On the maximal operator of a general Ornstein-Uhlenbeck semigroup

If $Q$ is a real, symmetric and positive definite $n\times n$ matrix, and $B$ a real $n\times n$ matrix whose eigenvalues have negative real parts, we consider the Ornstein--Uhlenbeck semigroup on $\mathbb{R}^n$ with covariance $Q$ and drift matrix $B$. Our main result says that the associated maximal operator is of weak type $(1,1)$ with respect to the invariant measure. The proof has a geometric gist and hinges on the"forbidden zones method"previously introduced by the third author.


Introduction
In this paper we prove a weak type (1, 1) theorem for the maximal operator associated to a general Ornstein-Uhlenbeck semigroup. We extend the proof given by the third author in 1983 in a symmetric context. Our setting is the following.
In R n we will consider the semigroup generated by the elliptic operator or, equivalently, where ∇ is the gradient and ∇ 2 the Hessian. Here Q = (q ij ) is a real, symmetric and positive definite n × n matrix, indicating the covariance of L. The real n × n matrix B = (b ij ) is negative in the sense that all its eigenvalues have negative real parts, and it gives the drift of L. The semigroup is formally H t = e tL , t > 0, but to write it more explicitly we first introduce the positive definite, symmetric matrices Q t = t 0 e sB Qe sB * ds, 0 < t ≤ +∞, (1.1) and the normalized Gaussian measures in R n γ t , with t ∈ (0, +∞], having density with respect to Lebesgue measure. Then for functions f in the space of bounded continuous functions in R n one has a formula due to Kolmogorov. The measure γ ∞ is invariant under the action of H t ; it will be our basic measure, replacing Lebesgue measure. We remark that H t t>0 is the transition semigroup of the stochastic process where W is a Brownian motion in R n with covariance Q.
We are interested in the maximal operator defined as Under the above assumptions on Q and B, our main result is the following.
Theorem 1.1. The Ornstein-Uhlenbeck maximal operator H * is of weak type (1, 1) with respect to the invariant measure γ ∞ , with an operator quasinorm that depends only on the dimension and the matrices Q and B.
In other words, the inequality holds for all functions f ∈ L 1 (γ ∞ ), with C = C(n, Q, B).
For large values of the time parameter, we also obtain a refinement of this result. Indeed, we prove in Proposition 6.1 that for large α > 0 and all normalized functions f ∈ L 1 (γ ∞ ). Here C = C(n, Q, B), and this estimate is shown to be sharp. It cannot be extended to H * , since the maximal operator corresponding to small values of t only satisfies the ordinary weak type inequality. This sharpening is not surprising, in the light of some recent results for the standard case Q = I and B = −I by Lehec [8]. He proved the following conjecture, recently proposed by Ball, Barthe, Bednorz, Oleszkiewicz and Wolff [2]: For each fixed t > 0, there exists a function ψ t = ψ t (α), with lim for all large α > 0 and all f ∈ L 1 (γ ∞ ) such that f L 1 (γ∞) = 1. Lehec proved this conjecture with ψ t (α) = C(t)/ √ log α independent of the dimension, and this ψ t is sharp. Our estimates depend strongly on the dimension n, but on the other hand we estimate the supremum over large t.
The history of H * is quite long and started with the first attempts to prove L p estimates. When H t t>0 is symmetric, i.e., when each operator H t is self-adjoint on L 2 (γ ∞ ), then H * is bounded on L p (γ ∞ ) for 1 < p ≤ ∞, as a consequence of the general Littlewood-Paley-Stein theory for symmetric semigroups of contractions on L p spaces [16,Ch. III].
It is easy to see that the maximal operator is unbounded on L 1 (γ ∞ ). This led, about fifty years ago, to the study of the weak type (1, 1) of H * with respect to γ ∞ . The first positive result is due to B. Muckenhoupt [13], who proved the estimate (1.3) in the one-dimensional case with Q = I and B = −I. The analogous question in the higher-dimensional case was an open problem until 1983, when the third author [15] proved the weak type (1, 1) in any finite dimension. Other proofs are due to Menárguez, Pérez and Soria [11] (see also [10,14]) and to Garcìa-Cuerva, Mauceri, Meda, Sjögren and Torrea [7]. Moreover, a different proof of the weak type (1, 1) of H * , based on a covering lemma halfway between covering results by Besicovitch and Wiener, was given by Aimar, Forzani and Scotto [1]. A nice overview of the literature may be found in [17,Ch.4].
In [4] the present authors recently considered a normal Ornstein-Uhlenbeck semigroup in R n , that is, we assumed that H t is for each t > 0 a normal operator on L 2 (γ ∞ ). Under this extra assumption, we proved that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure γ ∞ . This extends earlier work in the non-symmetric framework by Mauceri and Noselli [9], who proved some ten years ago that, if Q = I and B = λ(R − I) for some positive λ and a real skew-symmetric matrix R generating a periodic group, then the maximal operator H * is of weak type (1, 1).
In Theorem 1.1 we go beyond the hypothesis of normality. The proof has a geometric core and relies on the ad hoc technique developed by the third author in [15]. It is worth noticing that, while the proof in [4] required an analysis of the special case when Q = I and B = (−λ 1 , . . . , −λ n ), with λ j > 0 for j = 1, . . . , n, and then the application of factorization results, we apply here directly, avoiding many intermediate steps, the "forbidden zones" technique introduced in [15].
Since the maximal operator H * is trivially bounded from L ∞ to L ∞ , we obtain by interpolation the following corollary.
This result improves Theorem 4.2 in [9], where the L p boundedness of H * is proved for all p > 1 in the normal framework, under the additional assumption that the infinitesimal generator of H t t>0 is a sectorial operator of angle less than π/2.
In this paper we focus our attention on the Ornstein-Uhlenbeck semigroup in R n . In view of possible applications to stochastic analysis and to SPDE's, it would be very interesting to investigate the case of the infinite-dimensional Ornstein-Uhlenbeck maximal operator as well (see [5,18,3] for an introduction to the infinite-dimensional setting). The Riesz transforms associated to a general Ornstein-Uhlenbeck semigroup in R n will be considered in a forthcoming paper.
The scheme of the paper is as follows. In Section 2 we introduce the Mehler kernel K t (x, u), that is, the integral kernel of H t . Some estimates for the norm and the determinant of Q t and related matrices are provided in Section 3. As a consequence, we obtain bounds for the Mehler kernel. In Section 4 we consider the relevant geometric features of the problem, and introduce in Subsection 4.1 a system of polar-like coordinates. We also express Lebesgue measure in terms of these coordinates. Sections 5, 6, 7 and 8 are devoted to the proof of Theorem 1.1. First, Section 5 introduces some preliminary simplifications of the proof; in particular, we restrict the variable x to an ellipsoidal annulus. In Section 6 we consider the supremum in the definition of the maximal operator taken only over t > 1 and prove the sharp estimate (1.4). Section 7 is devoted to the case of small t under an additional local condition. Finally, in Section 8 we treat the remaining case and conclude the proof of Theorem 1.1, by proving the estimate (1.3) for small t under a global assumption.
In the following, we use the "variable constant convention", according to which the symbols c > 0 and C < ∞ will denote constants which are not necessarily equal at different occurrences. They all depend only on the dimension and on Q and B. For any two nonnegative quantities a and b we write a b instead of a ≤ Cb and a b instead of a ≥ cb. The symbol a ≃ b means that both a b and a b hold.
By N we mean 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 | · |.

The Mehler Kernel
For t > 0, the difference is a symmetric and strictly positive definite matrix. So is the matrix and we can define where we repeatedly used the fact that Q −1 ∞ − Q −1 t is symmetric. We now express the matrix D t in various ways.
and combining this with (2.5) we arrive at (i).
By means of (i) in this lemma, we can define D t for all t ∈ R, and they will form a one-parameter group of matrices. Now (ii) in Lemma 2.1 yields

Thus (2.4) may be rewritten as
where K t denotes the Mehler kernel, given by Here we introduced the quadratic form

Some auxiliary results
In this section we collect some preliminary bounds, which will be essential for the sequel. Proof. We make a Jordan decomposition of B * , thus writing it as the sum of a complex diagonal matrix and a triangular, nilpotent matrix, which commute with each other. This leads to expressions for e −sB * and e sB * , and since B * like B has only eigenvalues with negative real parts, we see that e −sB * e Cs and e sB * e −cs . (3.1) From (i) in Lemma 2.1, we now get the claimed upper estimates for D ±s . To prove the lower estimate for D s , we write The other parts of the lemma are completely analogous.
In the following lemma, we collect estimates of some basic quantities related to the matrices Q t .
Proof. (i) and (ii) Using (3.1), we see that for each t > 0 and for Since e sB * −1 = e −sB * e Cs , there is also a lower estimate Thus any eigenvalue of Q t has order of magnitude min(1, t), and (i) and (ii) follow.
(iii) From the definition of Q t and (3.1), we get (iv) Using now (ii) and (iii), we have (v) Since A 1/2 = A 1/2 for any symmetric positive definite matrix A, we consider as a consequence of (3.2). Inserting this and the simple estimate Using (2.1) and then the definition of Q ∞ , we observe that the last term can be written as Since Q −1/2 t e tB w 2 |w| 2 for t ≥ 1 by Lemmata 3.1 and 3.2 (ii), the proposition follows.
We finally give estimates of the kernel K t , for small and large values of t. When t ≤ 1, one has (Q −1 Lemma 3.4. For t ≥ 1 and x, u ∈ R n , we have Proof. This follows from (2.6), if we write u − D t x = D t (D −t u − x) and apply Proposition 3.3 with w = D −t u − x.
4. Geometric aspects of the problem 4.1. A system of adapted polar coordinates. We first need a technical lemma.
For all x in R n and s ∈ R, we have Proof. To prove (4.1), we use the definition of Q ∞ to write for any z ∈ R n Setting z = Q −1 ∞ x, we get (4.1). Further, (4.2) easily follows if we observe that Finally, we get by means of (4.2) and (4.1)

and (4.3) is verified.
We observe here that an integration of (4.2) leads to Fix now β > 0 and consider the ellipsoid As a consequence of (4.3), the map s → R(D s z) is strictly increasing for each 0 = z ∈ R n . Hence any x ∈ R n , x = 0, can be written uniquely as for somex ∈ E β and s ∈ R. We consider s andx as the polar coordinates of x.
Our estimates in what follows will be uniform in β.
Next, we shall write Lebesgue measure in terms of these polar coordinates. A normal vector to the surface E β at the pointx ∈ E β is N(x) = Q −1 ∞x , and the tangent hyperplane atx is N(x) ⊥ . For s > 0 the tangent hyperplane of the surface The scalar product of w and the tangent of the curve s → D sx at the point D sx is, because of (4.2) and (4.1), Thus the curve s → D sx is transversal to each surface D s E β . Let dS s denote the area measure of D s E β . Then Lebesgue measure is given in terms of our polar coordinates by where To see how dS s varies with s, we take a continuous function ϕ = ϕ(x) on E β and extend it to R n \ {0} by writing ϕ(D sx ) = ϕ(x). For any t > 0 and small ε > 0, we define the shell Ω t,ε = {D sx : t < s < t + ε,x ∈ E β }.
Then Ω t,ε is the image under D t of Ω 0,ε , and the Jacobian of this map is det D t = e −t tr B . Thus which we can rewrite as Since this holds for any ϕ, it follows that Together with (4.7), this implies the following result.
Proposition 4.2. The Lebesgue measure in R n is given in terms of polar coordinates (t,x) by We also need estimates of the distance between two points in terms of the polar coordinates. The following result is a generalization of Lemma 4.2 in [4], and its proof is analogous.
(ii) If also s (1) ≥ 0, then Proof. Let Γ : [0, 1] → R n \ {0} be a differentiable curve with Γ(0) = x (0) and Γ(1) = x (1) . It suffices to bound the length of any such curve from below by the right-hand sides of (4.8) and (4.9). For each τ ∈ [0, 1], we write The group property of D s implies that and so The vectorx ′ (τ ) is tangent to E β and thus orthogonal to N(x). Then (4.6) (with s = 0) implies that the angle between ∂ ∂s D s s=0x (τ ) andx ′ (τ ) is larger than some positive constant. It follows that where we also used the fact that, by (4.2), because of Lemma 3.1, we obtain from (4.11) Next, we derive a lower bound for s(0); assume first that s(0) < 0. The assumption R(x (0) ) > β/2 implies, together with Lemma 3.1, It follows that s(0) > −s, for somes with 0 <s < C, and this obviously holds also without the assumption s(0) < 0.

4.2.
The Gaussian measure of a tube. We fix a large β > 0. Define for x (1) ∈ E β and a > 0 the set This is a spherical cap of the ellipsoid E β , centered at x (1) . Observe that |x| ≃ √ β for x ∈ Ω, and that the area of Ω is |Ω| ≃ min (a n−1 , β (n−1)/2 ). Then consider the tube Z = {D sx : s ≥ 0,x ∈ Ω}. (4.13) Lemma 4.4. There exists a constant C such that β > C implies that the Gaussian measure of the tube Z fulfills Proof. Assuming β large enough, one has cβ > −2 tr B, and then the last integral is finite and no larger than C/β. The lemma follows.

Some simplifications
In this section, we introduce some preliminary simplifications and reductions in the proof of (1.3), i.e., of Theorem 1.1.
(1) We may assume that f is nonnegative and normalized in the sense that since this involves no loss of generality. (2) We may assume that α is large, α > C, since otherwise (1.3) and (1.4) are trivial.
(3) In many cases, we may restrict x in (1.

3) and (1.4) to the ellipsoidal annulus
To begin with, we can always forget the unbounded component of the complement of E α , since (4) When t > 1, we may forget also the inner region where R(x) < 1 2 log α.
since α is large. In other words, for any ( for all t > 1. Replacing α by Cα for some C, we see from (5.1) and (5.2) that we can assume x ∈ E α in the proof of (1.3) and (1.4), when the supremum of the maximal operator is taken only over t > 1. Before introducing the last simplification, we need to define a global region G = (x, u) ∈ R n × R n : |x − u| > 1 1 + |x| and a local region Notice that the definition of G and L does not depend on Q and B.
(5) When t ≤ 1 and (x, u) ∈ G, we shall see that (5.2) is still valid, and it is again enough to consider x ∈ E α . To prove this, we need a lemma which will also be useful later.
Proof. From the definition of G and (4.4) we get The lemma follows.
To verify now (5.2) in the global region with t ≤ 1, we recall from (3.3) that It follows from Lemma 5.1 that 3) The first inequality here implies that and (5.2) follows. If the second inequality of (5.3) holds, we have and we get the same estimate. Thus (5.2) is verified.
Finally, let and

The case of large t
In this section, we consider the supremum in the definition of the maximal operator taken only over t > 1, and we prove (1.4).
In particular, the maximal operator is of weak type (1, 1) with respect to the invariant measure γ ∞ .
Proof. We can assume that f ≥ 0. Looking at the arguments in Section 5, items (3) and (4), we see that it is suffices to consider points x ∈ E α . For both x and u we use the coordinates introduced in (4.5) with β = log α, that is, wherex,ũ ∈ E log α and s, s ′ ∈ R. From (3.4) we have for t > 1 and x, u ∈ R n . Since x ∈ E α and D −t u = D s ′ −tũ , we can apply Lemma 4.3 (i), getting In view of (4.3), the right-hand side here is strictly increasing in s, and therefore the inequality holds if and only if s > s α (x) for some functionx → s α (x), with equality for s = s α (x). Since α > 2 and f L 1 (γ∞) = 1, it follows that s α (x) > 0. For some C, the set of points x ∈ E α where the supremum in (6.1) is larger than Cα is contained in the set A(α) of points D sx ∈ E α fulfilling (6.2). We use Proposition 4.2 to estimate the γ ∞ measure of this set. Observe that H(0,x) ≃ |x| ≃ √ log α and that D sx ∈ E α implies s 1, so that also e −s tr B 1. We get where the last inequality follows from (4.3), since |D sx | 2 |x| 2 ≃ log α. Integrating in s, we obtain Now combine this estimate with the case of equality in (6.2) and change the order of integration, to get which proves Proposition 6.1.
Finally, we show that the factor 1/ √ log α in (6.1) is sharp.
Proposition 6.2. For any t > 1 and any large α, there exists a function f , normalized in L 1 (γ ∞ ) and such that Proof. Take a point z with R(z) = log α, and let f be (an approximation of) a Dirac measure at the point u = D t z. Then, as a consequence of (3.4), K t (x, u) ≃ exp(R(x)) in the ball B(D −t u, 1) = B(z, 1). We then have H t f (x) = K t (x, u) α in the set B = {x ∈ B(z, 1) : R(x) > R(z)}, whose measure is 7. The local case for small t Proposition 7.1. If (x, u) ∈ L and 0 < t ≤ 1, then Proof. In view of (3.3), it is enough to show that By (4.4), |u − x| |x − D t x| |u − x| t |x| ≤ t since (x, u) ∈ L, and (7.1) follows.
Proposition 7.2. The maximal operator H L * is of weak type (1, 1) with respect to the invariant measure γ ∞ .
Proof. The proof is standard, since Proposition 7.1 implies The supremum here defines an operator of weak type (1, 1) with respect to Lebesgue measure in R n . From this the proposition follows, cf. [7, Section 3].

The global case for small t
In this section, we conclude the proof of Theorem 1.1.
Proposition 8.1. The maximal operator H G * is of weak type (1, 1) with respect to the invariant measure γ ∞ . and that the B (ℓ) are pairwise disjoint. This would imply and thus also (8.2) and Proposition 8.1.
The sets B (ℓ) and Z (ℓ) will be introduced by means of a sequence of points x (ℓ) , ℓ = 1, . . . , ℓ 0 , which we define by recursion. To start, we choose as x (1) a point where the quadratic form R(x) takes its minimal value in the compact set However, should this set be empty, (8.2) is immediate. We now describe the recursion to construct x (ℓ) for ℓ ≥ 2. Like x (1) , these points will satisfy sup ε≤t≤1 K m t (x (ℓ) , u) f (u) dγ ∞ ≥ α.
All the x (ℓ) will be minimizing points of R(x). To avoid having them too close to one another, we will not allow x (ℓ) to be in any Z (ℓ ′ ) with ℓ ′ < ℓ. More precisely, assuming x (1) , . . . , x (ℓ) already defined, we will choose x (ℓ+1) as a minimizing point of R(x) in the set provided this set is nonempty. But if A ℓ+1 (α) is empty, the process stops with ℓ 0 = ℓ and (8.4) follows. We will see that this actually occurs for some finite ℓ. Now assume that A ℓ+1 (α) = ∅. In order to assure that a minimizing point exists, we must verify that A ℓ+1 (α) is closed and thus compact, although the Z (ℓ ′ ) are not open. To do so, observe that for 1 ≤ ℓ ′ ≤ ℓ, the minimizing property of x (ℓ ′ ) means that there is no point in A ℓ ′ (α) with R(x) < R(x (ℓ ′ ) ). Thus we have the inclusions

It follows that
A ℓ+1 (α) = A ℓ+1 (α) ∩ 1≤ℓ ′ ≤ℓ {x : R(x) ≥ R(x (ℓ ′ ) )} = ℓ We next verify that the sequence (x (ℓ) ) is finite. For ℓ < ℓ ′ , we have (8.11), and Lemma 4.3 (i) implies Since t ℓ ≥ ε, we see that the distance x (ℓ ′ ) − x (ℓ) is bounded below by a positive constant. But all the x (ℓ) are contained in the bounded set E α , so they are finite in number. Thus the set considered in (8.7) must be empty for some ℓ, and the recursion stops. This implies (8.4). We finally prove (8.5) . Observe that the forbidden zone Z (ℓ) is a tube as defined in (4.13), with a = A 2 3m √ t ℓ and β = R(x (ℓ) ). This value of β is large since x (ℓ) ∈ E α , and thus we can apply Lemma 4.4 to obtain We bound the exponential here by means of (8.9) and observe that R(x (ℓ) ) ∼ |x (ℓ) | 2 , getting As a consequence of (8.8), we obtain proving (8.5). This concludes the proof of Proposition 8.1.