Riesz transforms of a general Ornstein--Uhlenbeck semigroup

We consider Riesz transforms of any order associated to an Ornstein--Uhlenbeck operator $\mathcal L$, with covariance $Q$ given by a real, symmetric and positive definite matrix, and with drift $B$ given by a real matrix whose eigenvalues have negative real parts. In this general Gaussian context, we prove that a Riesz transform is of weak type $(1,1)$ with respect to the invariant measure if and only if its order is at most $2$.


Introduction
In this paper we are concerned with Riesz transforms of any order in a general Gaussian setting, in R n with n ≥ 1. An Ornstein-Uhlenbeck semigroup is determined by two n × n real matrices Q and B such that (h1) Q is symmetric and positive definite; (h2) all the eigenvalues of B have negative real parts.
Here Q and B indicate the covariance and the drift, respectively. We also introduce a family of covariance matrices Q t = t 0 e sB Q e sB * ds, t ∈ (0, +∞]. (1.1) and generated by the Ornstein-Uhlenbeck operator, defined below. Notice that dγ ∞ is the unique invariant probability measure with respect to the semigroup (H t ) t>0 ; its density is proportional to e −R(x) , where R(x) denotes the quadratic form In this general Gaussian framework, the Ornstein-Uhlenbeck operator L is given for functions f ∈ S (R n ) by where ∇ is the gradient and ∇ 2 the Hessian. Notice that −L is elliptic. We write D = (∂ x 1 , . . . , ∂ xn ) in R n and let α = (α 1 , . . . , α n ) ∈ N n \ {(0, . . . , 0)} denote a multiindex, of length |α| = n 1 α i . Then we can define the Gaussian Riesz transforms as R (α) = D α (−L ) −|α|/2 P ⊥ 0 , where P ⊥ 0 is the orthogonal projection onto the orthogonal complement in L 2 (γ ∞ ) of the eigenspace corresponding to the eigenvalue 0. Here the derivatives are taken in the sense of distributions. We will justify the introduction of negative powers of −L in Section 3.
When the order |α| of R (α) equals 1 or 2, we shall denote by R j and R ij the corresponding Riesz transforms, that is, for i, j ∈ {1, . . . , n} There exists a vast literature concerning the L p boundedness of Riesz transforms in the Gaussian setting, in both the strong and the weak sense. We will only mention the results that are most significant for this work; here 1 < p < ∞.
In the standard case, when Q and −B are the identity matrix, the strong type (p, p) of R (α) has been proved with different techniques in [24,16,29,31,10,13,22]; for a recent account of this case we refer to [32,Chapter 9]. Other proofs, holding in the more general case Q = I and B symmetric, may be found in [17,18]. G. Mauceri and L. Noselli have shown more recently that the Riesz transforms of any order are bounded on L p (γ ∞ ) in the general case (see [21,Proposition 2.3]). For some results in an infinite-dimensional framework, we refer to [8].
The problem of the weak type (1, 1) of R (α) is more involved than in the Euclidean context, where it is well known that a Riesz transform of any order associated to the Laplacian is of weak type (1,1). Indeed, in the standard Gaussian framework Q = −B = I, it is known that R (α) is of weak type (1,1) if and only if |α| ≤ 2 (see [25,30,11,1,9,26,27,28,14,12,2] for different proofs). In their paper [21], Mauceri and Noselli proved the weak type (1, 1) of the first-order Riesz transforms associated to an Ornstein-Uhlenbeck semigroup with covariance Q = I and drift B satisfying a certain technical condition. To the best of our knowledge, no result beyond this is known about the weak type (1,1), neither for first-order Riesz transforms associated to more general semigroups nor for higher-order Riesz operators.
In this paper we continue the analysis started in [4] and [5] of a general Ornstein-Uhlenbeck semigroup, with real matrices Q and B satisfying only (h1) and (h2). Our main result will be the following extension of the result in the standard case.
Theorem 1.1. The Riesz transform R (α) associated to the Ornstein-Uhlenbeck operator L is of weak type (1, 1) with respect to the invariant measure dγ ∞ if and only if |α| ≤ 2.
In particular, we shall prove the inequalities and for all i, j = 1, . . . , n and all functions f ∈ L 1 (γ ∞ ), with C = C(n, Q, B) > 0.
The plan of the paper is as follows. In Section 2, we introduce the Mehler kernel K t (x, u), which is the integral kernel of H t . Some estimates of this kernel are also given. As in [5], we introduce a system of polar coordinates which is essential in our approach, and we define suitable global and local regions. Section 3 deals with the definition of the negative powers of −L .
Then in Section 4 we explicitly write the kernels of R j and R ij as integrals with respect to the parameter t, taken over 0 < t < +∞. Section 5 contains bounds for those parts of these kernels which are given by integrals only over t > 1. In Section 6, several technical simplifications reducing the complexity of the proof of Theorem 1.1 are discussed. After this preparatory work, the proof of the theorem, which is quite involved and requires several steps, begins. In Section 7, we consider those parts corresponding to t > 1 of the kernels of R j and R ij , and prove a weak type estimate. Section 8 is devoted to the proof of the weak bounds for the local parts of the operators. Finally, in Section 9 we conclude the proof of the sufficiency part of Theorem 1.1, by proving the weak type estimates for the global parts, with the integrals restricted to 0 < t < 1. In Section 10, we establish the necessity statement in Theorem 1.1 by means of a counterexample.
In the following, the symbols c > 0 and C < ∞ will denote various constants, not necessarily the same at different occurrences. All of them depend only on the dimension n and on Q and B. With a, b > 0 we write a b instead of a ≤ Cb and a b instead of a ≥ cb. The relation a ≃ b means that both a b and a b hold.
By N we denote the set of all nonnegative integers. If A is an n × n matrix, we write A for its operator norm on R n with the Euclidean norm | · |. We let Observe that |x| Q is a norm on R n and that |x| Q ≃ |x|. Integral kernels of operators are always meant in the sense of integration with respect to the measure dγ ∞ .

Notation and preliminaries
It follows from (1.1) that for 0 < t < ∞ This difference and also are symmetric and strictly positive-definite matrices.
It is shown in [5, formula (2.6)] that for bounded and continuous functions f where the Mehler kernel K t is given by for x, u ∈ R n and t > 0. Here we use a one-parameter group of matrices We recall from [5, Lemma 2.1] that D t may be expressed in various ways. Indeed, for t > 0 one has We restate Lemma 3.1 in [5]. This also holds with D s replaced by e −sB or by e −sB * .
The following is part of [5,Lemma 3.2].
Lemma 4.1 in [5] says that for all x ∈ R n and s ∈ R one has Integration of (2.5) leads to again because of Lemma 2.1.
Proof. The upper estimate is an immediate consequence of (2.6). For the lower estimate, we write where we used (2.7) to estimate the numerator and Lemma 2.1 for the denominator.
The following implication will be useful as well. Since R(x) = |x| 2 Q /2 and |.| Q is a norm, We finally give estimates of the kernel K t , for small and large values of t. Combining (2.1) with Lemma 2.2 (iii) and (iv), we have (2.10) For t ≥ 1, we can use the norm |.| Q to write [5,Lemma 3.4] slightly more precisely. The proof of [5,Lemma 3.3] shows that for any w ∈ R n , and this leads to (2.11) For β > 0, let E β be the ellipsoid As in [5,Subsection 4.1], we introduce polar coordinates (s,x) for any point x ∈ R n , x = 0, by writing x = D sx (2.12) withx ∈ E β and s ∈ R.
The Lebesgue measure in R n is given in terms of (s,x) by where dS β denotes the area measure of E β . We refer to [5,Proposition 4.2] for a proof.
For any A > 0 we define global and local regions

On the definition of negative powers of −L
We start recalling the definition of Riesz transforms introduced by Mauceri and Noselli in [21] in a nonsymmetric context. For any nonzero multiindex α ∈ N n , the Riesz transform R (α) , of order |α|, on L 2 (γ ∞ ) is defined as Here the symbol I a denotes for any a > 0 a Gaussian Riesz potential given by Formally, I a corresponds to the negative power − L −a P ⊥ 0 . In fact, the definition of − L −a is the key point in order to define R (α) , since L is not self-adjoint in our general framework. Therefore, we shall now introduce in another way the Gaussian Riesz potentials, and prove the equivalence with (3.1) for a > 0.
In this section, we let L 2 (γ ∞ ) consist of complex-valued funtions. We first recall from [23] that the spectrum of −L is given by where λ 1 , . . . , λ r are the eigenvalues of the drift matrix B. In particular, 0 is an eigenvalue, and the corresponding eigenspace ker L is one-dimensional and consists of all constant functions, as proved in [23,Section 3]. Any other point in the spectrum of −L belongs to a fixed cone {z ∈ C : | arg z| < µ} with µ < π/2, since the same is true for the numbers −λ 1 , . . . , −λ r . We also recall that, given a linear operator L on some L 2 space, a number λ ∈ C is a generalized eigenvalue of L if there exists a nonzero u ∈ L 2 such that (L − λI) k u = 0 for some positive integer k. Then u is called a generalized eigenfunction, and those u span the generalized eigenspace corresponding to λ.
It is known from [23, Section 3] that the Ornstein-Uhlenbeck operator L admits a complete system of generalized eigenfunctions, that is, the linear span of the generalized eigenfunctions is dense in L 2 (γ ∞ ). Analogous L p results are obtained in [23, Theorem 3.1] but will not be used in our paper.
As will be shown in a forthcoming note by the authors [6], each generalized eigenfunction of −L with a nonzero eigenvalue is orthogonal to the space of constant functions, that is, to the kernel of −L . Thus the orthogonal complement of ker L in L 2 (γ ∞ ) coincides with the closure of the subspace generated by all generalized eigenfunctions with a nonzero generalized eigenvalue. We denote this subspace by L 2 0 (γ ∞ ), so that P ⊥ 0 is the orthogonal projection onto L 2 0 (γ ∞ ). The restriction of −L to the generalized eigenspace corresponding to an eigenvalue λ = 0 has the form where N is a nilpotent operator. Then for a ∈ R one would like to write the power (−L ) −a , restricted to the generalized eigenspace, as where the sum is finite and we use the Pochhammer symbol. But here λ ∈ C, and for noninteger a it is not obvious how to choose the value of λ −a . For a > 0 this can be done as follows. We define the argument arg λ to be in (−π/2, π/2). Proof. The restriction mentioned is the sum is again finite and the integral converges.
Here we make a complex change of variables, letting τ = tλ. We arrive at a complex integral where R λ is the ray t e i arg λ , t ∈ R + , going from 0 to ∞; also τ a−1 = |τ | a−1 e i(a−1) arg λ and λ −a = |λ| −a e −ia arg λ . It is not hard to see that we can move the integration to the positive real axis, getting This proves the proposition.
In [21,Lemma 2.2] it is proved that for each complex number a such that ℜa > 0 the Riesz potential (−L ) −a P ⊥ 0 is bounded on L 2 0 (γ ∞ ). Thus (−L ) −a P ⊥ 0 is entirely determined by its restrictions to the generalized eigenspaces, given by (3.2). To summarize, this means that by using these restrictions and taking a limit, we get a definition of (−L ) −a for a > 0, which is equivalent to (3.1).
Finally, let us comment on the fact that −L has real coefficients, although with Thus for nonreal λ, the generalized eigenspaces come in isomorphic pairs, and the isomorphism is complex conjugation. Conjugating the relation −L u = λ(I + N)u, we see that the restriction to the conjugate generalized eigenspace is given byλ(I +N ). The restriction of (−L ) −a to the sum of the two conjugate generalized eigenspaces is a real operator. Indeed, this sum is spanned by the functions ℜu and ℑu, with u in the generalized eigenspace with eigenvalue λ. For these real functions, we can use the expression (3.2) for (−L ) −a with a > 0, since and similarly for ℑu = −i(u −ū)/2. Thus we see that (−L ) −a P ⊥ 0 is a real operator.

Riesz transforms
We start this section with some technical lemmata.
The following lemma provides a different expression for P j .
3) and the expression for P j in (4.1), we get As a consequence of (4.1), Lemma 2.2 and Lemma 4.2, one has for all j ∈ {1, . . . , n} These integrals are absolutely convergent for all u = x, as seen from (2.10), (4.5), (4.6) and Lemma 2.1. In order to distinguish between small and large values of t, we split the integrals as and The proof of the next proposition is straightforward and so omitted. and The following estimates for R j,0 and R ij,0 result from (2.10), (4.5) and (4.6)

Some estimates for large t
In this section, we derive some estimates for R j,∞ and R ij,∞ , after some preparations.
This yields (5.1) in the case R(x) ≤ 1. Next, assume R(x) > 1. Then the integral is split at the point t 1 defined by Here we apply the polar coordinates (2.12) with β = R(x), writing x =x and u = D suũ , where s u ∈ R and R(ũ) = R(x). Then for 1 < t < t 1

and [5, Lemma 4.3 (ii)] applies, saying that
We conclude from (5.2) that For t > 1 ∨ t 1 , we deduce from Lemma 2.1 that We have proved Lemma 5.1.
Proof. We can delete the factor D −t u − x σ 1 in the integrand, by replacing the coefficient 1/2 of the exponential factor in K t by 1/4. Then this follows from Lemma 5.1.
For all x, u ∈ R n and for all i, j, k ∈ {1, . . . , n} the following estimates hold: Proof. It is enough to combine (2.11), Lemma 5.1 and Proposition 5.2 with (4.5) and (4.6). The quantities |D −t u − x| in the factors P j can be replaced by 1, because of the exponential factor in K t .
Then we can use a small part of the factor to get an extra factor exp (−c |ũ −x| 2 ) in the righthand sides of all the estimates in Lemma 5.1, Proposition 5.2, Corollary 5.3 and Proposition 5.4.

Some reductions and simplifications
This section is closely similar to Section 5 in [5]. When we prove (1.2) and (1.3), it is enough to take f ≥ 0 satisfying f L 1 (γ∞) = 1. From now on, we also assume that λ > 2, since otherwise (1.2) and (1.3) are obvious. The dγ ∞ measure of the set of points x satisfying R(x) > 2 log λ is so this set can be neglected in (1.2) and (1.3).
If also (x, u) ∈ G 1 , the same estimates hold for R j,0 and R ij,0 .
Proof. The first statement follows immediately from Proposition 5.4.
From (4.7) and (4.8) we see that both R j,0 (x, u) and R ij,0 (x, u) can be estimated by a sum of expressions of type where p, q ≥ 0. If here we integrate only over those t ∈ (0, 1) satisfying the first inequality in (6.1), we get at most for some C. For the remaining t, the second inequality of (6.1) holds, and the corresponding part of the integral is no larger than Obviously e R(x) (1 + |x|) C λ when R(x) < 1 2 log λ, and the proposition is proved. As a result of this section, we need only consider points x in the ellipsoidal annulus In particular, the operators with kernels R j,∞ and R ij,∞ are of weak type (1, 1) with respect to the invariant measure dγ ∞ .
Notice here that the estimate for R j,∞ is sharpened by a logarithmic factor. A similar phenomenon occurs for the related maximal operator; see [5].
Proof. Having fixed λ > 2, we use our polar coordinates with β = log λ and write x = D sx and u = D suũ , wherex,ũ ∈ E β . We restrict x to the annulus E λ , in view of Section 6. It is easily seen that this restriction is possible also with the logarithmic factor in the case of R j,∞ . Applying the estimates (5.6) and (5.7), we insert a factor exp (−c |ũ −x| 2 ), which is possible because of Remark 5.5. We also replace the factor 1 + |x| in (5.7) by √ log λ .
With σ ∈ {1, 2} we thus need to control the measure of the set The following lemma ends the proof of Proposition 7.1.
In view of (2.5), the function is strictly increasing in s. We conclude that the inequality holds if and only if s > s λ (x) for some functionx → s λ (x) ≤ ∞, with equality for s = s λ (x) < ∞. Notice also that if the point x = D sx is in A 2 (λ) and thus in E λ , then |s| < C because of Lemma 2.1. We use (2.13) to estimate the dγ ∞ measure of A 2 (λ). Since s stays bounded and |x| ≃ √ log λ, we obtain where the last inequality follows from (2.5), because |D sx | 2 ≃ log λ for |s| < C. Now integrate in s, to get We combine this estimate with the case of equality in (7.2) and change the order of integration, concluding that which proves Lemma 7.2.

The local case
In this section we define and estimate the local parts of the Riesz operators of orders 1 and 2.
Let η be a positive smooth function on R n × R n , such that η(x, u) = 1 if (x, u) ∈ L A and η(x, u) = 0 if (x, u) / ∈ L 2A , for some A ≥ 1. Here A will be determined later, in a way that depends only on n, Q and B. We can assume moreover that We introduce the global and local parts of the first-order Riesz transform R j by . For the second-order Riesz transforms, we simply repeat the above with the subscript j replaced by ij.
To prove the weak type (1, 1) of the operators R loc j and R loc ij , we shall verify that their kernels R j η and R ij η satisfy the standard Calderón-Zygmund estimates.
We first need a lemma.
where the constant C(δ, p, r) may also depend on n, Q, B and A.
Proof. Write Since |x − D t x| ≃ t|x| and (x, u) ∈ L 2A , the absolute value of the last term here is no larger than CAt. It follows that |u − D t x| 2 /t ≥ |u − x| 2 /t + ct|x| 2 − CA.
We now apply this to the integral in the lemma, and estimate exp (−cδt|x| 2 ) by Cδ −r/2 t −r/2 |x| −r . The integral is thus controlled by and the required estimate follows via the change of variables s = |u − x| 2 /t.

Proposition 8.2.
Let the function η be as above. For all (x, u) ∈ L 2A , x = u, and all j ∈ {1, . . . , n}, the following estimates hold: with implicit constants depending on n, B, Q and A. The same estimates hold for R ij , i, j ∈ {1, . . . , n}.
Proof. We start with R j .
(2) As a consequence of (8.1), one has for x = u Since item (1) above takes care of the last term here, it suffices to show that For R j,0 we get from (4.1), (4.3), (4.5), (4.6), combined with (2.10) In the last factor here, we can replace |u − D t x| by √ t, reducing slightly the factor 1/2 in the exponential expression. This will be done repeatedly in the sequel. We arrive at and Lemma 8.1 allows us to estimate this by e R(x) |u − x| −(n+1) as desired. For R j,∞ (5.4) and (5.5) imply that Here 1 + |x| |x − u| −1 |x − u| −(n+1) , and (8.2) is verified, proving (ii) as well.
(3) As in item (2), it suffices to estimate |∂ u ℓ R j (x, u)|. Because of (4.2), (4.4) and (4.5), we have where we proceeded much as in item (2). Similarly, as follows from Proposition 5.2. Items (i), (ii) and (iii) are proved for R j , and we now turn to R ij .
(2') As before, we need only consider the derivative ∂ x ℓ R ij (x, u) in the local region. From (4.1) and (4.3), we have For 0 < t < 1, we estimate the factors of type P i and ∆ ij here by means of (4.5) and (4.6). Then we use the exponential factor in K t to replace |u − D t x| by √ t, and apply Lemma 8.1. The result will be For t > 1, we use (4.5) and (4.6), getting because of Lemma 5.2.
(3') Applying (4.2) and (4.4), we have x, u)P j (t, x, u) + ∆ ji (t)) + Q −1 t e tB e i , e ℓ P j (t, x, u) + P i (t, x, u) Q −1 t e tB e j , e ℓ dt. Arguing as before, we conclude Further, the last step from Proposition 5.2. This completes the proof of Proposition 8.2.
We can now prove the weak type (1, 1) boundedness of the local parts of the Riesz transforms. In this section, we study the operators R glob j,0 and R glob ij,0 , with kernels 1 − η)R j,0 and 1 − η)R ij,0 , respectively. The function η was defined in the beginning of Section 8, depending on A. We have the following result. The estimates (4.7) and (4.8) show that to prove this proposition, it suffices to verify the weak type (1, 1) boundedness of the operator with kernel As is clear from Section 6, we need only consider the case |x| 1. This assumption will be valid in the rest of this section.

The sets
and together form a partition of (0, 1). For t ∈ I m (x, u),

The operator we need to consider has kernel
The operator whose kernel is Q m is of weak type (1, 1) with respect to dγ ∞ , with a quasinorm bounded by C 2 Cm for some C.
This proposition implies Proposition 9.1, since the factors exp (−c 2 2m ) in (9.1) will allow us to sum over m in the space L 1,∞ (γ ∞ ). Before proving Proposition 9.2, we make some preparations.
From now on, we fix m ∈ {0, 1, . . . }. If t ∈ I m (x, u), Lemma 2.3 implies and further If A is chosen large enough, depending only on n, Q and B, then Proof. If t ∈ I m (x, u) but t ≤ 2 −2m /|x| 2 , the two terms to the right in (9.2) are both bounded by 1/|x|, so that |x − u| < C/|x| for some C. Since we assume |x| 1, this will violate the hypothesis (x, u) ∈ G A , if A is large. The lemma follows. Proof. Because of our assumptions on t, (9.2) implies that |u − x| t|x| |x|, and so |u| |x|. This proves one of the inequalities in both (9.4) and (9.6). Aiming at (9.5), we write .
and (9.3) and our assumptions lead to Thus we can choose C 0 > 4 so large that |R(D t x)−R(u)| < (R(D t x)−R(x))/2, and the first estimate in (9.5) follows. In particular, R(u) > R(x), and so |u| |x|, which completes the proof of (9.4). We also obtain the remaining part of (9.6), by writing Finally, the second estimate in (9.5) is a trivial consequence of (9.6).
The lemma is proved.
Proof. For t < C 0 2 2m /|x| 2 , the estimate (9.2) implies which leads to Consider first the case |x| ≤ C 0 2 m . Then Q − m (x, u) e R(x) 2 Cm , and so Since this is uniform in u, the strong type follows for |x| < C 0 2 m .
To deal with points x with |x| > C 0 2 m , we introduce dyadic rings the last step since C 0 > 4. The triangle inequality now shows that u is in the extended ring where Ψ is given by Since Ψ(u) du 2 Cm , we can integrate in x to get Summing over i ≥ 0, we get The lemma follows.
Lemma 9.6. The operator with kernel Q + m is of weak type (1, 1) with respect to dγ ∞ , and its quasinorm is bounded by C 2 Cm .
Proof. The support of the kernel Q + m is contained in the set C m = (x, u) ∈ G A : I + m (x, u) = ∅ . We first sharpen (9.6) by restricting t further. Because of (9.3), (9.4) and (9.6), any and from (2.5) we know that ∂ t R(D t x) ≃ |x| 2 .
The size of this derivative shows that (9.8) can hold only for t in an interval of length at most C 2 m |x − u|/|x| 3/2 , call it I. We obtain, using (9.6) again, The global condition implies |x|/|x − u| |x| 2 , so that It will be enough to prove Lemma 9.6 with Q + m replaced by the kernel M m thus defined. With λ > 2 fixed, we assume x ∈ E λ . We use our polar coordinates with β = (log λ)/2, writing x = D sx and u = D suũ , wherex,ũ ∈ E β and s ≥ 0, s u ∈ R. If (x, u) ∈ C m , we take t ∈ I + m (x, u) and observe that R(D t x) > R(x) ≥ β. Then [5,Lemma 4.3 (i)] can be applied, giving the last step because of (9.6). We shall cover the ellipsoid E β with little caps, and start with E 1 . The small number δ > 0 will be specified below, depending only on n, Q and B. Define for e ∈ E 1 the cap Ω 1 e = E 1 ∩ B(e, δ). We cover E 1 with caps Ω 1 e with e ranging over a finite subset of E 1 , in such a way that the doubled capsΩ 1 e = E 1 ∩ B(e, 2δ) have C-bounded overlap. Since E β = √ β E 1 , we can scale these caps to get caps Ω β e = β Ω 1 e = E β ∩ B β e, β δ covering E β . Similarly,Ω β e = √ βΩ 1 e . For each x ∈ E λ , the pointx will belong to some cap Ω β e of the covering. In the proof of Lemma 9.6 we need only consider those u for whichũ is in the doubled cap Ω β e . The reason is that ifũ / ∈Ω β e , then |ũ −x| √ β δ ≃ |x|, and [5, Lemma 4.3 (i)] implies |u − D t x| |x| and also |u − x| |x|. This and the definition of I m (x, u) lead to |x| 2 m √ t and thus 1 + |x| 2 m . It follows that Since the last expression is independent of u and has integral this part of the kernel M m defines an operator which is of strong type (1, 1), with the desired bound. Thus we fix a cap Ω β e , assuming thatx ∈ Ω β e andũ ∈Ω β e . By means of a rotation, we may also assume that e is on the positive x 1 axis. Then we writex asx = (x 1 ,x ′ ) ∈ R × R n−1 , and similarlyũ = (ũ 1 ,ũ ′ ). If δ is chosen small enough, we will then have |x −ũ| ≃ |x ′ −ũ ′ |, (9.10) essentially because the x 1 axis is transversal to E β at the point of intersection √ β e. Further, the area measure dS β of E β will satisfy dS β (ũ) ≃ dũ ′ (9.11) inΩ β e , again if δ is small. We now recall Proposition 8 in [20]. This proposition is also applied in another framework in [3].
Applying (9.10), (9.9) and (9.12), we obtain This allows us to estimate M m in terms of the coordinates ξ and η : We must also express the Lebesgue measures dx and du in terms of ξ and η , with x and u restricted as before. By (2.13), dx ≃ e −s tr B |x| ds dS β (x) ≃ log λ ds dS β (x), the last step since x ∈ E λ implies s 1. Similarly, du ≃ √ log λ ds u dS β (ũ).
Lemmata 9.5 and 9.6 together imply Proposition 9.2 and also Proposition 9.1.
The off-diagonal kernel of R (α) is K t being the Mehler kernel as in (2.1). Repeated application of (4.1) in Lemma 4.1 implies that the derivative D α x K t (x, u) is a sum of products of the form K t (x, u) P (t, x, u) Q(t), where P (t, x, u) is a product of factors of type P j (t, x, u), and Q(t) is a product of factors of type ∆ ij (t). Since ∆ ij (t) does not depend on x, there will be nothing more. More precisely, consider a term in this sum where the derivatives falling on K t (x, u) are given by a multiindex κ, with κ ≤ α in the sense of componentwise inequalities. Then |α| − |κ| differentiations must fall on the P j (t, x, u) factors, and necessarily |α| − |κ| ≤ |κ|. This tells us that Q(t) must consist of |α| − |κ| factors and also that P (t, x, u) consists of N := |κ| − (|α| − |κ|) factors. It follows that |α| − |κ| = (|α| − N)/2. Thus we get products where the superscripts indicate the number of factors. Since |κ| can be any integer satisfying |α|/2 ≤ |κ| ≤ |α|, we see that N runs over the set of integers in [0, |α|] congruent with |α| modulo 2.
With η > 0 large, define where we mean the product of the matrix Q ∞ and (η, . . . , η) written as a column vector. Our f will be δ u 0 , and we will verify that the L 1,∞ (γ ∞ ) quasinorm of R (α) f tends to +∞ with η. Since δ u 0 can be approximated by L 1 functions in a standard way, this will disprove the weak type (1, 1) estimate for R (α) . We have For reasons that will become clear below, we fix a number t 0 ∈ (0, 1/2), independent of η and so small that (1, 1, . . . , 1), e t 0 B e j > 1/2, j = 1, . . . , n.
We are going to evaluate R (α) δ u 0 (x) when x is in the ball B x 0 , √ t 0 . Then we will have |x| ≃ |x 0 | ≃ |u 0 | ≃ η. From (2.10) we get an estimate of K t (x, u 0 ) for 0 < t < 1. There we want the exponent |u 0 − D t x| 2 /t to stay bounded when x ∈ B x 0 , √ t 0 and t is close to t 0 . Write which we must then make smaller than constant times √ t ≃ √ t 0 . Here because of Lemma 2.3. Thus we take t with |t − t 0 | < √ t 0 /|u 0 |, which implies t ≃ t 0 for large enough η. Further, |D t (x 0 − x)| ≃ |x 0 − x| < √ t 0 . Then |u 0 − D t x| √ t, and it follows that The first summand here is for for large η, by the definition of u 0 . Because of (10.3), this leads to and we observe that Q −1 ∞ x, e j does not depend on t. Next, we rewrite the product (10.2) by using (10.5) to expand the factor P (N ) . We will then get a sum of terms like (10.2) but where P (N ) is replaced by a product of powers of the two summands in (10.5). For N = |α| one of the terms in this sum will be the inequality coming from (10.6). Since N = |α|, the corresponding factor Q ((|α|−N )/2) (t) is 1. The positive quantity in (10.7) will give the divergence we need for the counterexample. We have to estimate the absolute values of all the other terms.
Thus each of the terms we must estimate is controlled by an expression of type We are now ready to estimate the integral in (10.1), at first taken only over the interval (0, 1). Here u = u 0 and x ∈ B x 0 , √ t 0 . The positive term described in (10.7) will, because of (10.4), give a contribution which is larger than a constant c times In order to estimate this integral, we write, recalling that D t x 0 = D t−t 0 u 0 , The first summand here satisfies for 0 < t < 1, in view of Lemma 2.3, |u 0 − D t−t 0 u 0 | ≃ |t − t 0 | |u 0 |, and the second summand is controlled by √ t 0 . Thus if |t − t 0 | > C/|u 0 | for some large C, we will have |u 0 − D t x| ≥ |t − t 0 | |u 0 | ≃ 1 + |t − t 0 | |u 0 |, so that |u 0 − D t x| 2 t This implies that e R(x) |x| N 1 What remains is Since N 1 < |α|, the last expression is less than e R(x) |x| |α|−2 , and we see that for large η the positive expression in (10.10) dominates over the effects of the other terms. We finally treat the integral over t > 1. For x ∈ B x 0 , √ t 0 and t > 1, (2.11), (4.5) and (4.6) imply the following three estimates |P j (t, x, u 0 )| e −ct |D −t u 0 − x| + |D −t u 0 | and |∆ ij (t)| e −ct . We can delete the factor |D −t u 0 − x| from the second of these formulas, if we reduce slightly the coeffient 1/2 in the first formula. Further, (10.11) An argument like (2.8) now leads to |D t 0 −t x 0 − x 0 | |x|, because here t 0 − t < −1/2 and so (2.5) implies that |x 0 | 2 Q −|D t 0 −t x 0 | 2 Q ≃ |x 0 | 2 Q . Since |x 0 −x| is much smaller than |x|, we conclude from (10.11) that |D −t u 0 − x| ≃ |x|. Moreover, |D −t u 0 | e −ct |u 0 | ≃ e −ct |x| by Lemma 2.1. Estimating the products in (10.2), we arrive at |D α x K t (x, u 0 )| e R(x) exp (−c |x| 2 ) e −ct |x| C , t > 1.