Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

In the Heisenberg group of dimension 2n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2n+1$$\end{document}, we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.

development of Harmonic Analysis, and has a very wide range of applications. This classical Calderón-Zygmund convolution operator is an isometry on L 2 (R n ), as seen by means of the Fourier transform. As is well known, it is bounded on L p (R n ) for 1 < p < +∞ and of weak type (1,1). There are many other aspects of the Riesz transform, such as endpoint boundedness H 1 −→ L 1 and L ∞ −→ B M O, dimension-free L pnorm bounds, weighted inequalities as well as the Riesz transforms associated to Schrödinger operators, etc. The literature on these topics is huge. In this paper, we restrict ourselves to the L p boundedness and the weak type (1, 1) (endpoint) estimate, in the Heisenberg setting defined below.
We now consider a smooth manifold M endowed with a measure dμ, a secondorder differential operator and a first-order operator ∇ such that the Green's formula holds; our Heisenberg setting is a special case of this. Then the (first-order) Riesz transform is defined as ∇(− ) −1/2 , with a suitable modification in the case where 0 is an eigenvalue of in L 2 (μ). Note that this operator is always bounded on L 2 (μ). Without the Fourier transform, the usual tools to study its L p boundedness are instead the Poisson semi-group, probabilistic Littlewood-Paley theory, the resolvent and in particular the heat semi-group and its kernel.
For the case 2 < p < +∞, the classical result is not always valid in this general setting. More precisely, for any fixed p 0 ≥ 2, there exists a Riemannian manifold of this type where the Riesz transform is bounded on L p for 2 < p < p 0 but unbounded for p > p 0 ; if p 0 > 2 it is unbounded also on L p 0 and not even of weak type ( p 0 , p 0 ). See [18,31] and [14][15][16][17][25][26][27] as well as [28] for some concrete examples. We also mention [8,9,13,20,21] and references therein, with characterizations under the assumptions of volume doubling and Gaussian heat kernel estimates.
For the opposite case 1 < p < 2, there are many known results but, to the best of our knowledge, no counterexample. In particular, an interesting dichotomy has been obtained in [4]: for any fixed p 0 ∈ (1, 2) and fixed n ≥ 1, either there is an L p 0 bound on the Riesz transform which is uniform over all complete Riemannian manifolds of dimension n, or there exists a complete n-dimensional Riemannian manifold in which the Riesz transform is unbounded on L p 0 .
Next, we focus on the weak type (1, 1) property for the Riesz transform. There exist numerous known specific results and several general sufficient conditions. We distinguish the following cases.
1. The standard assumption of Harmonic Analysis, namely the volume doubling property, holds. Then the most striking result is due to T. Coulhon and X. T. Duong. They showed in [18] that Gaussian estimates for the heat kernel imply the weak type (1, 1), without assuming Gaussian estimates for its gradient. Such gradient estimates are necessary in earlier works that use standard singular integral theory, but in general they do not hold in the setting of a complete Riemannian manifold. Furthermore, the method of Coulhon and Duong can be adapted to settings with sub-Gaussian heat kernel estimates and to even more general situations. See [17] and [37] for details. 2. The volume is not doubling but of at most polynomial growth in the sense of Nazarov-Treil-Volberg (cf. for example [47]). The best-known example seems to be the classical Ornstein-Uhlenbeck operator, studied by P.A. Meyer, R. Gundy, G. Pisier and many other authors. In particular, this example satisfies Bakry and Émery's condition 2 ≥ 0 (cf. [10]). The weak type (1, 1) estimate for the Riesz transform associated to the Ornstein-Uhlenbeck operator can be found in [22]; see also [48] for higher order Riesz transforms and other related operators. Recently, A. Hassell and A. Sikora provided another interesting example, the connected sum of a finite number of Riemannian manifolds with very strong geometric and analytic conditions. For this setting, they proved the weak type (1, 1) of the Riesz transform in [28,Theorem 7.1]. Their proof is based on spectral multipliers and the resolvent. One may naturally ask whether their result can be proved by means of the singular integral theory developed in [47]. 3. Manifolds of exponential volume growth without spectral gap. The typical case is the affine group, for which a partial result can be found in [50] and a suitable singular integral operator theory has been established in [29]. Moreover, this theory can be adapted to some other situations. 4. Manifolds of exponential volume growth with spectral gap. Under some additional assumptions such as the local volume doubling property and small-time Gaussian heat kernel estimates, the L p boundedness of the Riesz transform for 1 < p < 2 can be found in [18]. As for the weak type (1, 1) estimate, there is at present no adequate singular integral theory, as far as the authors know. However, in the setting of a symmetric space of the non-compact type, Anker obtained in [2] the weak type (1, 1) of the first-and second-order Riesz transforms. See also [6] for other examples. Recently, the present authors studied Riesz transforms associated to the Laplacian with drift in [34,36] and [35]. Also notice that in the papers [36] and [35], which treat the Laplacian with drift in Euclidean space, the setting can be seen as the direct product of a Euclidean space and the simplest weighted manifold on the real line satisfying exponential volume growth with spectral gap. Let us finally observe that the setting of the present paper is natural in this context, since it consists of a sub-Laplacian with drift in a typical sub-Riemannian manifold, the Heisenberg group. We refer the reader to [46] and [1] and references therein for further details about sub-Riemannian manifold.

Description of the Setting and Statements of Results
Let H n = R n × R n × R be the Heisenberg group of 2n + 1 real dimensions, with points written g = (x, y, t). The group law is (2.1) and the Haar measure dg on H n coincides with the (2n + 1)-dimensional Lebesgue measure. The vector fields are left invariant on H n and generate its Lie algebra. The associated sub-Laplacian is The drift is defined by a nonzero vector v = (a, b) ∈ R n ×R n , and the sub-Laplacian with this drift is Consider the homomorphism from H n to the multiplicative group R + It is easy to verify that v and μ v satisfy the Green's formula provided that f and w are smooth in H n and that f or w has compact support. Thus v is symmetric and has a negative-definite, self-adjoint extension in L 2 (dμ v ). Its spectral gap is positive and equals |v| 2 , since The heat semigroup (e h v ) h>0 generated by v is a diffusion semigroup to which the general Littlewood-Paley-Stein theory applies; see [51,Chap. III].
The Riesz transform R k = ∇ k (− v ) −k/2 of any order k ∈ {1, 2, . . .} can be expressed in terms of the heat semigroup; indeed This operator is bounded on L p (dμ v ) for 1 < p < +∞ and any k, as verified in Lohoué and Mustapha [44, Théorème 2(ii)]; see also the remarks about the Heisenberg group in Sect. 4 of the same paper. Observe further that the Green's formula implies that the first-order Riesz transform ∇(− v ) −1/2 is an isometry on L 2 (dμ v ).
Our results deal with the weak type (1, 1) of Riesz transforms and some other operators.

Theorem 1
The first-order Riesz transform R 1 is of weak type (1, 1) with respect to dμ v .
We do not know whether the same holds for the second-order transform. But we have the following negative result.
The maximal Littlewood-Paley-Stein operators related to v are given by In particular, H 0 is the maximal operator of the semigroup, and Littlewood-Paley-Stein theory implies that H 0 is bounded on L p (dμ v ), 1 < p < +∞. Actually, H k is bounded on these L p spaces for all k = 0, 1, . . . ; see for instance Lohoué's paper [40]. In Sect. 4.3, we give another proof of this result. As for weak type (1, 1), we prove the following result.
Theorem 3 For k = 0 and k = 1, the operator H k is of weak type (1, 1) with respect to dμ v , but not for k ≥ 2.
In this paper, we do not include the horizontal Littlewood-Paley-Stein functions, obtained by replacing ∇ k by differentiations with respect to t. However, we believe that they can be treated with similar methods, combined with arguments from [34] and [35].
We also mention the (first) Littlewood-Paley-Stein operator The operator H 1 is bounded on L p (dμ v ) for 1 < p < +∞. In the case 1 < p ≤ 2, this follows from results obtained in the setting of a manifold by Coulhon-Duong-Li [19, Theorems 1.2 or 1.3]; their arguments hold also in our case. Moreover, the boundedness for 2 < p < +∞ can be seen by adapting the techniques used in [40] or [9] to our setting. The weak type (1, 1) of H 1 can be proved essentially by the method used for our Theorem 1.
The proof of our Theorem 1 and that of the result for H 1 in Theorem 3 follow the same lines. The kernels of these operators are computed and estimated. In both cases, the local part of the operator is relatively simple to deal with. After several reductions, the arguments for the global parts boil down to an estimate for a maximal operator defined by taking convolutions with the characteristic functions of certain rectangles. This is Proposition 7, which is the fundamental point of our arguments.
The plan of this paper is as follows. After some preliminaries in Sect. 3, we prove the positive part of Theorem 3 in Sect. 4. The long arguments in Sect. 4.1 include Proposition 7, mentioned above and also used to prove Theorem 1 in Sect. 5. Finally, Sect. 6 contains the counterexamples needed for Theorem 2 and the negative part of Theorem 3.

Notation
We will often use complex notation for H n = C n × R. Setting z j = x j + iy j and z = (z 1 , . . . , z n ), we write points of H n as g = (z, t) instead of (x, y, t) whenever convenient. Further, we let |z| = n 1 |z j | 2 1/2 . From now on, we assume that |v| = 1 and that b = 0 in the expression for v, so that v = (a, 0) with a = (a 1 , . . . , a n ) ∈ S n−1 . This means no loss of generality, as seen via a dilation and an orthogonal transformation.
We denote by c > 0 and C < ∞ many different constants which only depend on n and the quantities k and p appearing in the statements of the theorems. By A B and A B we mean A ≤ C B and A ≥ cB, respectively, for positive quantities A and B. When both these inequalities hold, we write A ∼ B.

The Carnot-Carathéodory Distance
The Carnot-Carathéodory distance on H n will be denoted by d(., .). We write d(g) = d(g, o) where g ∈ H n and o = (0, 0, 0) is the origin of H n , and observe that d(g , g) = d(g −1 g ). Moreover, B(g, r ) denotes for g ∈ H n and r > 0 the ball {g ∈ H n ; d(g , g) < r }.

Lemma 4 For all points
This allows us to compute the μ v -measure of a ball. Observe that Proof The case r < 1 is clear, since the density of the measure is of constant order of magnitude in B(o, 1). So we assume r ≥ 1. By m(h), 0 < h < 2r , we denote the 2n−dimensional Lebesgue measure of the set Then From (3.5) and (3.8) we get which, considering the case where |x ⊥ | + |y| + √ |t| ≥ r , leads to This implies m(h) r 2n+1 .
Assume now that h < r /2, in order to get a better estimate of m(h). Then |a · x| = |r − h| ≥ r /2, and (3.8), (3.5) and (3.9) imply From this we obtain Inserting now our two estimates for m(h) in the integral in (3.7), we will get (3.11) This is the upper estimate for r ≥ 1 in the lemma. To get also the lower estimate, we let 1/4 < h < 1/2 and take |x ⊥ | < c √ r , |y| < c √ r and |t| < c r 3/2 . If the positive constant c here is small enough, Q(g) will be much smaller than h, and (3.4) will imply that the point g is in E h . It follows that the estimate (3.10) is sharp for these h. Now (3.7) gives the desired lower estimate, and the lemma is proved.

Semigroup Kernels
The heat semigroup (e h ) h>0 generated by the sub-Laplacian has a convolution kernel p h , in the sense that for suitable functions f . It is well known that p h has the form (cf. [24,30] or [45]) (3.12) We note the homogeneity property of p h The following sharp global estimate for p h , proved in [32, Théorème 1], will play an important role: for all h > 0 and (z, t) ∈ H n . We will also need sharp upper estimates for horizontal derivatives of p h , see [32, for k = 1, 2, . . . . Next, we consider the sub-Laplacian with drift. The corresponding semigroup for suitable functions f . It is explicitly given by (cf. [3, p. 4])

Proof of Theorem 3
The proof of the negative result for weak type (1, 1) and k ≥ 3 is deferred to Sect. 6.

Weak Type (1, 1) of H 1
and we can assume that these functions are nonnegative.
To write H 1 φ in terms of a convolution involving f , we first see from (3.16 Observe that the factor e a·(x−x ) is a function of (g ) −1 g. We now use (3.14) and (3.15) to estimate p h and |∇ p h | here. If we define K h (g) by the result will be We begin with the local part of H 1 . Thus we replace K h (g) in (4.3) by K loc h (g) = K h (g) χ {d(g)≤1} , and consider sup h>0 e −2 a·x f * K loc h (g).
To estimate the right-hand side of (4.1) for d(g) ≤ 1, we replace 4 by 8 in the last exponent. This allows us to eliminate the powers of d(g) 2 /h in the factors preceding the exponentials, and one finds that From (4.2) it follows that the local part of H 1 can be estimated in terms of the analogue of the Euclidean local gaussian maximal operator; notice that the local homogeneous dimension of our space is 2n +2. Since the measure μ v is locally doubling, this implies the weak type (1, 1) of the local part of H 1 . It remains to deal with the global part of H 1 , with kernel We must estimate sup Thus we assume that d(g) > 1 and observe that d(g) 1 + |z|, since also d(g) ≥ |z| because of (3.1).
To bound the right-hand side of (4.1), we first use the fact that to rewrite the exponentials. Hence, We shall estimate this product in a way that depends on the relative size of h and d(g).
If h < d(g) < 4h, d(g) > 1, the product of the factors preceding the exponential in (4.6) is controlled by (4.7) It follows that (4.8) If instead 1 < d(g) ≤ h, the factors preceding the exponential in (4.6) have a product controlled by h −n−1/2 d(g) n−1 1, and We conclude that then the last inequality in view of (3.1). It remains to consider the case d(g) ≥ 4h, d(g) > 1. Then One also has √ h d(g) since d(g) > 1, so that 1/ √ h d(g)/h. This allows us to estimate the non-exponential factors in (4.6) by constant times for some C. But these powers can be absorbed by the factor exp −c d(g) − c h −1 coming from (4.10). We conclude that in this case (4.11) The following simple lemma will allow us to restrict the kernel K glob h to a smaller set depending on h.

Lemma 6
If L ∈ L 1 (dg), the operator S defined by Proof It is enough to integrate S f (g) with respect to dμ v (g) and swap the order of integration.
Applying this lemma with L(g) = exp(−c d(g)), the estimate (4.2) together with (4.9) and (4.11) allows us to conclude that the operator obtained by multiplying K Now if a · x ≤ 1 then K glob (g) e −d(g) , and Lemma 6 implies that the corresponding part of the operator is of strong type (1, 1).

then (3.5) and (3.1) imply Q(g) d(g).
From (3.4), we see that K glob (g) is then dominated by e −c Q(g) e −c d(g) for some constants c. Again, Lemma 6 shows that the corresponding part of the operator is of strong type (1, 1).
What we need to consider is thus Defining for k = 1, 2, . . .
we get for each h > 1/4 and some C This leads to an estimate for our remaining operator saying that where Let g = (x, y, t) ∈ E k . Then √ |t| ≤ 2 k and so Thus there exists an m ∈ {1, . . . , k} for which For m > 1, we then have In both cases, one sees from (3.5) that Q(g) 2 m . Thus by (4.13) and this inequality holds trivially also if m = 1. For any g ∈ E k , this allows us to conclude that We now define operators For the right-hand side of (4.12), we then have (4.14) The following proposition will allow summation in m in the space L 1,∞ (dμ v ), and make the proof of the weak type (1, 1) of H 1 complete.
Proof Fixing m, we let 0 ≤ f ∈ L 1 (dg) and take λ > 0. Choosing a large > 0, we consider the level set To prove the proposition, we shall verify that with constants C independent of m, f , λ and . Now if g ∈ L m λ , we have for some k ∈ {m, m + 1, . . . } and this implies an upper bound for k, say k ≤ κ = κ( f , λ, ). Then which is a closed set since each f * χ E k,m is a continuous function. For each g ∈ L m λ , we let k(g) ∈ {m, m + 1, . . . , κ} be the maximal value of k for which e −2 a·x 2 −(n+1)k f * χ E k,m (g) ≥ λ.
We verify that the set L m λ is bounded. For g ∈ L m λ we have where E is the compact set Since f is integrable, the integral in (4.15) tends to 0 as d(g) → ∞. Thus L m λ is bounded and hence a compact set.
Together, these regions will be seen to cover the level set L m λ . Assume g i defined for 1 ≤ i < j, where j ∈ {1, 2, . . . }. Our idea is to choose g j as a point in L m λ but not in any region forbidden by the already selected points g i . Further, it should maximize k(g j ) and, secondly, maximize the quantity a · x ( j) .
More precisely, let provided the set L m λ \ 1≤i< j g i P −1 i is nonempty; otherwise the recursion ends. We choose g j as a point in the compact set such that a · x is maximal among the points of this set. To verify that A j is closed and thus compact, assume that g −→ g as −→ +∞ and that g ∈ A j . Then for all , and by continuity the same inequality holds at g. This means that k( g) ≥ k j ; thus k( g) = k j and g ∈ A j . (Here we actually verified that the function g −→ k(g) is upper semicontinuous.) Having thus defined the sequence (g j ), we observe that 1 ≤ i < j implies k j ≤ k i , and that if here k j = k i then a · x ( j) ≤ a · x (i) . We will verify the following three claims: and the sets g j E −1 k j ,m , j = 1, 2, . . . , are pairwise disjoint. (4.19) This would imply Proposition 7, since we would get In the third step here, we used the fact that g j ∈ L m λ . To verify (4.18), notice that Aiming at (4.19), we argue by contradiction and assume that g i E −1 k i ,m and g j E −1 k j ,m have a common point for some 1 ≤ i < j. Then there exist points To get the contradiction, it is enough to verify that the point g = ( x, y, t) is in P i , since g j cannot be in the forbidden region g i P −1 i . For the components in the a direction of these points, we have Here k j ≤ k i , and if this last inequality is strict, we conclude that a · x ≥ 0. But if k j = k i , then a ·x (i) ≥ a ·x ( j) because of the recurrence construction, and thus a · x ≥ 0 also in this case.
For the components orthogonal to a, we get (when n ≥ 2) In the same way, | y| ≤ 2 · 2 m/2 2 k i /2 . For the t coordinates, we have Since and similarly forx (i) , this implies It follows that g ∈ P i , and (4.19) is proved.
To verify (4.17), observe that for any j one has Because of (4. 19) and since f is integrable, this can only happen for a finite number of j, so the sequence (g j ) must be finite. This means that the set L (m) λ \ 1≤i< j g i P −1 i is empty for some j, which is (4.17). Proposition 7 is proved, and so is the weak type (1, 1) of H 1 . (1, 1) of H 0

Weak Type
Here one follows the argument just given for H 1 . The main difference will be that the factor (1 + |z|) −n−1 in (4.8) is now (1 + |z|) −n−3/2 .

L p -Boundedness of H k
Fix k ≥ 1 and p ∈ (1, +∞). We first consider small h and note that there exists a constant C > 1 such that This can be seen from the expression (3.16) and the classical gaussian estimates for the heat kernel and its derivatives on stratified groups; see Theorems IV.4.2 and IV.4.3 of [53]. Consequently, and H 0 is bounded on L p (μ v ) as pointed out in the Introduction. It remains to prove the L p boundedness of We use an argument inspired by [51, p. 75]. Let ε( p) = k/2 + 1/ p , where p denotes the conjugate exponent of p.
the convergence of the integral follows from Theorem IV.4.2 of [53]. By Hölder's inequality, this is majorized by In conclusion, we get 1. Using interpolation and the spectral gap, we see that e (s−1/2) v p→ p is exponentially decreasing as s → +∞. The boundedness of H k on L p (μ v ) follows.

Proof of Theorem 1
We let f (g) = φ(g) e 2 a·x as in Sect. 4.1.
The first-order Riesz transform is given by cf. (2.2). Except for a factor h −1 , the integrand here appeared in connection with the operator H 1 in the beginning of Sect. 4.1. Again, we have a kernel which is a function of (g ) −1 g multiplied by e −2a·x , and we arrive at a convolution, cf. (4.3). Indeed, for Moreover, (4.2) implies that | K h | h −1 K h , where K h is given by (4.1).
For the Riesz operator, we are thus led to the expression We shall now verify the convergence of the integral ∞ 0 | K h (g)| dh and estimate it, for all g = o. It will then follow that R 1 is given by (5.1) for all g / ∈ supp φ = supp f . Assume first that 0 < d(g) ≤ 2, so that a · x ≤ 2. From (4.1) we then see that dh.
To estimate this integral, one uses the exponential factor for h < d(g) 2 but not for other values of h, and finds that Using again (3.15), one also verifies that Assuming now d(g) > 1, we first consider the integral over d(g)/4 < h < d(g). For such h, we use (4.6) and the middle quantity in (4.7), and get The integral over 0 < h / ∈ (d(g)/4, d(g)) is controlled by e a·x−d(g)−cd(g) , as seen by means of the middle expressions in (4.9) and (4.11). Thus altogether To prove the weak type (1,1) of R 1 , we split the operator into a global and a local part. Choose a smooth function η ≥ 0 in H n satisfying η(g) = 1 if d(g) ≤ 1 and η(g) = 0 if d(g) ≥ 2. Then we define K glob h (g) = K h (g) (1 − η(g)) and The local part is R loc 1 = R 1 − R glob 1 . For g / ∈ supp f , the local part is given by where K loc h (g) = K h (g) η(g) satisfies the estimates (5.2) and (5.3), like K h . Notice that these are the standard estimates for singular integrals of Calderón-Zygmund type. By means of a suitable splitting of H n into pieces, it can be proved first that R loc 1 is bounded on L p (μ v ), 1 < p < ∞, and then that it is also of weak type (1,1), see [7, Lemma 5 p. 1316 f.].
Indeed, for the complement of this set we can apply Lemma 6 as in Sect. 4.1.
If the point g ∈ E is in the slice defined by 2 k−1 < a · x ≤ 2 k for some k ∈ {1, 2, . . . }, we combine (5.5) with (3.4) to conclude that With M k defined by (4.13), possibly with another value of the constant c, this means that From (4.14) and Proposition 7, we know that the operator is of weak type (1, 1), and the proof of Theorem 1 is complete.

Proof of Theorem 2
Let k ≥ 3. Instead of R k , it is enough to find a counterexample for (a · X) k (− v ) −k/2 . We will apply this operator to a function φ supported near the origin, and evaluate (a · X) k (− v ) −k/2 φ at points far away.
For large r > 0 we introduce the set If g ∈ r , Lemma 4 implies that |Q(g)| 1, and so

Lemma 8 If g ∈ r and r is large enough, then
The kernel p (v) h was introduced in Sect. 3.3. Before proving this lemma, we use it to construct the desired counterexample. Let φ be a nonnegative, continuous function supported in the ball B(o, ρ) for some small ρ, and satisfying φ dμ v = 1. When the point g is not in the support of φ, one has We define a subset of r by Then we can fix ρ > 0 such that (g ) −1 g ∈ r if g ∈ B(o, ρ) and g ∈ r , for any large r . This is seen from the group law (2.1), and ρ will depend only on n.
We now combine (6.3) with (2.2), where ∇ is replaced by a · X. With g ∈ r , we can swap the order of integration and obtain From Lemma 8, we conclude that for g ∈ r Since μ v ( r ) ∼ e 2r r n+1 and k ≥ 3 , this violates the weak type (1,1) for (a · X) k (− v ) −k/2 as r → +∞ and ends the proof of Theorem 2.

Proof of Lemma 8
We start by estimating the integral in (6.2) taken only over 0 < h / ∈ (d(g)/4, d(g)), in terms of the kernel K h introduced in the beginning of Sect. 4.1.
We will estimate I α,β , assuming the point g = (z, t) = (x, y, t) in r for some large r , and we take h ∈ (d(g)/4, d(g)). Then h ∼ d(g) ∼ r , and |z| 2 /h ∼ r and t/h ∼ √ r . We remark that Beals, Gaveau and Greiner [12, Sect. 2] compute a similar integral, by moving the contour of integration to a line in the complex plane; in our case this is not necessary.
As r → +∞, this contradicts the weak type (1, 1) inequality for H k when k ≥ 2. The proof of Theorem 3 is complete.
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