _{k=1}^{2n}\Vert \pi (X_k)f\Vert _{L^\infty }\right) 2^{-j} \\\le & {} Cr_0\Vert m_0\Vert _{L^\infty } \sum _{j=1}^{2n}\Vert \pi (X_j)f\Vert _{L^\infty } . \end{aligned}$$

\(\square \)

Thanks to Lemma 4.1 for \(f\in L^1({\mathbf {X}})\) and a regular kernel S of order 0 we define \(\pi (S)f\) in the sense of distribution setting

$$\begin{aligned} \langle \pi (S)f,\varphi \rangle = \langle f,\pi (\check{S})\varphi \rangle , \ \ \varphi \in C_c^\infty ({\mathbb {R}}^{n+1}). \end{aligned}$$

Lemma 4.2

There is a constant \(C>0\) such that for any regular kernel S of order 0 on \({\mathbb {H}}_n\) which has the form (4.1) we have

$$\begin{aligned} \Vert \pi (S)f\Vert _{H^1_{{\mathcal {L}}}}\le (c_1+C\Vert m_0\Vert _{C^1})\Vert f\Vert _{H^1_{{\mathcal {L}}}}. \end{aligned}$$
(4.12)

Proof

The proof is standard. For the sake of completeness we present its sketch. Without loss of generality we can assume that \(c_1=0\). Because \(\pi (S)\) maps continuously \(L^1({\mathbf {X}})\) to \({\mathcal {D}}'({\mathbb {R}}^{n+1})\), it suffices to prove that there is a constant \(C>0\) such that \(\Vert \pi (S)a\Vert _{H^1_{{\mathcal {L}}}}\le C\) for every atom \(a\in H^1_\mathrm{atom, \, \infty } ({\mathbf {X}})\). Fix a \((1,\infty )\)-atom a associated with a ball \(B({\mathbf {y}}_0, r_0)\). Since \(\sum _{j}\pi (m_j)a\) converges in \(L^2({\mathbf {X}})\), it converges in the \(L^1({\mathbf {X}})\) norm on \(B({\mathbf {y}}_0, 2r_0)\) as well. Note also that \(\pi (m_j)a({\mathbf {x}})=0\) for \(d({\mathbf {x}}, {\mathbf {y}}_0)>2C_1 2^{-j}+r_0\). If \(2r_0<d({\mathbf {x}},{\mathbf {y}}_0)\le 2C_1 2^{-j}+r_0\), then applying (1.2) and (4.5) we get

$$\begin{aligned} |\pi (m_j)a({\mathbf {x}})|= & {} \Big |\int _{B({\mathbf {y}}_0,r_0)} \Big ( \pi (m_j)({\mathbf {x}},{\mathbf {y}})- \pi (m_j)({\mathbf {x}},{\mathbf {y}}_0)\Big ) a({\mathbf {y}})\, d{\mathbf {y}}\Big |\nonumber \\\le & {} C \Vert m_0\Vert _{C^1} 2^jr_0|B({\mathbf {y}}_0, 2^{-j})|^{-1}. \end{aligned}$$
(4.13)

Hence, \(\sum _{j}\pi (m_j)a\) converges in \(L^1({\mathbf {X}})\) and \(L^2({\mathbf {X}})\) to \(\pi (S)a\) and \(\int \pi (S)a({\mathbf {x}})\, d{\mathbf {x}}=0\). Moreover,

$$\begin{aligned} | \pi (S)a({\mathbf {x}})|\le & {} C\Vert m_0\Vert _{C^1} \frac{r_0}{|B({\mathbf {y}}_0, d({\mathbf {x}},{\mathbf {y}}_0))|d({\mathbf {x}},{\mathbf {y}}_0)} \ \ \text { for } \ d({\mathbf {x}},{\mathbf {y}}_0)>2r_0,\\ \Vert \pi (S) a\Vert _{L^2({\mathbf {X}})}\le & {} C( \Vert m_0\Vert _{C^1}+c_1)|B({\mathbf {y}}_0, r_0)|^{-1\slash 2}. \end{aligned}$$

So, \(\pi (S)a\) can be written as \(\pi (S)a=\sum _{j} \lambda _j a_j \) with \(a_j\) being (1, 2)-atoms and \(\sum _j |\lambda _j |\le C\). \(\square \)

Let \(\psi \) be a Schwartz class function on \([0,\infty ) \) and \(dE_{L}\) and \(dE_{{\mathcal {L}}}\) be the spectral measures for L and \({\mathcal {L}}\) respectively. It is well known that the operator

$$\begin{aligned} \psi (L)f=\int _0^\infty \psi (\lambda ) \, dE_L(\lambda ) f \end{aligned}$$

is a convolution operator with a Schwartz class function on \({\mathbb {H}}_n\) denoted by the same symbol \(\psi (L)(\varvec{x})\) , that is, \(\psi (L)f(\varvec{x})=f*\psi (L)(\varvec{x})\) (see, e.g., [14]). Moreover, for every multi-index I and \(M\ge 0\) there is \(N>0\) and a constant \(C_{I,M,N}>0\) such that

$$\begin{aligned}&\sup _{\varvec{x}\in {\mathbb {H}}_n} (1+|\varvec{x}|)^M\big (|X^I\psi (L)(\varvec{x})|+|Y^I\psi (L)(\varvec{x})|\big )\nonumber \\&\quad \le C_{I,M,N}\sup _{\lambda >0} (1+\lambda )^{N} \left( \sum _{j=0}^N |\psi ^{(j)}(\lambda )|\right) . \end{aligned}$$
(4.14)

By homogeneity,

$$\begin{aligned} \psi (tL)(\varvec{x})=t^{-D\slash 2} \psi (L)(\delta _{t^{-1\slash 2}}\varvec{x}). \end{aligned}$$
(4.15)

Clearly, the operator

$$\begin{aligned} \psi ({\mathcal {L}}) f=\int _0^\infty \psi (\lambda )\, dE_{{\mathcal {L}}}(\lambda ) f \end{aligned}$$

is of the form

$$\begin{aligned} \psi ({\mathcal {L}})f({\mathbf {x}})=\int _{{\mathbf {X}}} \psi ({\mathcal {L}})({\mathbf {x}},{\mathbf {y}})f({\mathbf {y}})\, d{\mathbf {y}}, \end{aligned}$$

where

$$\begin{aligned} \psi ({\mathcal {L}})({\mathbf {x}},{\mathbf {y}})=\pi (\psi (L))({\mathbf {x}},{\mathbf {y}}). \end{aligned}$$
(4.16)

For detailed spectral properties of \({\mathcal {L}}\) we refer the reader to [19].

4.2 Riesz Transforms

The Riesz transforms \(R_j\), \(j=1,2,\ldots ,2n\), on the Heisenberg group \({\mathbb {H}}_n\) are defined by \(R_jf=X_jL^{-1\slash 2}f=\lim _{\varepsilon \rightarrow 0}c\int _{\varepsilon }^{\varepsilon ^{-1}} X_j e^{-tL}f \frac{dt}{\sqrt{t}}= \lim _{\varepsilon \rightarrow 0}f*(c\int _{\varepsilon }^{\varepsilon ^{-1}} X_j h_t \frac{dt}{\sqrt{t}}).\) By the Cotlar–Stein almost orthogonality principle the above limit defines a bounded operator on \(L^2({\mathbb {H}}_n)\). One can also prove that \(R_j\) are the principal valued convolution singular integral operators \(R_jf=f*\check{R}_j\), where \( R_j(\varvec{x})= -c\int _0^\infty Y_j h_t(\varvec{x})\frac{dt}{\sqrt{t}}\) are real-valued regular kernels of order 0.

Similarly the Riesz transforms \({\mathcal {R}}_j\) associated with the Grushin operator are defined by \({\mathcal {R}}_j=c\pi (X_j){\mathcal {L}}^{-1\slash 2}\), \(j=1,2,\ldots ,2n\). Clearly, \({\mathcal {R}}_j=\pi ( R_j)\). Thus \({\mathcal {R}}_j\) are Calderón–Zygmund operators on \({\mathbf {X}}\), which are bounded on \(L^p({\mathbf {X}})\), \(1<p<\infty \), and, by Lemma 4.2, bounded on \(H^1_{{\mathcal {L}}}\). For boundedness of \(\mathcal R_j\) on \(L^p(\mathbf X)\) see also [16].

5 \((\star )\) Property and Decomposition of \(BMO({\mathbf {X}})\)

Let \(\overrightarrow{S}=(S_1,S_2,\ldots ,S_d)\) be a system of regular kernels of order 0 on \({\mathbb {H}}_n\). We say that it fulfills condition \((\star )\) if for every unit vector \(\nu \in {\mathbb {R}}^d\) there are regular kernels \(T_j\) of order zero, \(\langle T_j,f\rangle =c_j f(0)+\mathrm{pv}\int _{{\mathbb {H}}_n} m^{\{j\}}(\varvec{x}) f(\varvec{x})\, d\varvec{x}\), \(j=1,2,\ldots ,d\), such that

$$\begin{aligned}({\star }) {\left\{ \begin{array}{ll} \sum _{j=1}^d S_j^*T_j=I,\\ \sum _{j=1}^d \nu _jT_j=0,\\ |c_j|\le C \ \ \mathrm{with } \ C\ \ \mathrm{independent\,\, of } \ \nu ,\\ |X^I m^{\{j\}}(\varvec{x})|+ |Y^I m^{\{j\}}(\varvec{x})| \le C_{I} \ \ \text {for}\ |\varvec{x}|=1, \ \mathrm{with } \ C_{I}\ \ \mathrm{independent\,\, of } \ \nu . \end{array}\right. } \end{aligned}$$

Theorem 5.1

(Christ and Geller [5, Sect. 6]) The system of the regular kernels of order zero \(\{\delta _0,R_1,R_2,\ldots ,R_{2n}\}\) on the Heisenberg group \({\mathbb {H}}_n\) fulfils condition \((\star )\).

A locally integrable function f on \({\mathbf {X}}\) is said to be an element of \(BMO({\mathbf {X}})\) if

$$\begin{aligned}\Vert f\Vert _{BMO({\mathbf {X}})} := \sup _{{\mathbf {y}}\in {\mathbf {X}}, \ r>0} \frac{1}{|B({\mathbf {y}},r)|} \int _{B({\mathbf {y}}, r)} |f({\mathbf {x}})-f_{B({\mathbf {x}},r)}|\, d{\mathbf {x}} <\infty , \end{aligned}$$

here \(f_{B({\mathbf {y}}, r)}=|B({\mathbf {y}},r)|^{-1}\int _{B({\mathbf {y}}, r)}f({\mathbf {x}})\, d{\mathbf {x}}\) denotes the mean value of f over \(B({\mathbf {y}},r)=\{ {\mathbf {x}}\in {\mathbf {X}}: d({\mathbf {y}}, {\mathbf {x}})<r\}\).

Our goal of this section is to prove the following theorem.

Theorem 5.2

Assume that \(S_j\), \(j=1,2,\ldots ,d\), is a system of operators satisfying \((\star )\). Then there is a constant \(C>0\) such that any compactly supported \(BMO({\mathbf {X}})\) function f can be written as

$$\begin{aligned} f=\sum _{j=1}^d \pi (S^*_j) \varvec{g}_j +\varvec{g}_0, \end{aligned}$$
(5.1)

with

$$\begin{aligned} \sum _{j=0}^d \Vert \varvec{g}_j\Vert _{\infty } \le C\Vert f\Vert _{BMO({\mathbf {X}})}. \end{aligned}$$
(5.2)

Moreover,

$$\begin{aligned} \sum _{j=0}^d \Vert \varvec{g}_j\Vert _{L^2({\mathbf {X}})}<\infty . \end{aligned}$$
(5.3)

For the proof of the theorem we follow methods presented in Christ and Geller [5] about decompositions of BMO functions on homogeneous Lie groups (see also the original Uchiyama’s proof [26] of constructive Fefferman–Stein decomposition of BMO functions on the Euclidean spaces).

There is no loss of generality to assume that \(\Vert f\Vert _{BMO({\mathbf {X}})}= \varepsilon \) with \(\varepsilon >0\) very small to be determined latter on.

Let us also emphasize that for any \(t>0\) the mapping \(f\mapsto f\circ \delta _t\) is an isometry on \(L^\infty \) and \(BMO({\mathbf {X}})\).

The main step of the proof of Theorem 5.2 is the following theorem.

Theorem 5.3

Assume that \(S_j\), \(j=1,2,\ldots ,d\), is a system of operators satisfying \((\star )\). Then there are constants constant \(C_8, C_9, \varepsilon _0 >0\) such that any \(BMO({\mathbf {X}})\) function f supported in any ball \(B({\mathbf {z}}_B, r)\) with \(\Vert f\Vert _{BMO({\mathbf {X}})}=\varepsilon <\varepsilon _0\), can be written as

$$\begin{aligned} f=\sum _{j=1}^d \pi (S^*_j) \widetilde{g}_j +\widetilde{g}_0+ f_1, \end{aligned}$$
(5.4)
$$\begin{aligned} \sum _{j=1}^d \Vert \widetilde{g}_j\Vert _{\infty } \le 3d, \ \ \Vert \widetilde{g}_0\Vert _{L^\infty }\le C_9\Vert f\Vert _{BMO({\mathbf {X}})}, \end{aligned}$$
(5.5)
$$\begin{aligned} \Vert f_1\Vert _{BMO({\mathbf {X}})}\le C_9 \varepsilon ^2, \ \ \mathrm{supp}\, f_1\subset B({\mathbf {z}}_B, C_8 r), \end{aligned}$$
(5.6)

Moreover,

$$\begin{aligned} \sum _{j=0}^d \Vert \widetilde{g}_j\Vert _{L^2({\mathbf {X}})}\le C_9| B({\mathbf {z}}_B,r)|^{1\slash 2} \Vert f\Vert _{BMO({\mathbf {X}})}. \end{aligned}$$
(5.7)

For the proof of Theorem 5.3 we adapt arguments of Christ and Geller [5]. For the convenience of the reader we present all the details in Appendixes 1 and 2.

Proof of Theorem 5.2

Fix \(0<\varepsilon <\varepsilon _0\) such that \(C_9\varepsilon <1\) and \(C_9C_8^{3\slash 2} \varepsilon <1\). Decompose f according to Theorem 5.3. If \(f_1=0\) we are done. Otherwise we apply Theorem 5.3 to the function \(\varepsilon \Vert f_1\Vert _{BMO({\mathbf {X}})}^{-1} f_1\) and obtain functions \(f_2\), \(\widetilde{g}_j^{\{1\}}\), \(j=0,1,\ldots ,d\), such that

$$\begin{aligned} f_1=\sum _{j=1}^d \pi (S^*_j) \widetilde{g}_j^{\{1\}} + \widetilde{g}_0^{\{1\}} + f_2, \end{aligned}$$
(5.8)
$$\begin{aligned} \sum _{j=1}^d \Vert \widetilde{g}_j^{\{1\}}\Vert _{L^\infty }\le & {} 3d\frac{\Vert f_1\Vert _{BMO({\mathbf {X}})}}{\varepsilon }\le 3dC_9\varepsilon , \nonumber \\ |\widetilde{g}_0^{\{1\}}\Vert _{L^\infty }\le & {} C_9\Vert f_1\Vert _{BMO({\mathbf {X}})}\le C_9^2\varepsilon ^2, \end{aligned}$$
(5.9)
$$\begin{aligned} \Vert f_2\Vert _{BMO({\mathbf {X}})}\le \frac{\Vert f_1\Vert _{BMO({\mathbf {X}})}}{\varepsilon } C_9\varepsilon ^2\le C_9^2\varepsilon ^3, \ \ \mathrm{supp}\, f_2\subset B({\mathbf {z}}_B , C_8^2 r), \end{aligned}$$
(5.10)
$$\begin{aligned} \sum _{j=0}^d \Vert \widetilde{g}_j^{\{1\}}\Vert _{L^2({\mathbf {X}})}\le & {} \frac{\Vert f_1\Vert _{BMO({\mathbf {X}})}}{\varepsilon } C_9| B({\mathbf {z}}_B,C_1r)|^{1\slash 2} \varepsilon \nonumber \\\le & {} C_9^2 |B({\mathbf {z}}_B, C_8r)|^{1\slash 2}\varepsilon ^2. \end{aligned}$$
(5.11)

Set \(\widetilde{g}_j^{\{0\}}=\widetilde{g}_j\), \(j=0,1,\ldots ,d\). Continuing this procedure we obtain sequences of functions \(\widetilde{g}^{\{n\}}_j\), \(j=0,1,\ldots ,d\), \(n=0,1,2,\ldots \), and \(f_n\) such that

$$\begin{aligned} f=\sum _{j=1}^d \pi ( S^*_j) \widetilde{g}_j^{\{0\}} +\widetilde{g}_0^{\{0\}}+ f_1, \end{aligned}$$
$$\begin{aligned} f_n=\sum _{j=1}^d \pi (S^*_j) g_j^{\{n\}} + g_0^{\{n\}} + f_{n+1}, \end{aligned}$$
$$\begin{aligned} \sum _{j=1}^d \Vert \widetilde{g}_j^{\{n\}}\Vert _{L^\infty } \le 3dC_9^n\varepsilon ^n, \ \ \sum _{j=0}^d \Vert \widetilde{g}_j^{\{n\}}\Vert _{L^2({\mathbf {X}})}\le C_9^{n+1} |B\left( {\mathbf {z}}_B, C_8^nr\right) |^{1\slash 2}\varepsilon ^{n+1}, \end{aligned}$$
$$\begin{aligned} \Vert f_{n+1}\Vert _{BMO({\mathbf {X}})} \le C_9^{n+1}\varepsilon ^{n+2}, \ \ \mathrm{supp}\, f_{n+1}\subset B\left( {\mathbf {z}}_B, C_8^{n+1}r\right) . \end{aligned}$$

Observe that

$$\begin{aligned} \Vert f_n\Vert _{L^2({\mathbf {X}})}\le & {} C' |B({\mathbf {z}}_B, C_1^n r)|^{1\slash 2} \Vert f_n\Vert _{BMO({\mathbf {X}})} \\\le & {} C'' C_8^{3n\slash 2} |B({\mathbf {z}}_B, r)|^{1\slash 2}C_2^n\varepsilon ^n\rightarrow 0, \ \ \ \text {as }\ n\rightarrow \infty , \end{aligned}$$
$$\begin{aligned} \sum _{n=0}^\infty \sum _{j=0}^d \Vert \widetilde{g}_j^{\{n\}}\Vert _{L^2({\mathbf {X}})} \le \sum _{n=0}^\infty C_9^{n+1} C_8^{3n\slash 2} |B({\mathbf {z}}_B, r)|^{1\slash 2} \varepsilon ^{n+1}. \end{aligned}$$

Putting \(\varvec{g}_j=\sum _{n=0}^\infty \widetilde{g}_j^{\{n\}}\) we obtain Theorem 5.2. \(\square \)

6 Proof of Theorem 1.2

Let \(VMO({\mathbf {X}})\) be the closure of the space of continuous functions with compact support in the BMO-norm. It is well-known (see [8, Theorem 4.1]) that \(VMO({\mathbf {X}})\) is a predual space to \(H^1_\mathrm{atom,\, \infty }({\mathbf {X}})\), that is, \(VMO({\mathbf {X}})^*=H^1_\mathrm{atom, \, \infty }({\mathbf {X}})\) in the sense that any functional \(\Phi \) on \(VMO({\mathbf {X}})\) is of the form \(\Phi (f)=\int f({\mathbf {x}})\overline{F({\mathbf {x}})}\, d{\mathbf {x}}\) for \(f\in C_c({\mathbf {X}})\), where \(F\in H^1_\mathrm{atom, \, \infty }({\mathbf {X}})\).

Assume firstly that \(F\in L^1({\mathbf {X}})\cap L^2({\mathbf {X}})\) and \({\mathcal {R}}_j F\in L^1({\mathbf {X}})\), \(j=1,2,\ldots ,2n\). If f is compactly supported continuous function on \({\mathbf {X}}\), then, according to Theorems 5.1 and 5.2, there are functions \(\varvec{g}_j \in L^\infty ({\mathbf {X}})\cap L^2({\mathbf {X}})\), \(j=0,1,2,\ldots ,2n\), such that \(\sum _{j=0}^{2n} \Vert \varvec{g}_j\Vert _\infty \le C \Vert f\Vert _{BMO({\mathbf {X}})}\) and \(f=\varvec{g}_0+\sum _{j=1}^{2n} {\mathcal {R}}_j^*\varvec{g}_j\). Hence,

$$\begin{aligned} \Big | \int _{{\mathbf {X}}} f({\mathbf {x}})\overline{ F({\mathbf {x}})}\, d{\mathbf {x}} \Big |= & {} \left| \int _{{\mathbf {X}}} \left( \varvec{g}_0+\sum _{j=1}^{2n} {\mathcal {R}}_j^*\varvec{g}_j\right) \overline{ F}\, d{\mathbf {x}}\right| \nonumber \\= & {} \left| \int \left( \varvec{g}_0({\mathbf {x}}) \overline{F({\mathbf {x}})}+\sum _{j=1}^{2n} \varvec{g}_j({\mathbf {x}})\overline{{\mathcal {R}}_j F({\mathbf {x}})}\right) \, d{\mathbf {x}}\right| \nonumber \\\le & {} \Vert f\Vert _{BMO({\mathbf {X}})}\left( \Vert F\Vert _{L^1({\mathbf {X}})} + \sum _{j=1}^{2n}\Vert {\mathcal {R}}_j F\Vert _{L^1({\mathbf {X}})}\right) . \end{aligned}$$
(6.1)

Thus, the integral \(f\mapsto \int f({\mathbf {x}})\overline{F({\mathbf {x}})}\, d{\mathbf {x}}\) has the unique extension to a bounded functional on \(VMO({\mathbf {X}})\) and, consequently, \(F\in H^1_{\mathrm{atom,\, \infty }}({\mathbf {X}})\) with

$$\begin{aligned} \Vert F\Vert _{H^1_\mathrm{atom,\, \infty }({\mathbf {X}})}\le C \left( \Vert F\Vert _{L^1({\mathbf {X}})}+\sum _{j=1}^{2n} \Vert {\mathcal {R}}_jF\Vert _{L^1({\mathbf {X}})}\right) . \end{aligned}$$

We now relax the assumption \(F\in L^2({\mathbf {X}})\) assuming only that \(F\in L^1({\mathbf {X}})\) with \({\mathcal {R}}_j F\in L^1({\mathbf {X}})\).

Lemma 6.1

Let S be a regular kernel of order zero on \({\mathbb {H}}_n\). Then there is a constant \(C>0\) such that for every \(t>0\)

$$\begin{aligned} \big \Vert \big [\pi (S),e^{-t{\mathcal {L}}}\big ]F\big \Vert _{L^1({\mathbf {X}})}\le C\Vert F\Vert _{L^1({\mathbf {X}})}, \end{aligned}$$
(6.2)

where \(\big [\pi ( S),e^{-t{\mathcal {L}}}\big ]= \pi ( S)e^{-t{\mathcal {L}}}- e^{-t{\mathcal {L}}}\pi ( S)\) is the commutator of \(\pi (S)\) and \(e^{-t{\mathcal {L}}}\).

We shall postpone the proof of the lemma to the end of the section.

Note that \(e^{-t{\mathcal {L}}} F\in L^2({\mathbf {X}})\) for \(F\in L^1({\mathbf {X}})\). Thus from Lemma 6.1 we conclude that

$$\begin{aligned} \Vert {\mathcal {R}}_j e^{-t{\mathcal {L}}}F\Vert _{L^1({\mathbf {X}})} \le C \Big (\Vert {\mathcal {R}}_j F\Vert _{L^1({\mathbf {X}})}+\Vert F\Vert _{L^1({\mathbf {X}})}\Big ) \end{aligned}$$
(6.3)

with a constant C independent of \(t>0\). The first part of the proof combined with (6.3) and Theorem 1.1 lead to

$$\begin{aligned} \Vert e^{-t{\mathcal {L}}} F\Vert _{H^1_{{\mathcal {L}}}}\le & {} C \left( \sum _{j=1}^{2n}\Vert {\mathcal {R}}_j e^{-t{\mathcal {L}}} F\Vert _{L^1({\mathbf {X}})}+\Vert e^{-t{\mathcal {L}}}F\Vert _{L^1({\mathbf {X}})}\right) \\\le & {} C \left( \sum _{j=1}^{2n}\Vert {\mathcal {R}}_j F\Vert _{L^1({\mathbf {X}})}+\Vert F\Vert _{L^1({\mathbf {X}})}\right) , \end{aligned}$$

because \(e^{-t{\mathcal {L}}}\) is uniformly bounded on \(L^1({\mathbf {X}})\). Since \({\mathcal {M}}_{{\mathcal {L}}} (e^{-t_1{\mathcal {L}}} F)({\mathbf {x}})\le {\mathcal {M}}_{{\mathcal {L}}} (e^{-t_2{\mathcal {L}}} F)({\mathbf {x}})\) for \(0<t_2<t_1\), we deduce from the Lebesgue monotone convergence theorem that \({\mathcal {M}}_{{\mathcal {L}}} F\in L^1({\mathbf {X}})\) and

$$\begin{aligned} \Vert {\mathcal {M}}_{{\mathcal {L}}} F\Vert _{ L^1({\mathbf {X}})}\le C \left( \sum _{j=1}^{2n}\Vert {\mathcal {R}}_j F\Vert +\Vert F\Vert _{L^1({\mathbf {X}})}\right) . \end{aligned}$$

This completes the proof of the first inequality of (1.6).

The proof of the second inequality in (1.6) is standard and follows from the fact that \({\mathcal {R}}_j\) are Calderón–Zygmund operators (see Sect. 4). We omit the details.

Proof of Lemma 6.1

By the homogeneity it suffices to prove the lemma for \(t=1\). Recall that

$$\begin{aligned} \int _{{\mathbf {X}}} H_t({\mathbf {x}},{\mathbf {z}})\, d{\mathbf {z}}=\int _{{\mathbf {X}}} H_t({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {z}}=1, \end{aligned}$$

where \(H_t({\mathbf {x}},{\mathbf {y}})\) denote the integral kernel for \(e^{-t{\mathcal {L}}}\). Let \(m_j\) be as in Sect. 4.1. Set

$$\begin{aligned} M_j=\int \Big |\int \big (H_1({\mathbf {x}},{\mathbf {z}})\pi (m_j)({\mathbf {z}},{\mathbf {y}})-\pi (m_j)({\mathbf {x}},{\mathbf {z}})H_1({\mathbf {z}},{\mathbf {y}})\big )\, d{\mathbf {z}}\Big |\, d{\mathbf {x}}. \end{aligned}$$

If \(j\le 0\), then

$$\begin{aligned} M_j\le & {} \int \Big |\int H_1({\mathbf {x}},{\mathbf {z}})\big (\pi (m_j)({\mathbf {z}},{\mathbf {y}})-\pi (m_j)({\mathbf {x}},{\mathbf {y}})\big ) d{\mathbf {z}}\Big |\, d{\mathbf {x}}\\&+\, \int \Big |\int \big (\pi (m_j)({\mathbf {x}},{\mathbf {y}})-\pi (m_j)({\mathbf {x}},{\mathbf {z}})\big ) H_1({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {z}}\Big |\, d{\mathbf {x}}\\= & {} J_1+J_2. \end{aligned}$$
$$\begin{aligned} J_1\le & {} \int \Big |\int _{d({\mathbf {z}},{\mathbf {y}})\le 8C_12^{-j}} H_1({\mathbf {x}},{\mathbf {z}})\big (\pi (m_j)({\mathbf {z}},{\mathbf {y}})-\pi (m_j)({\mathbf {x}},{\mathbf {y}})\big ) d{\mathbf {z}}\Big |\, d{\mathbf {x}} \\&+\, \int \Big |\int _{d({\mathbf {z}},{\mathbf {y}})> 8C_12^{-j}} H_1({\mathbf {x}},{\mathbf {z}})\big (\pi (m_j)({\mathbf {z}},{\mathbf {y}})-\pi (m_j)({\mathbf {x}},{\mathbf {y}})\big ) d{\mathbf {z}}\Big |\, d{\mathbf {x}}\\= & {} J_{11}+J_{12}. \end{aligned}$$

By (4.5), (1.2) and (2.13), we have

$$\begin{aligned} J_{11}\le C \int _{d({\mathbf {z}},{\mathbf {y}})\le 8C_12^{-j}} \int H_1({\mathbf {x}},{\mathbf {z}})\frac{2^{j} d({\mathbf {x}},{\mathbf {z}})}{|B({\mathbf {y}},2^{-j})|}\, d{\mathbf {x}} \, d{\mathbf {z}}\le \nonumber C2^j.\\ \end{aligned}$$
(6.4)

Using (4.4) and (2.13) and (4.5), we obtain

$$\begin{aligned} J_{12}= & {} \int \int _{d({\mathbf {z}},{\mathbf {y}})> 8C_12^{-j}} H_1({\mathbf {x}},{\mathbf {z}})|\pi (m_j)({\mathbf {x}},{\mathbf {y}})| d{\mathbf {z}}\, d{\mathbf {x}}\nonumber \\\le & {} \int \int _{d({\mathbf {x}},{\mathbf {z}})> C_12^{-j}} H_1({\mathbf {x}},{\mathbf {z}})|\pi (m_j)({\mathbf {x}},{\mathbf {y}})| d{\mathbf {z}}\, d{\mathbf {x}} \le C2^j. \end{aligned}$$
(6.5)

Similarly,

$$\begin{aligned} J_2\le & {} \int _{d({\mathbf {x}},{\mathbf {y}})\le 8C_12^{-j}}\Big | \int \big (\pi (m_j) ({\mathbf {x}},{\mathbf {y}})-\pi (m_j)({\mathbf {x}},{\mathbf {z}})\big ) H_1({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {z}}\Big |\, d{\mathbf {x}}\\&+\,\int _{d({\mathbf {x}},{\mathbf {y}}) > 8C_12^{-j}}\Big | \int \big (\pi (m_j) ({\mathbf {x}},{\mathbf {y}})-\pi (m_j)({\mathbf {x}},{\mathbf {z}})\big ) H_1({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {z}}\Big |\, d{\mathbf {x}}=J_{21}+J_{22}\\ \end{aligned}$$

Again, applying (1.2), (4.5), (2.13) and the doubling property of the measure, we get

$$\begin{aligned} J_{21}&\le C\int _{d({\mathbf {x}},{\mathbf {y}})<8C_12^{-j}} \int _{{\mathbf {X}}} \frac{d({\mathbf {y}},{\mathbf {z}}) 2^{j}}{|B({\mathbf {x}},2^{-j})|}H_1({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {z}}\, d{\mathbf {x}}\le C2^{j}. \end{aligned}$$
(6.6)

If \(d({\mathbf {x}},{\mathbf {y}}) > 8C_12^{-j}\) then \(\pi (m_j)({\mathbf {x}},{\mathbf {y}})=0\). Hence, thanks to (2.13) and (4.5), we have

$$\begin{aligned} J_{22}\le & {} \int _{d({\mathbf {x}},{\mathbf {y}}) > 8C_12^{-j}}\int _{d({\mathbf {x}},{\mathbf {z}})<2C_12^{-j}}|\pi (m_j)({\mathbf {x}},{\mathbf {z}})|H_1({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {z}}\, d{\mathbf {x}}\nonumber \\\le & {} \int _{d({\mathbf {z}},{\mathbf {y}}) > 2C_12^{-j}}\int _{{\mathbf {X}}}|\pi (m_j)({\mathbf {x}},{\mathbf {z}})|H_1({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {x}}\, d{\mathbf {z}} \le C2^{j}. \end{aligned}$$
(6.7)

If \(j>0\), then using (4.3) and Corollary 2.5 we arrive to

$$\begin{aligned} M_j\le & {} \int \Big | \int \big (H_1({\mathbf {x}},{\mathbf {z}})-H_1({\mathbf {x}},{\mathbf {y}})\big )\pi (m_j)({\mathbf {z}},{\mathbf {y}})\, d{\mathbf {z}}\Big |d{\mathbf {x}}\nonumber \\&+\,\int \Big |\int \pi (m_j)({\mathbf {x}},{\mathbf {z}})\big (H_1({\mathbf {z}},{\mathbf {y}})-H_1({\mathbf {x}},{\mathbf {y}})\big )d{\mathbf {z}}\Big | \,d{\mathbf {x}}\nonumber \\\le & {} C \int \Big |\int _{d({\mathbf {z}},{\mathbf {y}})<2C_12^{-j}} \frac{2^{-j\gamma }}{|B({\mathbf {x}},1)|}e^{-c_0d({\mathbf {x}},{\mathbf {y}})^2}|\pi (m_j)({\mathbf {z}},{\mathbf {y}})|\, d{\mathbf {z}}\,d{\mathbf {x}}\nonumber \\&+\, C \int \Big |\int _{d({\mathbf {x}},{\mathbf {z}})<2C_12^{-j}}|\pi (m_j)({\mathbf {x}},{\mathbf {z}})| \frac{2^{-j\gamma }}{|B({\mathbf {y}},1)|}e^{-c_0d({\mathbf {x}},{\mathbf {y}})^2}\, d{\mathbf {z}}\,d{\mathbf {x}}\nonumber \\\le & {} C2^{-j\gamma }. \end{aligned}$$
(6.8)

Finally, (6.4)–(6.8) imply \(\sum _{j=-\infty }^\infty M_j\le C<\infty \), which completes the proof of the Lemma. \(\square \)