Abstract
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.
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1 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 \(\mathbb {R}^n\) we will consider the semigroup generated by the elliptic operator
or, equivalently,
where \(\nabla \) is the gradient and \(\nabla ^2\) the Hessian. Here \(Q = (q_{ij})\) is a real, symmetric and positive definite \(n\times n\) matrix, indicating the covariance of \(\mathcal L\). The real \(n\times n\) matrix \(B = (b_{ij})\) is negative in the sense that all its eigenvalues have negative real parts, and it gives the drift of \(\mathcal L\).
The semigroup is formally \( \mathcal H_t=e^{t\mathcal L}\), \({t> 0}\), but to write it more explicitly we first introduce the positive definite, symmetric matrices
and the normalized Gaussian measures \(\gamma _t \) in \(\mathbb {R}^n\), with \(t\in (0,+\infty ]\), having density
with respect to Lebesgue measure. Then for functions f in the space of bounded continuous functions in \(\mathbb {R}^n\) one has
a formula due to Kolmogorov. The measure \(\gamma _\infty \) is invariant under the action of \(\mathcal H_t\); it will be our basic measure, replacing Lebesgue measure.
We remark that \(\big ( \mathcal H_t \big )_{t> 0}\) is the transition semigroup of the stochastic process
where W is a Brownian motion in \(\mathbb {R}^n\) 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 \(\mathcal H_*\) is of weak type (1, 1) with respect to the invariant measure \(\gamma _\infty \), 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 \(f\in L^1 (\gamma _\infty )\), 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 \(\alpha >0\) and all normalized functions \(f\in L^1 (\gamma _\infty )\). Here \(C=C(n,Q,B)\), and this estimate is shown to be sharp. It cannot be extended to \(\mathcal H_*\), 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, proposed by Ball, Barthe, Bednorz, Oleszkiewicz and Wolff [2]:
For each fixed \(t>0\), there exists a function \(\psi _t=\psi _t(\alpha )\), with \(\displaystyle \lim _{\alpha \rightarrow +\infty } \psi _t (\alpha )=0\), satisfying
for all large \(\alpha >0\) and all \(f\in L^1(\gamma _\infty )\) such that \( \Vert f\Vert _{L^1( \gamma _\infty )}=1\). Lehec proved this conjecture with \(\psi _t (\alpha )={C(t)}/{\sqrt{\log \alpha }}\) independent of the dimension, and this \(\psi _t\) 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 \(\mathcal H_*\) is quite long and started with the first attempts to prove \(L^p\) estimates. When \(\big ( \mathcal H_t \big )_{t> 0}\,\) is symmetric, i.e., when each operator \(\mathcal H_t \) is selfadjoint on \(L^2 (\gamma _\infty )\), then \(\mathcal H_*\) is bounded on \(L^p (\gamma _\infty )\) for \(1<p\le \infty \), 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 \(L^1 (\gamma _\infty )\). This led, about fifty years ago, to the study of the weak type (1, 1) of \(\mathcal H_*\) with respect to \(\gamma _\infty \). The first positive result is due to B. Muckenhoupt [13], who proved the estimate (1.3) in the onedimensional case with \(Q=I\) and \(B=I\). The analogous question in the higherdimensional 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 GarcìaCuerva, Mauceri, Meda, Sjögren and Torrea [7]. Moreover, a different proof of the weak type (1, 1) of \(\mathcal H_*\), 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 \(\mathbb {R}^n\), that is, we assumed that \(\mathcal H_t\) is for each \(t> 0\) a normal operator on \(L^2 (\gamma _\infty )\). Under this extra assumption, we proved that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure \(\gamma _\infty \). This extends earlier work in the nonsymmetric framework by Mauceri and Noselli [9], who proved that if \(Q=I\) and \(B=\lambda (RI)\) for some positive \(\lambda \) and a real skewsymmetric matrix R generating a periodic group, then the maximal operator \( \mathcal H_* \) 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 \(B=(\lambda _1, \ldots , \lambda _n)\), with \(\lambda _j>0\) for \(j=1, \ldots , n\), 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 \(\mathcal H_* \) is trivially bounded from \(L^{\infty }\) to \(L^{\infty }\), we obtain by interpolation the following corollary.
Corollary 1.2
The Ornstein–Uhlenbeck maximal operator \(\mathcal H_*\) is bounded on \(L^p (\gamma _\infty )\) for all \(p>1\).
This result improves Theorem 4.2 in [9], where the \(L^p\) boundedness of \(\mathcal H_*\) is proved for all \(p>1\) in the normal framework, under the additional assumption that the infinitesimal generator of \(\big ( \mathcal H_t \big )_{t> 0}\,\) is a sectorial operator of angle less than \(\pi /2\).
In this paper we focus our attention on the Ornstein–Uhlenbeck semigroup in \(\mathbb {R}^n\). In view of possible applications to stochastic analysis and to SPDE’s, it would be very interesting to investigate the case of the infinitedimensional OrnsteinUhlenbeck maximal operator as well (see [3, 6, 18] for an introduction to the infinitedimensional setting). The Riesz transforms associated to a general Ornstein–Uhlenbeck semigroup in \(\mathbb {R}^n\) have been studied in the authors’ paper [5].
The scheme of the paper is as follows. In Sect. 2 we introduce the Mehler kernel \(K_t(x,u)\), that is, the integral kernel of \(\mathcal H_t\). Some estimates for the norm and the determinant of \(Q_t\) and related matrices are provided in Sect. 3. As a consequence, we obtain bounds for the Mehler kernel. In Sect. 4 we consider the relevant geometric features of the problem, and introduce in Sect. 4.1 a system of polarlike 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, Sect. 5 introduces some preliminary simplifications of the proof; in particular, we restrict the variable x to an ellipsoidal annulus. In Sect. 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 Sect. 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 \(C<\infty \) 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 \(a\lesssim b\) instead of \(a \le C b\) and \(a \gtrsim b\) instead of \(a \ge c b\). The symbol \(a\simeq b\) means that both \(a\lesssim b\) and \(a \gtrsim b\) hold.
By \(\mathbb {N}\) we mean the set of all nonnegative integers. If A is an \(n\times n\) matrix, we write \(\Vert A\Vert \) for its operator norm on \(\mathbb {R}^n\) with the Euclidean norm \(\cdot \).
2 The Mehler kernel
For \(t>0\), the difference
is a symmetric and strictly positive definite matrix. So is the matrix
and we can define
Then formula (1.2), the definition of the Gaussian measure and some elementary computations yield
where we repeatedly used the fact that \(Q_\infty ^{1} Q_t^{1}\) is symmetric. We now express the matrix \(D_t\) in various ways.
Lemma 2.1
For all \(x\in \mathbb {R}^n\) and \(t>0\) we have

(i)
\( D_t = Q_\infty e^{tB^*} Q_\infty ^{1} \);

(ii)
\(D_t = e^{tB} + Q_t e^{tB^*}Q_\infty ^{1}\).
Proof

(i)
The formulae (2.1) and (1.1) imply
$$\begin{aligned} Q_\infty  Q_t= e^{tB} Q_\infty e^{tB^*} \end{aligned}$$(2.5)(see also [12, formula (2.1)]). From (2.3) and (2.2) it follows that
$$\begin{aligned} {D_t = Q_\infty ( Q_\infty Q_t)^{1}\, e^{tB},}\end{aligned}$$and combining this with (2.5) we arrive at (i).

(ii)
Multiplying (2.5) by \( e^{tB^*} Q_\infty ^{1}\) from the right, we obtain
$$\begin{aligned} Q_\infty e^{tB^*}Q_\infty ^{1} Q_t e^{tB^*}Q_\infty ^{1} = e^{tB}, \end{aligned}$$and (ii) now follows from (i).
\(\square \)
By means of (i) in this lemma, we can define \(D_t\) for all \(t\in \mathbb {R}\), and they will form a oneparameter 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
for \(x,u\in \mathbb {R}^n\). Here we introduced the quadratic form
3 Some auxiliary results
In this section we collect some preliminary bounds, which will be essential for the sequel.
Lemma 3.1
For \(s>0\) and for all \(x\in \mathbb {R}^n\) the matrices \( D_{s }\) and \( D_{s }= D_{s }^{1}\) satisfy
and
This also holds with \( D_{s }\) replaced by \(e^{sB}\) and \(e^{sB^*}\).
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
From (i) in Lemma 2.1, we now get the claimed upper estimates for \(D_{\pm s}\). To prove the lower estimate for \(D_s\), we write
The other parts of the lemma are completely analogous. \(\square \)
In the following lemma, we collect estimates of some basic quantities related to the matrices \(Q_t\).
Lemma 3.2
For all \(t>0\) we have

(i)
\(\det { \, Q_t} \simeq (\min (1,t))^{n}\);

(ii)
\(\Vert Q_t^{1}\Vert \simeq (\min (1,t))^{1}\);

(iii)
\(\Vert Q_\infty Q_t \Vert \lesssim e^{ct}\);

(iv)
\(\Vert Q_t^{1}Q_\infty ^{1} \Vert \lesssim {t}^{1}\,{e^{ct}}\);

(v)
\(\Vert \left( Q_t^{1}Q_\infty ^{1} \right) ^{1/2}\Vert \lesssim {t^{1/2}}\, e^{Ct} \).
Proof
(i) and (ii) Using (3.1), we see that for each \(t>0\) and for all \(v\in \mathbb {R}^n\)
Since \(\Vert \left( e^{s B^*}\right) ^{1}\Vert = \Vert e^{s B^*}\Vert \lesssim e^{Cs}\), 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 \(\Vert A^{1/2}\Vert = \Vert A\Vert ^{1/2}\) for any symmetric positive definite matrix A, we consider \((Q_t^{1}Q_\infty ^{1})^{1}\), which can be rewritten as
It follows from (2.5) that \((Q_\infty  Q_t)^{1}= e^{tB^*} Q_\infty ^{1}e^{tB},\)
so that
as a consequence of (3.2). Inserting this and the simple estimate \(\Vert Q_t \Vert \lesssim t\) in (3.2), we obtain \(\Vert (Q_t^{1}Q_\infty ^{1})^{1}\Vert \lesssim t e^{Ct}\), and (v) follows. \(\square \)
Proposition 3.3
For \(t\ge 1\) and \(w\in \mathbb {R}^n\), we have
Proof
By (2.3) and Lemma 2.1 (i) we have
Since \(Q_\infty Q_t^{1}=I+(Q_\infty Q_t)Q_t^{1}\), this leads to
Here \( \langle Q_\infty ^{1} w, w\rangle \simeq w^2\). Using (2.1) and then the definition of \( Q_\infty \), we observe that the last term can be written as
Since \(\big Q_t^{1/2} e^{tB} w \big ^2 \lesssim w^2 \) for \(t\ge 1\) by Lemmata 3.1 and 3.2 (ii), the proposition follows.\(\square \)
We finally give estimates of the kernel \(K_t\), for small and large values of t. When \(t\le 1\), one has \(\Vert (Q_t^{1}Q_\infty ^{1})^{1/2}\Vert \simeq t^{1/2} \) and \(\Vert (Q_t^{1}Q_\infty ^{1})^{1/2}\Vert \simeq t^{1/2}\), by (iv) and (v) in Lemma 3.2. Combined with (2.6), this implies
Lemma 3.4
For \(t\ge 1\) and \(x,u\in \mathbb {R}^n\), we have
Proof
This follows from (2.6), if we write \(uD_t\, x= D_t (D_{t}\,u x)\) and apply Proposition 3.3 with \(w=D_{t}\,u x\). \(\square \)
4 Geometric aspects of the problem
4.1 A system of adapted polar coordinates
We first need a technical lemma.
Lemma 4.1
For all x in \(\mathbb {R}^n\) and \(s\in \mathbb {R}\), we have
Proof
To prove (4.1), we use the definition of \(Q_\infty \) to write for any \(z\in \mathbb {R}^n\)
Setting \(z=Q_\infty ^{1} x \), 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. \(\square \)
We observe here that an integration of (4.2) leads to
Fix now \(\beta >0\) and consider the ellipsoid
As a consequence of (4.3), the map \(s\mapsto R(D_s z)\) is strictly increasing for each \(0 \ne z\in \mathbb {R}^n\). Hence any \(x\in \mathbb {R}^n,\, x\ne 0\), can be written uniquely as
for some \({\tilde{x}}\in E_\beta \) and \(s\in \mathbb {R}\). We consider s and \({\tilde{x}}\) as the polar coordinates of x. Our estimates in what follows will be uniform in \(\beta \).
Next, we shall write Lebesgue measure in terms of these polar coordinates. A normal vector to the surface \(E_\beta \) at the point \({\tilde{x}} \in E_\beta \) is \(\mathbf{{N}} ({\tilde{x}})=Q_\infty ^{1}{\tilde{x}} \), and the tangent hyperplane at \({\tilde{x}}\) is \(\mathbf{{N}}({\tilde{x}})^\perp \). For \(s > 0\) the tangent hyperplane of the surface \(D_sE_\beta = \{D_s\, {\tilde{x}}: {\tilde{x}}\in E_\beta \}\) at the point \(D_s\,{\tilde{x}}\) is \(D_s(\mathbf{{N}}({\tilde{x}})^\perp )\), and a normal to \(D_sE_\beta \) at the same point is \(w = (D_s^{1})^* (\mathbf{{N}}({\tilde{x}}))=D_{s}^* Q_\infty ^{1}{\tilde{x}} = Q_\infty ^{1}e^{sB} {\tilde{x}}\).
The scalar product of w and the tangent of the curve \(s \mapsto D_s\,{\tilde{x}}\) at the point \(D_s\,{\tilde{x}}\) is, because of (4.2) and (4.1),
Thus the curve \(s \mapsto D_s\,{\tilde{x}}\) is transversal to each surface \(D_sE_\beta \). Let \(dS_s\) denote the area measure of \(D_sE_\beta \). 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 \(\varphi =\varphi ({\tilde{x}}) \) on \(E_\beta \) and extend it to \({\mathbb {R}}^n \setminus \{0 \}\) by writing \(\varphi (D_s\, {\tilde{x}}) =\varphi ({\tilde{x}}) \). For any \(t>0\) and small \(\varepsilon >0\), we define the shell
Then \( \Omega _{t,\varepsilon }\) is the image under \(D_t\) of \( \Omega _{0,\varepsilon }\), and the Jacobian of this map is \(\det D_t = e^{t{{\,\mathrm{tr}\,}}B}\). Thus
which we can rewrite as
Now we divide by \(\varepsilon \) and let \(\varepsilon \rightarrow 0\), getting
Since this holds for any \(\varphi \), it follows that
Together with (4.7), this implies the following result.
Proposition 4.2
The Lebesgue measure in \({\mathbb {R}}^n\) is given in terms of polar coordinates \((t, {\tilde{x}})\) 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.
Lemma 4.3
Fix \(\beta > 0\). Let \(x^{(0)},\; x^{(1)}\in \mathbb {R}^n \setminus \{ 0\} \) and assume \( R(x^{(0)}) > \beta /2\). Write
with \(s^{(0)}\), \(s^{(1)}\in \mathbb {R}\) and \({\tilde{x}}^{(0)},\; {\tilde{x}}^{(1)} \in E_\beta \).

(i)
Then
$$\begin{aligned} \big x^{(0)}  x^{(1)}\big  \gtrsim c\, \big {\tilde{x}}^{(0)}  {\tilde{x}}^{(1)}\big  .\end{aligned}$$(4.8) 
(ii)
If also \(s^{(1)} \ge 0\), then
$$\begin{aligned} \big x^{(0)}  x^{(1)}\big  \gtrsim c\,\sqrt{\beta }\,s^{(0)} s^{(1)}. \end{aligned}$$(4.9)
Proof
Let \(\Gamma : [0,1] \rightarrow \mathbb {R}^n \setminus \{0\}\) be a differentiable curve with \(\Gamma (0) =x^{(0)}\) and \(\Gamma (1) =x^{(1)}\). It suffices to bound the length of any such curve from below by the righthand sides of (4.8) and (4.9).
For each \(\tau \in [0,1]\), we write
with \({\tilde{x}}(\tau ) \in E_\beta \) and \({\tilde{x}}{(i)}= \tilde{x}^{(i)}\), \(s{(i)}= s^{(i)}\) for \(i=0,1\). Thus
The group property of \(D_s\) implies that
and so
with
The vector \({\tilde{x}}' (\tau )\) is tangent to \(E_\beta \) and thus orthogonal to \(\mathbf{{N} }({\tilde{x}})\). Then (4.6) (with \(s=0\)) implies that the angle between \(\frac{\partial }{\partial s} {D_{s}}_{\big s=0} \,{\tilde{x}} (\tau )\) and \({\tilde{x}}'(\tau )\) is larger than some positive constant. It follows that
where we also used the fact that, by (4.2),
Since
because of Lemma 3.1, we obtain from (4.10)
Next, we derive a lower bound for s(0); assume first that \(s{(0)} < 0\). The assumption \( R(x^{(0)}) > \beta /2\) implies, together with Lemma 3.1,
It follows that
for some \({\tilde{s}}\) with \(0<{\tilde{s}}<C\), and this obviously holds also without the assumption \(s(0)<0\).
Assume now that \( s(\tau ) > {\tilde{s}}1\) for all \(\tau \in [0,1]\). Then (4.11) implies
and
Integrating these estimates with respect to \(\tau \) in [0, 1], we immediately see that one can control the length of \(\Gamma \) from below by the righthand sides of (4.8) and (4.9).
If instead \( s(\tau ) \le {\tilde{s}}1\) for some \(\tau \in [0,1]\), we can proceed as in the proof of Lemma 4.2 in [4]. More precisely, since the image s([0, 1]) contains the interval \([{\tilde{s}}1, \max (s(0), s(1))]\), we can find a closed subinterval I of [0, 1] whose image s(I) is exactly the interval \([{\tilde{s}}1, \max (s(0), s(1))]\). Thus we may use (4.11) to control the length of \(\Gamma \) by
Here
and (4.8) follows. Under the additional hypothesis \(s(1)\ge 0\) of (ii), we have
Then
and (4.9) follows. \(\square \)
4.2 The Gaussian measure of a tube
We fix a large \(\beta > 0\). Define for \(x^{(1)}\in E_\beta \) and \(a>0\) the set
This is a spherical cap of the ellipsoid \(E_\beta \), centered at \(x^{(1)}\). Observe that \(x \simeq \sqrt{\beta }\) for \(x \in \Omega \), and that the area of \(\Omega \) is \(\Omega \simeq \min \,( a^{n1},\beta ^{(n1)/2}) \). Then consider the tube
Lemma 4.4
There exists a constant C such that \(\beta > C\) implies that the Gaussian measure of the tube Z fulfills
Proof
Proposition 4.2 yields, since \(H(0,{\tilde{x}}) \simeq {\tilde{x}}\simeq \sqrt{\beta }\),
By (4.3) we have
which implies
Assuming \(\beta \) large enough, one has \(c\beta > 2{{\,\mathrm{tr}\,}}B\), and then the last integral is finite and no larger than \(C/\beta \). The lemma follows. \(\square \)
5 Simplifications
In this section, we introduce some preliminary simplifications and reductions for 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
$$\begin{aligned} \Vert f\Vert _{L^1( \gamma _\infty )}=1, \end{aligned}$$since this involves no loss of generality.

(2)
We may assume that \(\alpha \) is large, \(\alpha > C\), 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
$$\begin{aligned} {\mathcal E_\alpha }=\left\{ x \in \mathbb {R}^n:\, \frac{1}{2} \log \alpha \le R(x) \le 2 \log \alpha \, \right\} . \end{aligned}$$To begin with, we can always forget the unbounded component of the complement of \(\mathcal E_\alpha \), since
$$\begin{aligned}&\gamma _\infty \{ x\in \mathbb {R}^n : \,R( x)> 2 \log \alpha \}\nonumber \\ {}&\quad \lesssim \int _{ R(x)> 2 \log \alpha } \exp (R(x) )\, dx\, \lesssim (\log \alpha )^{(n2)/2}\,\exp ( { 2 \log \alpha }) \lesssim \frac{1}{\alpha }. \end{aligned}$$(5.1) 
(4)
When \(t>1\), we may forget also the inner region where \(R(x)<\frac{1}{2} \log \alpha \). Indeed, from (3.5) we get, if \((x,u)\in \mathbb {R}^n\times \mathbb {R}^n\) with \(R(x) < \frac{1}{2} \log \alpha \),
$$\begin{aligned} K_t (x,u) \lesssim \, e^{R(x)}< \sqrt{\alpha } < \alpha , \end{aligned}$$since \(\alpha \) is large. In other words, for any \((x,u)\in \mathbb {R}^n\times \mathbb {R}^n\)
$$\begin{aligned} R(x) < \frac{1}{2} \log \alpha \; \qquad \Rightarrow \;\qquad K_t (x,u) \lesssim \alpha , \end{aligned}$$(5.2)for all \(t> 1\).
Replacing \(\alpha \) by \(C\alpha \) for some C, we see from (3) and (4) that we can assume \(x \in {\mathcal E_\alpha }\) in the proof of (1.3) and (1.4), when the supremum in the maximal operator is taken only over \(t> 1\).
Before introducing the last simplification, we need to define a global region
and a local region
Notice that the definition of G and L does not depend on Q and B.

(5)
When \(t\le 1\) and \((x,u) \in G\), we shall see that (5.2) is still valid, and it is again enough to consider \(x\in {\mathcal E_\alpha }\).
To prove this, we need a lemma which will also be useful later.
Lemma 5.1
If \((x,u)\in G\) and \(0<t\le 1\), then
Proof
From the definition of G and (4.4) we get
The lemma follows. \(\square \)
To verify now (5.2) in the global region with \(t\le 1\), we recall from (3.4) that
It follows from Lemma 5.1 that
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
6 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).
Proposition 6.1
For all functions \(f\in L^1 (\gamma _\infty )\) such that \(\Vert f\Vert _{L^1( \gamma _\infty )}=1\),
In particular, the maximal operator
is of weak type (1, 1) with respect to the invariant measure \(\gamma _\infty \).
Proof
We can assume that \(f\ge 0\). Looking at the arguments in Sect. 5, items (3) and (4), we see that it suffices to consider points \(x\in {\mathcal E_\alpha }\). For both x and u we use the coordinates introduced in (4.5) with \(\beta =\log \alpha \), that is,
where \({\tilde{x}}, {\tilde{u}} \in E_{\log \alpha }\) and \(s,s' \in \mathbb {R}\).
From (3.5) we have
for \(t> 1\) and \(x,u\in \mathbb {R}^n\). Since \(x \in {\mathcal E_\alpha }\) and \(D_{t}\,u= D_{s't}\, \tilde{u}, \) we can apply Lemma 4.3 (i), getting
so that
In view of (4.3), the righthand side here is strictly increasing in s, and therefore the inequality
holds if and only if \(s> s_\alpha ({\tilde{x}})\) for some function \({\tilde{x}}\mapsto s_\alpha ({\tilde{x}})\), with equality for \(s=s_\alpha ({\tilde{x}})\). Since \(\alpha >2\) and \(\Vert f\Vert _{L^1 (\gamma _\infty )}=1\), it follows that \(s_\alpha ({\tilde{x}})>0\).
For some C, the set of points \(x\in {\mathcal E_\alpha }\) where the supremum in (6.1) is larger than \(C\alpha \) is contained in the set \(\mathcal A(\alpha )\) of points \(D_s\, {\tilde{x}}\in {\mathcal E_\alpha }\) fulfilling (6.2). We use Proposition 4.2 to estimate the \(\gamma _\infty \) measure of \(\mathcal A(\alpha )\). Observe that \(H(0,{\tilde{x}}) \simeq {\tilde{x}}\simeq \sqrt{\log \alpha }\) and that \(D_s\, {\tilde{x}}\in {\mathcal E_\alpha }\) implies \(s\lesssim 1\), so that also \(e^{s{{\,\mathrm{tr}\,}}B} \lesssim 1\). We get
where the last inequality follows from (4.3), since \( D_s\, {\tilde{x}}^2\gtrsim {\tilde{x}}^2\simeq \log \alpha . \) 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. \(\square \)
Finally, we show that the factor \(1/\sqrt{\log \alpha }\) in (6.1) is sharp.
Proposition 6.2
For any \(t> 1\) and any large \(\alpha \), there exists a function f normalized in \(L^1 (\gamma _\infty )\) and such that
Proof
Take a point z with \(R(z)=\log \,\alpha \), and let f be (an approximation of) a Dirac measure at the point \(u=D_t z\). Then, as a consequence of (3.5), \(K_t (x,u) \simeq \exp (R( x))\) when x is in the ball \(B(D_{t}\,u,1)=B(z,1)\). We then have \(\mathcal H_t f(x)=K_t (x,u)\gtrsim \alpha \) in the set \(\mathcal B=\{x\in B(z,1)\!: R(x)>R(z) \}\), whose measure is
\(\square \)
7 The local case for small t
Proposition 7.1
If \((x,u) \in L \) and \(0<t\le 1\), then
Proof
In view of (3.4), it is enough to show that
We write
By (4.4),
since \((x,u) \in L\), and (7.1) follows. \(\square \)
Proposition 7.2
The maximal operator \(\mathcal H_*^{L} \) is of weak type (1, 1) with respect to the invariant measure \(\gamma _\infty \).
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 \(\mathbb {R}^n\). From this the proposition follows, cf. [7, Section 3]. \(\square \)
8 The global case for small t
In this section, we conclude the proof of Theorem 1.1.
Proposition 8.1
The maximal operator \(\mathcal H_*^{G}\) is of weak type (1, 1) with respect to the invariant measure \(\gamma _\infty \).
Proof
We take f and \(\alpha \) as in items (1) and (2) of Sect. 5. Then item (5) tells us that we need only consider \(\mathcal H_*^{G} f(x)\) for \(x\in \mathcal E_\alpha \).
For \(m\in \mathbb {N}\) and \(0<t\le 1\), we introduce regions \(\mathcal S^{m}_t\). If \(m>0\), we let
If \(m=0\), we replace the condition \(2^{m1} \sqrt{t}< uD_t\, x \le 2^{m} \sqrt{t}\) by \( uD_t\, x  \le \sqrt{t}\). Note that for any fixed \(t\in (0,1]\) these sets form a partition of G.
In the set \({\mathcal S^{m}_t}\) we have, because of (3.4),
Then setting
one has, for all \((x,u)\in G\) and \(0<t<1\),
Hence, it suffices to prove that for \(m = 0,1,\dots \)
for large \(\alpha \) and some C, since this will allow summing in m in the space \(L^{1,\infty }(\gamma _\infty )\).
Fix \(m\in {\mathbb {N}}\) and assume that \((x,u) \in S_t^m\) for some \(t \in (0,1]\), so that \(uD_t\, x \le 2^{m}\sqrt{t}\). Then Lemma 5.1 leads to
Consequently, a point \(x\in \mathcal E_\alpha \) satisfies
as soon as there exists a point u with \(\mathcal K_t^{m}(x,u)\ne 0\), and then \(t\ge \varepsilon >0\) for some \(\varepsilon =\varepsilon (\alpha ,m) >0\). Hence the supremum in (8.1) will be the same if taken only over \(\varepsilon \le t\le 1\), and it follows that this supremum is a continuous function of \(x\in {\mathcal E_\alpha }\).
To prove (8.1), the idea, which goes back to [15], is to construct a finite sequence of pairwise disjoint balls \(\big (\mathcal B^{(\ell )}\big )_{\ell =1}^{\ell _0}\) in \(\mathbb {R}^n\) and a finite sequence of sets \(\big (\mathcal Z^{(\ell )}\big )_{\ell =1}^{\ell _0}\) in \(\mathbb {R}^n\), called forbidden zones. These zones will together cover the level set in (8.1). We will then verify that
that for each \(\ell \)
and that the \( \mathcal B^{(\ell )}\) are pairwise disjoint. This would imply
and thus also (8.1) and Proposition 8.1.
The sets \(\mathcal B^{(\ell )}\) and \(\mathcal Z^{(\ell )}\) will be introduced by means of a sequence of points \(x^{(\ell )}, \ell =1,\ldots , \ell _0\), 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, (8.1) is immediate.
We now describe the recursion to construct \(x^{(\ell )}\) for \(\ell \ge 2\). Like \(x^{(1)}\), these points will satisfy
Once an \(x^{(\ell )},\;\ell \ge 1 \), is defined, we can thus by continuity choose \(t_\ell \in [\varepsilon , 1] \) such that
Using this \(t_\ell \), we associate with \(x^{(\ell )}\) the tube
Here the constant \(A>0\) is to be determined, depending only on n, Q and B.
All the \(x^{(\ell )}\) will be minimizing points of R(x). To avoid having them too close to one another, we will not allow \(x^{(\ell )}\) to be in any \(\mathcal Z^{(\ell ')}\) with \(\ell ' < \ell \). More precisely, assuming \(x^{(1)}, \dots , x^{(\ell )}\) already defined, we will choose \(x^{(\ell +1)}\) as a minimizing point of R(x) in the set
provided this set is nonempty. But if \(\mathcal A_{\ell +1} (\alpha )\) is empty, the process stops with \(\ell _0=\ell \) and (8.3) follows. We will see that this actually occurs for some finite \(\ell \).
Now assume that \(\mathcal A_{\ell +1} (\alpha ) \ne \emptyset \). In order to assure that a minimizing point exists, we must verify that \(\mathcal A_{\ell +1} (\alpha )\) is closed and thus compact, although the \(\mathcal Z^{(\ell ')}\) are not open. To do so, observe that for \(1\le \ell ' \le \ell \), the minimizing property of \(x^{(\ell ')}\) means that there is no point x in \(\mathcal A_{\ell '} (\alpha ) \) with \(R(x) < R(x^{(\ell ')})\). Thus we have the inclusions
It follows that
For each \(\ell ' = 1,\dots , \ell \) we have
and this set is closed. It follows that \( \mathcal A_{\ell +1} (\alpha )\) is compact, and a minimizing point \(x^{(\ell +1)}\) can be chosen. Thus the recursion is well defined.
We observe that (8.2) applies to \(t_\ell \) and \(x^{(\ell )}\), and \(x^{(\ell )}\) is large, so
Further, we define balls
Because of the definitions of \(\mathcal K_t^m\) and \({\mathcal S^{m}_t}\), the inequality (8.5) implies
It remains to verify the claimed properties of \(\mathcal B^{(\ell )}\) and \(\mathcal Z^{(\ell )}\). The arguments below follow the lines of the proof of Lemma 6.2 in [4], with only slight modifications.
Lemma 8.2
The balls \(\mathcal B^{(\ell )}\) are pairwise disjoint.
Proof
Two balls \(\mathcal B^{(\ell )}\) and \(\mathcal B^{(\ell ')}\) with \(\ell <\ell '\) will be disjoint if
By means of our polar coordinates with \(\beta =R(x^{(\ell )})\), we write
for some \({\tilde{x}}^{(\ell ')}\) with \(R({\tilde{x}}^{(\ell ')})= R(x^{(\ell )})\) and some \(s \in \mathbb {R}\). Note that \(s\ge 0\), because \(R(x^{(\ell ')})\ge R( x^{(\ell )})\). Since \(x^{(\ell ')}\) does not belong to the forbidden zone \( \mathcal Z^{(\ell )}\), we must have
We first assume that \({t_{\ell '}} \ge M\, 2^{4m} \, t_\ell \), for some \(M=M(n, Q,B)\ge 2\) to be chosen. Lemma 4.3 (ii) implies
the last step by our assumption. Using again the assumption and then (8.7), we get
Fixing M suitably large, we obtain (8.9) from the last two formulae.
It remains to consider the case when \({t_{\ell '}} < M\, 2^{4m}\, t_\ell \). Then
Applying this to (8.10), we obtain (8.9) by choosing A so that \(A/\sqrt{M}\) is large enough. \(\square \)
We next verify that the sequence \((x^{(\ell )})\) is finite. For \(\ell <\ell '\), we have (8.10), and Lemma 4.3 (i) implies
Since \(t_\ell \ge \varepsilon \), we see that the distance \(\left x^{(\ell ')} x^{(\ell )} \right \) is bounded below by a positive constant. But all the \( x^{(\ell )}\) are contained in the bounded set \( {\mathcal E_\alpha }\), so they are finite in number. Thus the set considered in (8.6) must be empty for some \(\ell \), and the recursion stops. This implies (8.3).
We finally prove (8.4) . Observe that the forbidden zone \(\mathcal Z^{(\ell )}\) is a tube as defined in (4.12), with \(a=A\, 2^{3m} \sqrt{t_\ell }\) and \(\beta =R(x^{(\ell )})\). This value of \(\beta \) is large since \(x^{(\ell )} \in {\mathcal E_\alpha }\), and thus we can apply Lemma 4.4 to obtain
We bound the exponential here by means of (8.8) and observe that \(R(x^{(\ell )}) \sim x^{(\ell )}^2\), getting
As a consequence of (8.7), we obtain
proving (8.4). This concludes the proof of Proposition 8.1. \(\square \)
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V. Casarino and P. Ciatti were partially supported by GNAMPA (Project 2018 “Operatori e disuguaglianze integrali in spazi con simmetrie”) and MIUR (PRIN 2015 “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis"). This research was carried out while the third author was a visiting scientist at the University of Padova, Italy, and he is grateful for its hospitality.
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Casarino, V., Ciatti, P. & Sjögren, P. On the maximal operator of a general Ornstein–Uhlenbeck semigroup. Math. Z. 301, 2393–2413 (2022). https://doi.org/10.1007/s0020902202986w
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DOI: https://doi.org/10.1007/s0020902202986w