On Hardy and BMO Spaces for Grushin Operator

We study Hardy and BMO spaces associated with the Grushin operator. We first prove atomic and maximal functions characterizations of the Hardy space. Further we establish a version of Fefferman–Stein decomposition of BMO functions associated with the Grushin operator and then obtain a Riesz transforms characterization of the Hardy space.


Grushin Operator
On R n+1 = R n × R we consider the Grushin operator The operator L is homogeneous of degree 2 with respect to the dilations . It is well known L is a hypoelliptic operator. It is related to the Heisenberg group H n . Actually the Grushin operator L is the image of a sub-Laplacian L associated to H n under a representation π acting on functions on R n+1 . In fact we make use of this relation to prove some crucial estimates on some kernels related to L. The control distance on R n+1 associated with L is defined by where D = {ψ ∈ W 1,∞ (R n+1 ) : n j=1 (|∂ x j ψ| 2 + |x j ∂ x ψ| 2 ) ≤ 1}. It is homogeneous, that is, d(δ s x, δ s y) = sd(x, y) and behaves like: d(x, y) ∼ |x − y | + |x −y | |x |+|y | if |x − y | 1/2 ≤ |x | + |y |, |x − y | 1/2 if |x − y | 1/2 > |x | + |y |, (1.1) see, e.g., [18,20] for details. Clearly, Let B(x, r ) = {y ∈ X : d(x, y) < r } denote the ball with center x and radius r > 0 in the metric d(x, y) and |B(x, r )| be its Lebesgue measure volume. Then |B(x, r )| ∼ r n+1 max{r, |x |} ∼ r n+1 (r + |x |) (1.3) and, consequently, , R ≥ r > 0. (1.4) The homogeneity of the distance d implies |B(δ s x, sr)| = s n+2 |B(x, r )|. (1.5) The space X = R n × R equipped with the Lebesgue measure dx and the distance d(x, y) is the space of homogeneous type in the sense of Coifman-Weiss [7]. It is well known (see e.g., [20]) that −L generates a semigroup of self-adjoint linear operators e −tL on L 2 (X) which has the form where the heat kernel H t (x, y) satisfies the Gaussian upper bound estimates (see (2.13)).

L
Let M L f (x) = sup t>0 |e −tL f (x)| be the maximal function associated with the semigroup e −tL . The upper Gaussian estimates (2.13) imply that M L is bounded on L p (X) for 1 < p ≤ ∞ and of weak-type (1,1). We define the Hardy space Now we define atoms associated to the homogeneous space X.  where the infimum is taken over all decompositions f = λ j a j , λ j ∈ C, a j are (1, q)-atoms for H 1 (X).

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

Riesz Transforms
The system of Riesz transforms R j , j = 1, 2, . . . , 2n, associated with L is defined by This formal definition has a precise meaning and the operators R j are Calderón-Zygmund operators on X. Moreover, R j are well-defined in the sense of distributions on L 1 (X) (see Sect. 4). Our second main result is the following theorem.

Theorem 1.2 An L 1 (X) function F belongs to H 1
L if and only if R j F ∈ L 1 (X) for j = 1, 2, . . . , 2n. Moreover, there is a constant C > 0 such that (1.6) The theory of the classical real Hardy spaces on 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 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 R n to homogeneous nilpotent Lie groups. Let us emphasise that our proof of the Riesz transforms characterization of Hardy space H 1 L associated with the Grushin operator (see Theorem 1.2) takes an inspiration from [5]. We make use of the relation between L and L via the already mentioned representation π and transfer the methods of [5] into the space of homogeneous type 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 L we consider here is a special example of operators of the form L = − x − |x | 2 x , (x , x ) ∈ R n × R m . It seems likely the methods we present here combined with [5] and relation of L with the Heisenberg-Reiter groups (see, e.g., [18]) will allow to develop the theory of Hardy spaces for L.

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(−tL). 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 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 L is presented in Sect. 6.

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

Heisenberg Group
The Heisenberg group H n is a Lie group with the underlying manifold R 2n+1 = R n × R n × R and the group multiplication where x · y is the standard inner product in R n . We shall also denote the elements of the Heisneberg group by x = (x, y, t). Then The Lebesgue measure d x on R 2n+1 turns out to be the bi-invariant Haar measure on H n . Clearly, H n is a homogeneous nilpotent Lie group with dilations δ s (x, y, t) = (sx, sy, s 2 t).
We fix a homogeneous norm on H n to be so called Koranyi norm given by The function H n x → |x| ∈ R + ∪{0} is smooth away from the origin, homogeneous of degree one, that is, |δ s x| = s|x|, and symmetric (|x| = | − x|). Moreover, |x y| ≤ |x| + | y|. Clearly, |(x, y, t)| ∼ |x| + |y| + |t| 1/2 . The homogeneous dimension of H n is denoted by D and in our case D = 2n + 2. We choose the standard basis of the left-invariant vector fields: and the corresponding right-invariant vector fields: Obviously, for j = 1, 2, . . . , n we have We apply the usual notation for higher order derivatives (see [12]).
Let L = − 2n k=1 X 2 k denote the left-invariant sub-Laplacian on H n . It is wellknown (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

Unitary Representation
We define the unitary representation of H n on L 2 (X) by [19]). It is easy to see that Hence, π(L) = L.
Proof In the proof we will frequently use, without any comment, the formulas (2.1), (2.6), and (1.1). Assume that d(x, y) > C 1 R with C 1 being large. If |x − y | > R, then π(F)(x, y) = 0. Thus for the remaining part of the proof we assume that |x − y | ≤ R. We shall consider two cases.
Thus π(F)(x, y) = 0 if C 1 is large and, consequently, so is C.

Heat kernel for L
The kernels of the semigroups e −sL and e −sL are related by Let us also note that thanks to the homogeneity of L one has

Proposition 2.3 (Gaussian bounds for H s )
There are constants c, C > 0 such that 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 (2.14) Using (2.12), Lemmas 2.1 and 2.2, we obtain Proof Fix y ∈ X and s > 0 and set F(x) = H s (x, y). Now, using (2.5) and the same arguments we have used in the proof of (2.13), we obtain Finally, by the definition of the distance d, we have Proof By the homogeneity it suffices to prove the estimate for s = 1. To this end

Proof of Theorem 1.1
Proof To prove the theorem we use Uchiyama's results [25]. For this purpose we set where the infimum is taken over all closed balls B containing x and y. Then ρ is a quasi-distance such that ρ(x, y) |B(x, d(x, y))| for all x, y ∈ X and |B ρ (x, r )| r for every x ∈ X and r > 0, (3.1) where B ρ (x, r ) denotes the closed quasi-ball with center x and radius r (see, e.g. [1, Lemma 6.4] for the proof). Define the new kernel K r (x, y) by The kernel K r (x, y) satisfies the following assumptions of Uchiyama's theorem, which are stated in conditions (3.3)-(3.5) below.
• The on-diagonal lower estimate: • Upper estimate: for every δ >0, • Hölder estimate: there exist C 3 > 0, δ >0, such that 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 max,K r as the set of all 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. It is easy to prove that there exists a constant c ≥ 1 such that if r = |B(x, √ t)|, then The above inclusions imply that the atomic Hardy spaces for d(x, y) and ρ(x, 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.

Homogeneous Kernels
A tempered distribution S on H n is said to be a regular kernel of order r ∈ R if S coincides with a C ∞ function m(x) away from the origin and satisfies for any f ∈ S(H). Any regular kernel of order r gives rise to the convolution operator which will be denoted by the same symbol S. Any tempered distribution S on H n which is a regular kernel of order 0 is of the form Then any regular kernel S of order 0 can be written as Thus, by the Cotlar-Stein lemma, The space of convolution operators with regular kernels of order 0 is an algebra with involution. Clearly, Consequently, π(S) is a Calderón-Zygmund operator with the associated kernel which satisfies There is a constant C > 0 such that for any regular kernel S of order 0 on H n and for every function Proof Note that (1.2) and (4.5) imply and Hence, applying (4.10), (4.11) and (1.4) we obtain To estimate J 2 we use (1.2) together with (4.11) and get Thanks to Lemma 4.1 for f ∈ L 1 (X) and a regular kernel S of order 0 we define π(S) f in the sense of distribution setting

Lemma 4.2
There is a constant C > 0 such that for any regular kernel S of order 0 on H n which has the form (4.1) we have 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 π(S) maps continuously Fix a (1, ∞)-atom a associated with a ball B(y 0 , r 0 ). Since j π(m j )a converges in L 2 (X), it converges in the L 1 (X) norm on B(y 0 , 2r 0 ) as well. Note also that π(m j )a(x) = 0 for d(x, y 0 ) > 2C 1 2 − j +r 0 . If 2r 0 < d(x, y 0 ) ≤ 2C 1 2 − j + r 0 , then applying (1.2) and (4.5) we get (4.13) Hence, j π(m j )a converges in L 1 (X) and L 2 (X) to π(S)a and π(S)a(x) dx = 0. Moreover, So, π(S)a can be written as π(S)a = j λ j a j with a j being (1, 2)-atoms and j |λ j | ≤ C. Let ψ be a Schwartz class function on [0, ∞) and d E L and d E L be the spectral measures for L and L respectively. It is well known that the operator is a convolution operator with a Schwartz class function on H n denoted by the same symbol ψ(L)(x) , that is, ψ(L) f (x) = f * ψ(L)(x) (see, e.g., [14]). Moreover, for every multi-index I and M ≥ 0 there is N > 0 and a constant C I,M,N > 0 such that (4.14) By homogeneity, Clearly, the operator where ψ(L)(x, y) = π(ψ(L))(x, y). (4.16) For detailed spectral properties of L we refer the reader to [19].

Riesz Transforms
The Riesz transforms R j , j = 1, 2, . . . , 2n, on the Heisenberg group H n are defined . By the Cotlar-Stein almost orthogonality principle the above limit defines a bounded operator on L 2 (H n ). One can also prove that R j are the principal valued convolution singular integral operators Similarly the Riesz transforms R j associated with the Grushin operator are defined by R j = cπ(X j )L −1/2 , j = 1, 2, . . . , 2n. Clearly, R j = π(R j ). Thus R j are Calderón-Zygmund operators on X, which are bounded on L p (X), 1 < p < ∞, and, by Lemma 4.2, bounded on H 1 L . For boundedness of R j on L p (X) see also [16].

( ) Property and Decomposition of B M O(X)
Let − → S = (S 1 , S 2 , . . . , S d ) be a system of regular kernels of order 0 on H n . We say that it fulfills condition ( ) if for every unit vector ν ∈ R d there are regular kernels T j of order zero, A locally integrable function f on X is said to be an element of Our goal of this section is to prove the following theorem. π(S * j )g j + g 0 , 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 f B M O(X) = ε with ε > 0 very small to be determined latter on.
Let us also emphasize that for any t > 0 the mapping f → f • δ t is an isometry on L ∞ and B M O(X).
The main step of the proof of Theorem 5.2 is the following theorem.
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.

Lemma 6.1 Let S be a regular kernel of order zero on H n . Then there is a constant
C > 0 such that for every t > 0 π(S), e −tL F L 1 (X) ≤ C F L 1 (X) , (6.2) where π(S), e −tL = π(S)e −tL − e −tL π(S) is the commutator of π(S) and e −tL .
We shall postpone the proof of the lemma to the end of the section.
Note that e −tL F ∈ L 2 (X) for F ∈ L 1 (X). Thus from Lemma 6.1 we conclude that with a constant C independent of t > 0. The first part of the proof combined with (6.3) and Theorem 1.1 lead to we deduce from the Lebesgue monotone convergence theorem that M L F ∈ L 1 (X) and 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 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.
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Appendix 1: Chang-Fefferman Decomposition
Our goal in this section is to prove a version of Chang-Fefferman decomposition of compactly supported B M O(X) functions. Then we shall establish some properties of the decomposition. We borrow main ideas from [5].

Chang-Fefferman Decomposition
Lemma 7.2 Suppose that N 1 ∈ N is given. Then every f ∈ B M O(X) with compact support can be decomposed to where C I and C are independent of f . Remark 7.3 Let us emphasize that the condition (7.7) replaces the condition a Q k α (x)W (x) dx = 0 for all polynomials W of homogeneous degree <2N 1 , (7.11) in the Chang-Fefferman decomposition for the classical B M O spaces and for the BMO spaces on homogeneous Lie groups. Actually (7.7) implies (7.11) by the integration by parts, since L N 1 W (x) = 0 for every polynomial W of homogeneous degree < 2N 1 .
Proof of Lemma 7.2 For fixed N 1 let φ, ψ, η ∈ S(R) be real valued functions such that Here φ and η denote the Fourier transforms of φ and η respectively. Then there are Schwartz class functions φ( t x). The same holds for η(t √ L)(x). Moreover, it follows from (2.13), (7.13), and the finite propagation of the fundamental solution of the wave equation that the functions φ(t √ L)(x) and η(t √ L)(x) are compactly supported, that is, there is a constant C > 0 such that see [10,21] for details. Consequently, by Lemma 2.1, Additionally, with the same estimates on π( Thus (7.6) and (7.7) hold. Now (7.8) follows from (7.4) and (7.15). Observe that |B(y, t)| ∼ |Q k α | for (y, t) ∈ Q k α × (δ k , δ k−1 ). Thus from (7.15) and (7.16) we conclude |π(X I ) a Q k α (x)| which gives (7.9). Finally, for fixed Q ∈ B, So to finish the proof of (7.10) it suffices to note that for any ball B ⊂ X. This fact has a standard proof, which for the reader convenience we present here. First from (7.12) and (2.7) one gets Fix a ball B and decompose Further, thanks to (7.15), φ(t √ L) f 2 (x) = 0 for x ∈ B and 0 < t ≤ diam B. Thus (7.23) is proved.
Let us remark that for λ Q defined in (7.18) we have Lemma 7.4 Let N ∈ Z + be given. Then there exist constants N 1 ∈ Z + and C N > 0 such that for any regular kernel S of order 0 of the form (4.1) and any a Q satisfying (7.7)-(7.9) we have Proof Thanks to (7.7)-(7.9) without loss of generality we may assume that c 1 = 0. In order to prove (7.27) it suffices, by (7.7)-(7.9), (4.5) and Lemma 4.1, to consider d(x, z Q ) > 16C 4 a 0 (Q). Then, integration by parts leads to where in the last inequality we have used (1.4). The proof of (7.27) is complete.
Assume first that d(x, z Q ) > 16C 4 a 0 (Q). Let g ∈ C ∞ c (X) be such that Then, integrating by parts and using (1.2) together with (4.8) and (7.9), we obtain where in the last inequality we have applied (1.4).

Auxiliary Functions
Let N 0 be a large integer. For a compactly supported f ∈ B M O(X) define where λ Q are scalars from the Chang-Fefferman decomposition (see Lemma 7.2). Fix 0 < κ < 1 − δ, where δ is from Theorem 7.1 and set Then, Easily, if N 0 is sufficiently large, then Lemma 7.6 (Christ and Geller [5]) If N 0 is sufficiently large, then for every compacty supported B M O(X)-function f one has Proof The proof is same as of [5,Lemma 3.3].

Lemma 7.7 There is a constant C such that if f = λ Q a Q is the Chang-Fefferman decomposition of a B M O(X)-function f given by the proof of Lemma 7.2 such that
Proof There is a constant M 1 such that for k ≤ 0 the number of Q ∈ B k such that λ Q = 0 is bounded by M 1 with M 1 independent of x 0 . For such Q, |Q| ∼ |B(x 0 , δ k )|. So, by (7.18) and (7.16), which implies the lemma.

Appendix 2: Proof of Theorem 5.3
Proof of Theorem 5.3 Using dilations we may assume without loss of generality that f is supported by B(z B , 1). By the Chang-Fefferman decomposition given in the proof of Lemma 7.2 we have It follows from Lemma 7.7 that Thus, in farther consideration we shall deal with the function f 0 = (Q)≤1 λ Q a Q with λ Q , a Q satisfying (7.7)-(7.10). Remark that there is a constant C 10 independent of z B such that that if λ Q = 0 in the decomposition of f 0 , then Following [5] our task is to construct, by induction, for each integer l ≥ −1 functions − → h l , − → g l and − → E l on X taking values in C d such that The proofs of the above will be based on the following (simultaneously established) properties of − → h l , − → g l , and − → E l : for all x, y ∈ X, (8.11) where in the first inequality we have used (8.10) and (8.14). Recall that | − → g l−1 (x)| ≡ 1.
where in the last inequality for the first summand we have used (8.23) while for the second one we have applied (8.25). Now from Lemma 7.6 we obtain (8.16). Thus the construction of the functions − → h l , − → g l and − → E l satisfying (8.3)-(8.5) and (8.9)-(8.16) is complete.
It remains to prove (8.6)-(8.8). First, (7.29) combined with (8.11) and (8.13) imply Now for nonnegative integers s 1 ≤ s 2 let Q = Q s 1 ,s 2 = s 2 l=s 1 B l . In virtue of (8.26), Applying two times the Cauchy-Schwarz inequality we obtain Observe that for fixed integer k ≥ 0 and fixed Q ∈ Q, It follows from (1.4) and Theorem 7.1 that there is a constant C > 0 such that for every j, k ≥ 0 and every P ∈ Q the number of Q ∈ Q such that (P) = δ k (Q), d(z Q , z P ) ≤ 2 j (Q) is bounded by C2 3 j . Therefore, |φ(t √ L) f (y)| 2 dy dt t . (8.27) where in the last inequality we have used (7.18). From the spectral theorem we easily conclude that Let k be such that δ k = (Q). From (8.15) and Lemma 7.6 we obtain On the other hand we conclude from (8.16) that for x, y ∈Q we have the following estimate on the finite sum: