Weak Type (1,1) Bounds for Some Operators Related to the Laplacian with Drift on Real Hyperbolic Spaces

The setting of this work is the n-dimensional hyperbolic space R+×Rn-1$\mathbb {R}^{+} \times \mathbb {R}^{n-1}$, where the Laplacian is given a drift in the R+$\mathbb {R}^{+}$ direction. We consider the operators defined by the horizontal Littlewood-Paley-Stein functions for the heat semigroup and the Poisson semigroup, and also the Riesz transforms of order 1 and 2. These operators are known to be bounded on Lp,1<p<8$L^{p},\; 1<p<\infty $, for the relevant measure. We show that most of the Littlewood-Paley-Stein operators and all the Riesz transforms are also of weak type (1,1). But in some exceptional cases, we disprove the weak type (1,1).


Introduction
Let (M, μ) be a σ -finite measure space, and let {T t } t>0 denote a symmetric diffusion semigroup on (M, μ) in the sense of [20]. The horizontal Littlewood-Paley-Stein function of order k ∈ N + associated to {T t } t>0 is defined by +∞ 0 s k ∂ k ∂s k T s f (x) 2 ds for f ∈ L p (μ), 1 ≤ p < +∞, and the related maximal function of order k ∈ N is sup s>0 s k ∂ k ∂s k T s f (x) , x ∈ M.
By the general Littlewood-Paley theory of Stein [20], these operators are bounded on L p (μ) for all 1 < p < + ∞. Here k ≥ 1 for g k and h k , but k ≥ 0 for G k and H k . In R n , it is obvious that g k and h k are not bounded on L 1 . But they are of weak type (1, 1), as follows from the classical vector-valued singular integral operator theory; see for instance Stein [21,Ch. IV]. These results can be generalized to complete Riemannian manifolds satisfying the doubling volume property and an on-diagonal heat kernel upper estimate; see [3], where the weak type (1, 1) of g 1 , g 2 and h 1 is proved. Moreover, it is not hard to see that the arguments of [3, pp. 50-52] are valid for g k and h k of higher order. In this setting, one can also show that the H k are of weak type (1, 1) by using basic properties of the centered Hardy-Littlewood maximal function.
If the manifolds considered have exponential volume growth, no doubling condition is satisfied. This situation is worse, since there is no adequate theory of singular integrals. There are some weak type (1, 1) results for H 0 in that case; see for example [1,2,4,14,16] and references therein. To our knowledge, there exists only one result about g k : in Anker's paper [1], the weak type (1, 1) inequality of g 1 is obtained for noncompact symmetric spaces, and it is clear that the argument (see [1, pp. 290-291]) also works for g k of higher order and in the setting of harmonic AN groups in the sense of [5]. The cases of h k and H k (k ≥ 1) seem to be more difficult: there is no known result even for h 1 or H 1 .
The main purpose of this paper is to exhibit some manifolds of exponential volume growth in which h 1 and H 1 are of weak type (1, 1). As a consequence, we give for symmetric spaces of noncompact type and rank one an affirmative answer to a problem left open in [1,Remark (1), pp. 278-279]. In the manifolds considered, we also treat Riesz transforms ∇(− ) −1/2 and (− ) −1/2 ∇, and analogous second-order operators. For these, a few weak type (1, 1) results have been obtained in the settings of noncompact symmetric spaces (see for example [1]), harmonic AN groups (see [2]), affine groups (see for example [18] and [10]) and the Laplacian with drift on euclidean spaces (see [16]).
The setting of this paper will be a weighted manifold based on the real hyperbolic space H n of dimension n ≥ 2. Here H n is considered as R + × R n−1 endowed with the measure dμ(y, x) = y −n dydx and the distance d((y, x), (y , x )) = arcosh On H n we consider the vector fields The gradient and its norm are given by The Laplacian on H n is Given α ∈ R, we replace dμ by dμ α = y α dμ and get H (n,α) = (H n , dμ α ), which is a weighted manifold as defined in [9,Definition 3.17,p. 67]. The corresponding Laplacian is obtained by adding to H n a drift term in the y coordinate, Notice that H (n,α) has exponential volume growth, which affects the behavior of the Hardy-Littlewood maximal function; see [11,12] and [13]. Moreover, it is stochastically complete, see [9,Theorem 11.8,p. 303]. It follows that the operators g k , G k , h k and H k are bounded on L p (μ α ), 1 < p < ∞.
For the Riesz transforms of any order on H (n,α) , α = n − 1, the boundedness on L p (μ α ), 1 < p < ∞, is proved by Lohoué and Mustapha [17], or can be deduced from this paper. We shall focus on the weak type (1, 1) boundedness. The case α = n − 1 is special, since H (n,n−1) is the affine group with the right-invariant Haar measure. In that case, explicit formulas for the operator kernels are available, and the weak type (1, 1) boundedness properties of the Riesz transforms are known; see [19] and references there. Notice that this is the only case where − H (n,α) has no spectral gap.
From now on, we exclude the affine group and assume that α = n − 1, except in Remark 3 below.
The following are our main results. The measure we use on H (n,α) is always dμ α .

Remark 1
It is easier to obtain the weak type (1, 1) continuity for the first-order vertical Littlewood-Paley-Stein functions, i.e., those defined in terms of derivatives with respect to the x i instead of t.

Remark 2
The results of Theorem 1 remain valid for harmonic AN groups in the sense of [5]. This is because for such a group the heat kernel has an explicit expression, given in [2, Theorem 5.9, p. 664], and an observation in [15,Section 6] makes it possible to control it in terms of the heat kernels of H (n,0) , with several values of n. One can then use [2, (5.26) Proposition, p. 667] and follow the method of our paper. The details are left to the reader. Notice that harmonic AN groups, as Riemannian manifolds, have constant negative Ricci curvature, see for example [2, p. 647]. They include all symmetric spaces of noncompact type and rank one, but most of them are not symmetric spaces.

Remark 3
It is worth observing that in the excluded case α = n − 1, our methods, together with the Hopf-Dunford-Schwartz maximal ergodic theorem, can be used to show that H k is of weak type (1, 1). Moreover, our proof for the weak type (1, 1) of G k in Section 4 applies also in this case. But we do not know whether g k and h k are also of weak type (1, 1) when α = n − 1.
(a) On H (n,α) , the first-order Riesz transforms are of weak type (1, 1). (b) The same holds for the second-order Riesz transforms The structure of this paper is as follows. After some preliminaries in Section 2, we prove Theorem 2 in Section 3. The proof of Theorem 1 fills the remaining sections. First the case of G k is treated in Section 4, in a more general setting. Then Section 5 contains the kernel estimates needed for the estimates of h 1 , H 0 , g k and H 1 in Sections 7 and 8. Some relations between different Littlewood-Paley-Stein functions are obtained in Section 6. Finally, Section 9 proves the negative results in Theorem 1.

Notation and Preliminaries
Symbols like ∼ and will have their usual meaning, with implicit constants depending only on n and α.
It will be convenient to write The subordination formula connects the Poisson and heat semigroups by du.
We denote by p (n) t the heat kernel of H n and by p (n,α) t that of H (n,α) . The two are related by To simplify notations in the sequel, we let .
as easily verified. Observe that when these derivatives are applied to a function of r.
It is well known that the heat kernel of H n and the kernels of other operators which are given as functions of H n depend only on r. We shall write Theorem 5.7.2, p. 179, of [6] says that for all r ≥ 0 and t > 0 a result obtained by Davies and Mandouvalos [7]. Letting The space H (n,α) has the local doubling property. Indeed, the distance formula (1.1) implies that a ball B(Y, s) with Y = (y, x) ∈ H n and a small radius s > 0 is, up to a small error, given by the inequality It follows that μ α (B(Y, s)) ∼ y α s n , (2.10) uniformly in Y and s ∈ (0, 1], and this implies the local doubling.

The Local Part
Considering the two integrals in (3.4) and (3.3), we observe that the exponents appearing in the integrands can be written respectively, since ρ(n, α) > 0. When r ≤ 1, it follows that both integrands will be very small for t >> 1 and for t << r 2 . From this and (2.10), we conclude Using (2.9) with n replaced by n + 2 and (2.5), one obtains by similar computations Thus the local parts of the Riesz kernels are standard Calderón-Zygmund kernels in H (n,α) , which implies the weak type (1, 1).

The Global Part
In this subsection, r > 1.
Invoking again the expression in (3.5), it easily follows that the order of magnitude of the integrals in In that interval 1 + r + t ∼ r, and one finds Here we suppress the power of r. What we need is then the following lemma.

Lemma 3
For n − 1 = α ∈ R, the operator defined by integration against the kernel Proof We can estimate T by applying (1.1) to the exponential factor, since e r ∼ cosh r. The result will be Consider first the case α < n − 1.
Since the last expression here is monotone in the variable y, we can argue as in Strömberg's paper [22]. This means observing for λ > 0 that |Tf (Y)| > λ implies y > y 0 (x), where y 0 (x) satisfies which ends this case.
The T 1 part here gives rise to a strong type (1, 1) operator, since The T 2 part requires a longer argument. We first observe that T 2 ∼ y −α for |x − x | < y.
The following lemma will end the proof of Lemma 3, since it will allow summation in j .

Q j ((y, x), (y , x ))f (y , x ) dμ α (y , x ) > λ,
(3.7) where Z y,x is the cylinder Consider the family Z of all cylinders Z y,x with (y, x) ∈ H n which verify (3.8). We will mimic the ordinary proof of the weak type (1, 1) inequality for the standard maximal function in R n . Notice that μ α (Z y,x ) ∼ 2 (n−1)j y α , and that for any Z y,x ∈ Z the inequality (3.8) implies We shall define recursively a sequence (Z k ) ∞ 1 of pairwise disjoint cylinders in Z. At each step, we shall choose a Z y,x with y essentially as large as possible, among the cylinders disjoint with those already chosen. Let first Z 1 = Z y 1 ,x 1 be any cylinder in Z verifying From Eq. 3.9 we see that this supremum is finite. Assuming Z 1 , ..., Z k−1 already defined, we let Z k = Z y k ,x k be any cylinder in Z disjoint with Z 1 , ..., Z k−1 and verifying y k > 1 2 sup {y : ∃x ∈ R n−1 such that Z y,x ∈ Z and Z y,x is disjoint with Z 1 , ..., Z k−1 }.
(3.10) Should the set here be empty, the procedure stops. Since the Z k are pairwise disjoint, Eq. 3.9 implies In particular, both μ α (Z k ) and y k will tend to 0 as k → ∞, and so the supremum in Eq. 3.10 also tends to 0. Now assume Z y,x is a cylinder in Z which is not among the Z k . Then Z y,x must intersect some Z k , since otherwise y would be less than the supremum in Eq. 3.10 for each k. Let Z k be the first cylinder in the sequence which intersects Z y,x . It follows from the choice of Z k that y k > y/2.
But then the enlarged cylinder 3Z k = Z 3y k ,x k will contain the point (y, x). That means that the union set ∪ k 3Z k contains all points (y, x) in the level set defined by Eq. 3.7. The μ α measure of this level set is then at most (3.12) where the last step comes from Eq. 3.11. Lemma 4 is proved with ε = α − n + 1, and Lemma 3 follows. With this, Theorem 2(a) is proved for the the Riesz transforms X j (− H (n,α) ) − 1 2 .
For 1 ≤ j ≤ n − 1, one finds that (3.14) We will thus get an expression for this kernel analogous to Eq. 3.2, with only the last summand, and with X j cosh r = y ∂ cosh r/∂x j replaced by −y ∂ cosh r/∂x j .
The case j = 0 is only slightly more complicated, and one finds Here we get two terms like those in Eq. 3.2, but with X 0 cosh r = y ∂ cosh r/∂y replaced by −y ∂ cosh r/∂y and with X 0 (yy It is now easy to verify that the arguments given above for the local and global parts remain valid for the operators (− H (n,α) ) − 1 2 X j and lead to the weak type (1, 1) estimate. Part (a) of Theorem 2 is proved. For Part (b), one can follow the pattern of the proof of Part (a), and we leave the details to the reader.
This ends the proof of Theorem 2.

Weak Type (1, 1) of G k for a General Symmetric Diffusion Semigroup
Let {e t } t>0 be a symmetric diffusion semigroup and {e −t √ − } t>0 the corresponding Poisson semigroup. Stein's argument (see [20, pp. 48-49]) leads to the weak type (1, 1) inequality for G k . Indeed, the subordination formula (2.3) can be written as Using integration by parts, we obtain Letting we have Then for k ∈ N, t > 0 and f ∈ L 1 The change of variable λ = t 2 √ u shows that the last integral equals Thus for all t > 0 and r ≥ 0. From Eqs. 5.1 and 2.7, we conclude After some computations, we get from Eqs. 5.1, 2.9 and 2.8 This yields and In general, let P j (h 1 , h 2 , h 3 ) be the real homogeneous polynomials of degree j on R 3 defined recursively for j = 0, 1, . . . by With j = 0, 1, . . . , we then have For even dimensions, Eqs. 5.5 and 5.7 hold only with an error term, which we shall estimate. From Eq. 5.2, we get for j = 0, 1, . . .
and Eq. 5.9 will be a consequence of the case j = 1 of the following lemma. where the implicit constant depends only on j, α and n.
Before the proof of this lemma, we finish that of Proposition 5. The arguments for Eq. 5.10 are similar to those for Eq. 5.9 just given. Instead of the factor (s 2 − r 2 )/t 2 in Eq. 5.12, we will now have and Lemma 6 can be applied again. To prove Eq. 5.11, one uses the estimate valid for s ≥ r ≥ 0, since P j is a homogeneous polynomial of degree j . We omit the details. This ends the proof of Proposition 5.
Proof of Lemma 6 We start with the case r ≥ 1. Since for s ≥ r ≥ 1, we have From Eqs. 5.1, 5.18 and 5.4, we get Since y/2y ≤ cosh r and ρ(n) + ρ(n, α) ≥ α/2, we have If r/t 1, then Eq. 6.5 is immediate from this and Eq. 6.6. If r/t is large, we will get in Eq. 6.6 a factor exp(−cr 2 /t) with some c > 0. It allows us to replace r in the polynomial factors in Eq. 6.6 by √ t, and Eq. 6.5 follows again. For f ∈ L 1 and Y ∈ H n , the estimate (6.5) implies that lim t→+∞ ∂ k ∂t k e t H (n,α) f (Y) = 0. We claim that also Indeed, with P (n,α) t (Y, Y ) denoting the Poisson kernel, the subordination formula (4.1) implies that for t > 1 Following the argument in [ The subordination formula (2.3) implies We estimate the inner integral here by using the Cauchy-Schwarz inequality, getting The change of variable λ = t 2 /4u shows that the last inner integral equals . As a result, since r ≥ 1, and 2r ∞ ρ(n,α) r λ n−3 e −rλ/2 λ 4 dλ r 1−n−2 1.
We have verified (7.1) and thus the weak type (1, 1) of h 1 . The weak type (1, 1) of H 0 follows from the above together with Eq. 6.4 and the Hopf-Dunford-Schwartz maximal ergodic theorem.
Finally we consider the weak type (1, 1) of g k , k ≥ 1, proceeding as in the case of h 1 above. The local part causes no problems, and for the part at infinity, it is enough to verify the following estimate: and Eq. 7.2 follows. (1, 1)  The result will follow if we show that W 1 + W 2 e −(n−1+|n−1−α|)r .

Weak Type
To estimate the integral W 2 , we apply Eq. 5.18 together with Eq. 5.4 and observe that here |r/2t − ρ(n, α)| > ρ(n, α)/2, by the choice of C 0 . The result is The argument for H k , k ≥ 2, is complete, and so is the proof of Theorem 1.