Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Let $v \ne 0$ be a vector in $\R^n$. Consider the Laplacian on $\R^n$ with drift $\Delta_{v} = \Delta + 2v\cdot \nabla$ and the measure $d\mu(x) = e^{2 \langle v, x \rangle} dx$, with respect to which $\Delta_{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood-Paley-Stein functions associated with the heat and the Poisson semigroups.


Introduction
Consider the weighted manifold R (n,v) , defined as R n with the Euclidean distance and the measure dµ(x) = e 2 v,x dx. Here v = (v 1 , · · · , v n ) ∈ R n \ {0} is fixed and ., . denotes the inner product on R n . Notice that large balls in this space have exponential volume growth. With R (n,v) , we associate the Laplacian with drift We denote by ∇ the usual gradient operator in R n . Notice that the Green formulas holds with respect to the measure µ, that is, provided f and w are smooth and f or w has compact support. Thus ∆ v has a selfadjoint extension in L 2 (R n , dµ).
We shall consider Riesz transforms and Littlewood-Paley-Stein functions of any order in R (n,v) . These operators are defined and studied in many general settings, such as Lie groups, symmetric spaces and other Riemannian manifolds. Their L p boundedness properties for 1 < p < ∞ have been well studied and are known in several cases. We refer the reader to [1], [5]- [12], [18], [21]- [25] and references therein. In particular, the classical results are not always valid on manifolds. Interesting counterexamples can be found in [18], [9], [7] and [8] for Riesz transforms of order one, and in [14], [1] and [12] for those of order two. The setting of the Ornstein-Uhlenbeck semigroup in Euclidean space is considered in [13] and [26]; see also [2]. There the measure is Gaussian, thus finite but not doubling.
The weak type (1,1) property of these operators is more difficult and less known in spaces of exponential volume growth. The main reason is that the existing methods to treat singular integrals are insufficient. Theorem 1.2 of [9] is a weaker estimate. But some results have been established; see [14] and [27] for the affine group, which is a typical case without spectral gap. Some other groups and spaces are treated in [3], [4] and [17]. Further, [20] and [19] deal with the Laplacian with drift.
Returning to our setting R (n,v) , we mention that Lohoué and Mustapha [25] proved that the Riesz transforms ∇ k (−∆ v ) −k/2 of any order k are bounded on L p , 1 < p < ∞. Their setting and results are actually more general. In [20], the authors and Y.-R. Wu showed that the first-order Riesz transform ∇(−∆ v ) −1/2 is of weak type (1,1) in R (n,v) . Here and in the sequel, L p and weak L p estimates in R (n,v) always refer to the measure µ.
For R (n,v) we observe that ∂/∂x i commutes with ∆ v and thus with any negative power of −∆ v , so that the factors ∇ k and (−∆ v ) −k/2 can be written in any order. We will study the weak type (1, 1) property of ∇ k (−∆ v ) −k/2 . But instead of ∇ k , we will use a general homogeneous differential operator of order k ≥ 1, with constant coefficients, not all 0. Our Riesz operator will thus be Letting ∂ v denote differentiation along the vector v, we can write D as a sum where D ′ k−i is a constant coefficient operator of order k − i involving only differentiation in directions orthogonal to v. The maximal order of differentiation along v is then q = max {i : D ′ k−i = 0} ∈ {0, . . . , k}, and this quantity turns out to be significant.
Our result about R D is the following. such that for all f ∈ L(1 + ln + L) q 2 −1 (µ) and all λ > 0, we have This inequality is sharp in the sense that q cannot be replaced by any smaller number.
Let (e t∆v ) t>0 denote the heat semigroup on R (n,v) , which is a symmetric diffusion semigroup in the sense of [28] (the conservation property can be justified by Theorem 11.8 in [16], and the other properties are obvious). Further, let (e −t √ −∆v ) t>0 denote the Poisson semigroup. For f ∈ C ∞ 0 , we define the vertical Littlewood-Paley-Stein functions associated with the operator D from (1.1) as and The L p (1 < p < +∞) boundedness of G D and H D is easy to verify.
Theorem 2 (a) The operator H D is of weak type (1, 1) if and only if q ≤ 1. When q > 1, there exists a constant C = C(v, D) such that for all f ∈ L(1 + ln + L) q 2 − 3 4 (µ) and all λ > 0 The operator G D is of weak type (1, 1) if and only if q ≤ 2. When q > 2, there exists a constant C = C(v, D) such that for all f ∈ L(1 + ln + L) q 2 −1 (µ) and all λ > 0 In (1.3) and (1.4), q cannot be replaced by any smaller number.
Consider now the horizontal Littlewood-Paley-Stein functions and related maximal operators, defined for f ∈ C ∞ 0 by Here k ≥ 1 for h k and g k , but k ≥ 0 for H k and G k . These operators are bounded on L p for 1 < p < +∞ in the setting of a general symmetric semigroup with the contraction property, see [28]. The weak type (1, 1) property of G k , k ≥ 0, in a general setting is obtained in [19]. For the other three operators, we have the following endpoint estimates. We remark that the weak type (1, 1) of H 0 was obtained in [20,Theorem 2].

Theorem 3
The operators h 1 , H 1 and g k with k ≥ 1 are of weak type (1, 1). For k ≥ 2, h k and H k are not of weak type (1, 1); however, there exists a constant C = C(v, k) such that for all f ∈ L(1 + ln + L) (1.5) and for all f ∈ L(1 + ln + L) In these two estimates, the exponents cannot be replaced by any smaller numbers.
Our estimates, in particular those involving Orlicz spaces, are optimal and go beyond earlier known results.
The structure of this paper is as follows. Section 2 contains estimates for the kernels of the Riesz transforms, which are used in Section 3 to prove Theorem 1. A fundamental tool here is Proposition 7 that is also needed in the later sections. The two parts of Theorem 2 are proved in Sections 4 and 5. Section 6 gives estimates for the time derivatives of the heat kernel, which are applied in the proof of Theorem 3 in Section 7.
In the following, C denotes various constants which depend only on n and D in Sections 2 -5, and only on n and k in Sections 6 -7. By A B, we mean A ≤ CB with such a C (we say that A is controlled by B), and A ∼ B stands for A ≤ CB and B ≤ CA.
We have for any x ∈ R n and r ≤ 1, It follows that our space has the local doubling property, The heat kernel p t (x, y) in our setting is defined through for suitable functions f . It is explicitly given by (cf. [16] p. 258) for all t > 0 and x = (x 1 , x ′ ), y = (y 1 , y ′ ) ∈ R × R n−1 .
Our results include the simpler case n = 1, but the proofs are written for n ≥ 2. Except for Proposition 7, we leave it to the reader to see how the arguments simplify for n = 1.

Estimates of the Riesz kernels
With D, k and q as in Theorem 1, we write R D (x, y) for the kernel of the Riesz transform R D = D(−∆ e 1 ) −k/2 . The multiindex α will be split as α = (α 1 , α ′ ).
We state and prove local and global estimates for R D (x, y).

Proposition 4
The kernel R D (x, y) satisfies the local standard estimates and This estimate is sharp in the sense that there exists a ball B ⊂ R n−1 , depending only on n and D, such that We will need the following integral estimate.

Lemma 6
With ν ∈ R and a > 0, let (a) Then for a ≤ 2 and this estimate remains true if B ν (a) is replaced by for any ℓ ≥ a 3/4 . In this lemma, the implicit constants depend on ν but not on a.
Lemma 6 can be proved either with elementary estimates or by evaluating the integral in terms of Bessel functions according to the 12th equality of §3.471 in [15]. The latter option also requires an easy estimate of those parts of the integral left out in the last statement.
We will also use a trivial extension of this lemma. Let Q(t) = ν q ν t ν be any finite linear combination of real powers of the variable t > 0. Then Lemma 6 (b) implies that To prepare for the proof of the two propositions, we deduce an expression for R D (x, y). From (1.8) it follows that the kernel of (−∆ e 1 ) −k/2 is Further, Recall that the Hermite polynomial of degree j ∈ N is defined by and H α = ⊗ i H α i for any multiindex α. It is well known that the H α are orthogonal with respect to the Gaussian measure e −|x| 2 dx in R n and that the leading term of H j (s) is 2 j s j . The definition of H α implies that The kernel R D (x, y) is a linear combination of terms ∂ α x (−∆ e 1 ) −k/2 (x, y), with α = (α 1 , α ′ ) a multiindex of length |α| = k and with α 1 ≤ q. When we differentiate the expression in (1.8), some differentiations with respect to x 1 may fall on the factor e −x 1 . Those differentiations falling on the factor e − |x−y| 2 4t will be given by a multiindexα = (α 1 , α ′ ), with 0 ≤α 1 ≤ α 1 . We then see from (2.7), with α replaced byα, that ∂ α x (−∆ e 1 ) −k/2 (x, y) will be a sum of terms whereα 1 runs from 0 to α 1 and the terms have positive coefficients coming from Leibniz' formula and from the factors (4π) −n/2 and 1/Γ(k/2). Expanding the Hermite polynomials here, we obtain a sum of terms proportional to 9) and the sum is now taken also over a multiindex (γ 1 , γ ′ ) with 0 ≤ γ 1 ≤α 1 and 0 ≤ γ ′ ≤ α ′ (componentwise ordering).
Proof of Proposition 4. Here |x − y| ≤ 1. If the exponent of t in the integral in (2.9) is negative, we see from Lemma 6 (a) that the modulus of the expression (2.9) is controlled by For other values of the exponent, the bound e −2x 1 |x − y| −n also follows. This is the first standard estimate (2.1). To obtain also (2.2), it is enough to trace the argument given, with a differentiation also in y. ✷ Proof of Proposition 5. For |x − y| > 1, Lemma 6 (b) implies that the expression (2.9) equals constant times whose modulus is controlled by In this argument, we will neglect all terms which are much smaller than the right-hand side of (2.4). From (2.10) and (2.11), we then see that in (2.8) we need only take α 1 = q, which implies |α ′ | = k − q, and γ 1 =α 1 . The latter equality means that in the Hermite polynomial Hα 1 in (2.8), we consider only the leading term, which is Instead of (2.8) we will now have Applying (2.5) to the integral here, we see that (2.8) amounts to a positive constant times Here we can replace the powers of |x − y| by the same powers of Summing over 0 ≤α 1 ≤ q, we conclude that for some b α > 0. Here α = (q, α ′ ) and |α| = k, and if we sum over such α with the coefficients from (1.1), the result will be Since the a α do not all vanish, the orthogonality property of the Hermite polynomials implies that the polynomial given by the sum here is not identically 0. To finish the proof of (2.4), we need only take a closed ball B where this polynomial does not vanish. ✷

Proof of Theorem 1
We split R D into a part at infinity and a local part R loc The local part is easy to treat. Because of the local doubling property, we can use the method of localization. In view of Proposition 4, standard Calderón-Zygmund singular integral theory gives the weak type (1, 1) of R loc D . To estimate R ∞ D , we start with that part defined by the restriction uniformly in y. It follows that this part of the operator is of strong type (1, 1).
To treat the opposite case x 1 − y 1 > 1, we write the first exponent in (2.3) as and observe that in this case Thus Proposition 5 implies that for It follows that this part of |R D (x, y)| is controlled by the kernel Let for κ ∈ R and suitable functions f To prove Theorem 1 in the case q ≤ 2, it is enough to show the following result.
The case n = 1 is trivial, since then In what follows, we suppose that n ≥ 2, so that where we may assume 0 ≤ f ∈ L 1 (dµ). It will be convenient to use g(y) = f (y)e 2y 1 ∈ L 1 (dy) instead of f . When |x ′ − y ′ | √ x 1 − y 1 , the exponential in the integrand is essentially 1. This indicates the most important part of the operator, dealt with in the following proposition.

Proposition 8 The operator
T g(x) = e −2x 1 We first verify that this proposition implies Proposition 7. Let for j = 1, 2, . . . then In this integral, the exponential is less than exp(−c2 j ) for some c > 0, so that Scaling the variables x ′ and y ′ by a factor 2 j , we can control this expression in terms of the operator T , still with a rapidly decreasing coefficient. One can then apply Proposition 8 and use [29,Lemma 2.3] to sum in weak L 1 . The result will be the weak type (1, 1) of V 2 , which is Proposition 7.
Proof of Proposition 8. We cover R n−1 and R n with lattices of unit cubes where m = (m 2 , . . . , m n ) ∈ Z n−1 , and In this proof, the word "cube" will always refer to cubes of these lattices. Given 0 ≤ g ∈ L 1 (dy) and λ > 0, we must estimate the µ measure of the level set L λ = {x : T g(x) > λ}. Since T g(x) ≤ e −2x 1 g(y) dy, the coordinate x 1 is bounded above on L λ . Without restriction, we can then assume that L λ is contained in the left half-plane {x : x 1 ≤ 0}.
The aim of the following lemma is to discretize the variable x 1 .

Lemma 9
Let i ≤ 0 and assume that the cube Q i,m intersects the level set L λ . Theñ Proof. Take a point z ∈ Q i,m ∩ L λ , so that λ < T g(z) = e −2z 1 Here i < 1 + z 1 and z 1 − y 1 > 1, so that i − y 1 < 1 + z 1 − y 1 < 2(z 1 − y 1 ) and thus (z 1 − y 1 ) (1−n)/2 < 2 (n−1)/2 (i − y 1 ) (1−n)/2 . Further, the region of integration in (3.6) is contained in that in (3.5), because Since also e −2z 1 ≤ e 2 e −2i , it follows that T g(z) < e 2 2 (n−1)/2T g(i, x ′ ), which proves the lemma. ✷ The region of integration in (3.5) is contained in the set For each cube Q i,m , we define a family of "forbidden" cubes as easily verified. The exponential behavior of µ implies that For each i ≤ 0, we will select some of the cubes Q i,m , namely those for which m is in a set A i ⊂ Z n−1 . These A i will be defined by recursion. We will have the inclusion (3.8) Because of (3.7), this implies that When we make the selection, i.e., define the A i , we will consider theD i,m as forbidden regions, where no selection is allowed. Indeed, we will select for each i = 0, −1, −2, . . . those cubes Q i,m which intersect the level set and are not in F j,ℓ for any already selected cube Q j,ℓ . In this way, cubes selected at different steps will be far from each other.
More precisely, we first let Assuming for some i < 0 the sets A i+1 , . . . A 0 already defined, we then let The inclusion (3.8) is immediate from this construction. For each i ≤ 0, we define which is a union of cubes contained in {y : y 1 ≤ i − 1}. In order to replace the D i by pairwise disjoint sets, we let E 0 = D 0 and Then we let g i = g χ E i and observe that g = 0 j=i g j in D i . This together with Lemma 9 implies that, for i ≤ 0 and The integral in y ′ in this expression can be seen as a convolution in R n−1 . That leads to an estimate which is insufficient for our purpose, essentially because several values of i are used for the same g j . To do better, we will replace i by j in the iterated integral in (3.10). Thus we first claim that if y, x ′ , m, i and j are as in (3.10). Then we observe that an immediate consequence of (3.10) and (3.11) is that if x ′ ∈ Q m for some m ∈ A i , then where we also made the regions of integration larger. Before we use (3.12) to finish the proof of the lemma, we verify the claim (3.11). Then we can obviously assume that g j (y) = 0, which implies y ∈ E j ⊂ D j . The definition of D j says that y 1 < j − 1 and |y ′ − z ′ | < √ j − y 1 + √ n − 1 for some z ′ ∈ Q ℓ and some ℓ ∈ A j . Clearly |z ′ − ℓ| ≤ √ n − 1, so the triangle inequality leads to The inner integral in (3.10) is taken over those y ′ satisfying (3.14) Since also |x ′ − m| ≤ √ n − 1, (3.13) and (3.14) imply On the other hand, the cube Q i,m was selected and thus not forbidden by the selected cube Q j,ℓ , i.e., Q i,m / ∈ F j,ℓ . This means that |m − ℓ| ≥ 4 √ j − y 1 + 4 √ n − 1. Combining this inequality with (3.15), we get 4 √ j − i < 2 √ j − y 1 , so that j − i < (j − y 1 )/4. Then we can write and thus This implies (3.11), and (3.12) also follows. We now integrate the inequality (3.12) in x ′ over the unit cube Q m , to conclude that The inner part of this expression, is the value at x ′ of the convolution in R n−1 of g j (y 1 , .) and the characteristic function of the ball {y ′ ∈ R n−1 : |y ′ | < √ j − y 1 + √ n − 1}, essentially normalized in L 1 (R n−1 ) by the first factor. Thus uniformly in j − y 1 > 1. Combining (3.16) with (3.9), we have In the last expression here, we have a double sum over all m ∈ A i and i ≤ j. The corresponding family of cubes Q m is pairwise disjoint, since m ∈ A i implies Q i ′ ,m ∈ F i,m for i ′ < i by the definition of F i,m . Thus the double sum of integrals over Q m is no larger than the left-hand side of (3.17). As a result, This ends the proof of Proposition 8. ✷ Consider now the case q > 2 in Theorem 1. The main part of the kernel R D (x, y) is controlled by the kernel V q defined in (3.2). Therefore, the estimate (1.2) is a consequence of the following proposition.
Proof. We have for h ∈ L 1 (µ) Here we assume h ≥ 0 and apply the elementary inequality easily proved by separating the cases b ≤ 10(1 + ln + a) and b > 10(1 + ln + a). Here C 0 depends only on κ. Letting a = h(y) and b = |x − y|, we get dy.
The integral here equals and to see that it is finite, we split it in two: In I 1 , we can estimate the second and third exponential factors by 1 and e 3z 1 /8 , respectively, and integrate first in z ′ . In I 2 , we have |z| < 3|z ′ |/2 and thus |z ′ | 2 /|z| > 2|z ′ |/3, so that the second exponential in I 2 is less than exp(−|z ′ |/6). It is then enough to estimate the third exponential by exp(z 1 /8 + |z ′ |/8).
It now follows that for some constant C 1 .

Proof of Theorem 2 (a)
Before proving this theorem, we find local and global estimates for t k/2 D x p t (x, y) in the space L 2 (dt/t). They will be analogous to Propositions 4 and 5, with similar proofs. and .
For |x − y| ≤ 1, we see from Lemma 6 (a) that the modulus of (4.5) is controlled by if the exponent of t in the integral is negative. But in the opposite case, this is also true. Thus (4.1) is proved. To complete the proof of Proposition 11, one verifies (4.2) in a similar way.
We now introduce a local part H loc D and a global part H ∞ D of this operator, defined by restricting the integration in y here to the region |y − x| ≤ 1 and |y − x| > 1, respectively. Thus To deal with the local part, we apply the method of localization as done for the Riesz transforms, but now using vector-valued singular integral theory and Proposition 11.
For H ∞ D , we start by applying Minkowski's integral inequality, getting dµ(y).
We can now follow the argument for R ∞ D in the proof of Theorem 1, replacing (2.3) by (4.3). In particular, that part of H ∞ D given by x 1 −y 1 ≤ 1 is seen to be of strong type (1, 1). Moreover, the kernel of the remaining part is controlled by V κ (x, y) with κ = q + 1/2. This leads to the weak type (1, 1) estimate of Theorem 2 (a) and also to (1.3), in view of Propositions 7 and 10.
For the sharpness parts of Theorem 2 (a), we let f = χ B(0,1) . Let |α| = k and take points y and x with |y| < 1 and x 1 > 0 large but x ′ / |x| bounded, which implies x 1 = |x| + O(1). We will then estimate t k/2 ∂ α x p t (x, y), and we need to be more precise than in the preceding argument. Because of (1.8) and (2.7), this quantity is the sum of taken overα 1 = 0, . . . , α 1 . Hereα = (α 1 , α ′ ) as before. The remark after the end of the proof of Propositions 11 and 12 shows that we need only consider the case α 1 = q, which implies |α ′ | = k − q. For the same reason, we may replace the polynomial Then we can write where the dots indicate negligible terms. Restricting t by t − |x| 2 < |x| 3/4 , we see that We now insert this last expression for the Hermite polynomial in (4.9), and observe that |α| =α 1 + |α ′ | =α 1 + k − q. Then the only factor in (4.9) which depends onα 1 will be the binomial coefficient. Since we get by summing (4.9) inα 1 The next step is to sum these expressions in α, with the coefficients from (1.1). The result is that for y, x and t as described above, The sum here is P x ′ / 2|x| for some nonzero polynomial P in n−1 variables. Thus we can find a ball B ⊂ R n−1 in which P does not vanish. Further, since |y| < 1 and t ∼ |x|. With f = χ B(0,1) , we then get by integrating in y where as before x 1 > 0 is large, x ′ / 2|x| ∈ B and t − |x| 2 < |x| 3/4 . Hence, (4.10) After the change of variable τ = 2t, we can apply the last part of Lemma 6 (b) to conclude that the right-hand side of (4.10) equals a positive constant times We define Ω η by (3.19), though B is not the same as in Section 3. If η is large, x ∈ Ω η implies x ′ / 2|x| ∈ B. It now follows from (4.10) and (4.11) that for large η in Ω η . This implies the "only if" part of Theorem 2 (a) and the sharpness of q; cf. the very last part of Section 3. We leave it to the reader to check that all the neglected terms above can be disregarded. Theorem 2 (a) is proved. ✷

Proof of Theorem 2 (b)
To prove the claimed boundedness properties of G α , we copy the reasoning used for Theorems 1 and 2 (a). We must estimate the kernel t k D x P t (x, y) in the space L 2 (dt/t).
The subordination formula says that the Poisson kernel P t is given by This will also hold with P t (x, y) and p u (x, y) replaced by their derivatives D x P t (x, y) and D x p u (x, y), respectively. Using Minkowski's integral inequality, we conclude that 2 dt t 1 2 du.
The last inner integral here is seen to equal 2 k Γ(k + 1)u k+1 , and so The right-hand side in (5.1) is like the expression for R D (x, y) in (2.6), except for the modulus signs. In Section 2, the equality (2.6) was used to prove Propositions 4 and 5.
Tracing those arguments, one sees that the inequality (5.1) is sufficient to imply estimates similar to (2.1) and (2.3) for ( +∞ 0 |t k D x P t (x, y)| 2 dt/t) 1/2 . Taking a derivative also in y, one obtains the analog of (2.2). This leads to the weak type of G α for q ≤ 2 and also to (1.4), as seen from the proof of Theorem 1 in Section 3.
It remains to verify the sharpness parts of Theorem 2 (b). Recall that D = |α|=k a α ∂ α .
We will find an f ∈ L 1 (µ) and a ball B ⊂ R n−1 such that if x 1 > 0 is large and one has From this estimate, the sharpness parts of Theorem 2 (b) will follow, cf. the last few lines of Section 3.
To estimate D x P t (x, y), we consider with |α| = k and α 1 ≤ q. From (1.8) we see that ∂ α x P t (x, y) is a sum of terms whereα = (α 1 , α ′ ) as before, and the sum is taken overα 1 = 0, . . . , α 1 . Using (2.7), we see that this expression equals We now apply (2.5) to the integral here. It follows that for large |x − y| the quantity (5.3) equals Again we consider points x for which x 1 > 0 is large and x ′ / √ x 1 stays bounded.
Further, we assume |y| < 1 and restrict t by √ x 1 < t < 2 √ x 1 . Some Taylor expansions then lead to |x − y| = x 1 (1 + O(x 1 −1 )) and Using (5.5), we conclude that the expression (5.4) now equals We can replace the Hermite polynomial Hα 1 x 1 /2 here by its leading term, which is 2α 1 /2 xα 1 /2 1 . Since |α| =α 1 + |α ′ |, this will make the exponents of 2 and x 1 in (5.6) independent ofα 1 . When we then sum inα 1 = 0, . . . , α 1 , the binomial coefficients in (5.6) will sum up to 2 α 1 , and as a result Next, we integrate against f (y) dµ(y) with f = χ B(0,1) /µ(B(0, 1)). Summing also over α, we get For large x 1 , the largest terms in the sum here are those where |α ′ | takes its minimal value k − q, so that α 1 = q. We introduce the nonzero polynomial Since x ′ / √ 2x 1 stays bounded, it follows that We can now integrate in t and conclude that The integral here is of order of magnitude x k+1 1 , and so To obtain (5.2), it is now enough to let B be a closed ball in R n−1 in which the polynomial P does not vanish.
This ends the proof of Theorem 2 (b). ✷ 6 Sharp estimates for ∂ k ∂t k p t This section is a preparation for the proof of Theorem 3. We take k derivatives with respect to t of Those derivatives which fall on the last exponential here will produce a factor |x−y| 2 4t 2 − 1. The derivative will also appear. We see that where the factor q k is given by for a polynomial Q k in three variables, whose coefficients depend only on n and k.
To estimate q k , we examine the terms of this polynomial. Consider a term obtained by letting exactly m of the k differentiations fall on the exponential e −t−|x−y| 2 /4t . If, moreover, the number of differentiations falling on a power of |x−y| 2 4t 2 − 1 is p, the resulting term of Q k will contain p factors − |x−y| 2 2t 3 and m − p factors |x−y| 2 4t 2 − 1. The remaining k − m − p differentiations will produce factors −t −1 . We conclude that q k is a sum of expressions of the form with C m,p > 0. Here p + m ≤ k and p ≤ m; thus 2p ≤ k so that p ≤ [ k 2 ]. This implies the upper estimate the sum taken over 0 ≤ p ≤ [ k 2 ] and 0 ≤ m ≤ k − p. We also need a lower estimate. Let η > 0 be large and define Σ η = {x; η − 1 < x 1 < η and √ η < x i < 2 √ η, i = 2, · · · , n}.
Lemma 13 For x ∈ Σ η , |y| < 1 and with c 1 = c 1 (n, k) small enough and η large enough, one has (−1) [ k 2 ] t k ∂ k ∂t k p t (x, y) e −2η η k−n 2 . (6.5) Proof. Assuming x, y and t as in the lemma, we can write where c 1 < ζ < 2c 1 . Some simple computations will lead to |x − y| = η (1 + O(η −1 )) and ]. Moreover, if c 1 is small enough and η large enough, these values of m and p will make the expression (6.6) much larger than any other admissible values of m and p, in absolute value. We conclude that then We also need an estimate of the value of p t (x, y) for these x, y, t, and write the sum of the exponents of e in (6.1) as Thus (6.1) implies that p t (x, y) ∼ e −2η t − n 2 .
We now combine this with (6.2) and (6.7), and take η large after fixing a small c 1 . Then (6.5) follows. ✷

Proof of Theorem 3
For the local parts of the operators in Theorem 3, one can use the method of localization, since R (n,v) has the local doubling property. Standard vector-valued singular integral theory then gives the weak type (1, 1) of the local parts of g k , h k and H k . The parts at infinity of these operators can all be estimated by the method used for Theorem 1, in the following way.
Consider first the part at infinity of h k , given by dµ(y).
We estimate the inner integral here. and use Minkowski's integral inequality, getting As in the last part of Section 3, this implies that h k is not of weak type (1, 1) for k ≥ 2 and that (1.5) is sharp.
The case of H k is even simpler, and this ends the proof of Theorem 3. ✷