CONICAL SQUARE FUNCTIONS AND NON-TANGENTIAL MAXIMAL FUNCTIONS WITH RESPECT TO THE GAUSSIAN MEASURE

Abstract We study, in L(R; γ) with respect to the gaussian measure, nontangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in L-norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.


Introduction
Gaussian harmonic analysis, understood as the study of objects associated with the gaussian measure dγ(x) = (2π) −n/2 exp − 1 2 |x| 2 dx on R n , and the Ornstein-Uhlenbeck operator Lf (x) = −∆f (x) + x • ∇f (x) on function spaces such as L 2 (R n ; γ), has recently gained new momentum following the development, by Mauceri and Meda [9], of an atomic Hardy space H 1 at (R n ; γ), on which various functions of L give rise to bounded operators.Harmonic analysis in L p (R n ; γ) has been relatively well established for some time, with results such as the boundedness of Riesz transforms going back to the work of Meyer and Pisier in the 1980's.The p = 1 case, however, has always proven to be difficult.Over the last 30 years, some weak type (1,1) estimates have been obtained, while others have been disproved (see the survey [13]).The proofs of these results rely on subtle decompositions and estimates of kernels.Until the seminal Mauceri-Meda paper appeared in 2007, a large part of euclidean harmonic analysis, such as end point estimates using Hardy and BMO spaces, seemed to have no gaussian counterpart.Gaussian harmonic analysis in L 2 (R n ; γ) is relatively straightforward given the fact that the Ornstein-Uhlenbeck operator is diagonal with respect to the basis of Hermite polynomials.The L p (R n ; γ) case, with 1 < p < ∞, is harder but still manageable through kernel estimates.The end points p = 1 and p = ∞, however, usually require techniques such as Whitney coverings and Calderón-Zygmund decompositions, for which the non-doubling nature of the gaussian measure, has, so far, not been overcome.Mauceri and Meda's paper [9], though, indicates a possible strategy.The authors used the notion of admissible balls which goes back to the work of Muckenhoupt [11].These are balls B(x, r) with the property that r ≤ a min(1, 1 |x| ) for some fixed admissibility parameter a > 0. On these admissible balls, the gaussian measure turns out to be doubling.The idea is then to follow classical arguments using admissible balls only.This is easier said than done.Indeed, admissible balls are very small when their centre is far away from the origin, whereas tools such as Whitney decompositions of open sets require the size of balls to be comparable to their distance to the boundary of the set, hence possibly very large.This may be why, although it contains many breakthrough results, Mauceri and Meda's paper [9] does not yet give a full theory of H 1 and BMO spaces for the gaussian measure.For instance, the boundedness of key operators such as maximal functions, conical square functions (area integrals), and above all Riesz transforms, is still missing.In fact, while this paper was in its final stages, Mauceri, Meda, and Sjögren [10] proved that Riesz transforms (more precisely some Riesz transforms, see their paper for the details) are bounded from the Mauceri-Meda Hardy space H 1 at (R n ; γ) into L 1 (R n ; γ) only in dimension one.This suggests that the 'correct' gaussian Hardy space H 1 (R n ; γ) should be a modification of theirs.
In this paper, we take another step towards a satisfying H 1 (R n ; γ) theory by studying, in L 1 (R n ; γ), non-tangential square functions and maximal functions.These are gaussian analogues of the sublinear operators which, in the euclidean setting, are the cornerstones of the real variable theory of H 1 (R n ).In the gaussian context, non-tangential maximal functions were first introduced in an unpublished work by Fabes and Forzani, who studied a gaussian counterpart of the Lusin area integral.Their L p -boundedness was shown subsequently by Forzani, Scotto, and Urbina [6].Our definition is an averaged version of a non-tangential maximal function from a subsequent paper of Pineda and Urbina [12].The additional averaging adds some technical difficulties, but experience has shown (see e.g.[7]) that such averaging can be helpful in Hardy space theory and its applications (to boundary value problems, for instance).
Here we prove a change of aperture formula for the maximal function in the spirit of one of the key estimates of Coifman, Meyer, and Stein [3].We then show that the non-tangential square function is controlled by the non-tangential maximal function.Such estimates are central in Hardy space theory (see for instance [4], [5]).Thus, the purpose of this article is twofold.On the one hand, it contributes to the development of dyadic techniques in gaussian harmonic analysis, i.e., methods and results based on gaussian analogues of the decomposition of R n into dyadic cubes, the related covering lemmas, and the corresponding H 1 and weak type (1,1) estimates.This makes the paper technical in nature, but we believe that the techniques developed here will find more applications, as gaussian harmonic analysis becomes more geometric and relies less on euclidean (after a change of variables) estimates of the Mehler kernel.On the other hand, this article gives some of the results required in the development of a gaussian Hardy space theory.When completed, such a theory will not only be satisfying from a pure harmonic analytic perspective, but it should also be applicable to stochastic partial differential equations (SPDE).Given the success of Hardy space techniques in deterministic PDE, one can think that a gaussian analogue would similarly have applications to non-linear SPDE and stochastic boundary value problems.Now let us state the main result of this paper.We set m(x) := min 1, 1 |x| and let Γ a x (γ) := (y, t) ∈ R n × (0, ∞) : |y − x| < t < am(x) denote the admissible cone with parameter a > 0 based at the point x ∈ R n .We denote by (e −tL ) t≥0 the Ornstein-Uhlenbeck semigroup acting on L p (R n ; γ) for 1 < p < ∞ (see the survey [13] and the references therein).For test functions u ∈ C c (R n ) and admissibility parameter a > 0 we consider the conical square function , and the non-tangential maximal function .
The main result of this paper reads as follows: Theorem 1.1.For each a > 0 there exists an a ′ > 0 such that the conical square function S a is controlled by the non-tangential maximal function T * a ′ , in the sense that By using the truncated cones Γ a x , we are only averaging over admissible balls in the definition of the operators.The idea is, of course, to exploit the doubling property of the gaussian measure on these balls.This makes the operators "admissible".The reader should notice, however, that they are not local, in the sense that their kernels are not supported in a region of the form {(x, y) ∈ R 2n ; |x − y| ≤ a 1+|x|+|y| }.Moreover, they can not be written as sums of local operators.This is due to a lack of off-diagonal estimates, that is a crucial difference between the Ornstein-Uhlenbeck semigroup and the heat semigroup.
Acknowledgement.We are grateful to the anonymous referee for valuable suggestions that led to simplifications and generalisations of various arguments.

A covering lemma
In this section we introduce partitions of R n into "admissible" dyadic cubes and use them to prove a covering lemma which will be needed later on.
We begin with a brief discussion of admissible balls.Let For a > 0 we define The balls in B a are said to be admissible at scale a.It is a fundamental observation of Mauceri and Meda [9] that admissible balls enjoy a doubling property: Lemma 2.1 (Doubling property).Let a, τ > 0. There exists a constant d = d a,τ,n , depending only on a, τ , and the dimension n, such that if B 1 = B(c 1 , r 1 ) ∈ B a and B 2 = B(c 2 , r 2 ) have non-empty intersection and r 2 ≤ τ r 1 , then γ(B 2 ) ≤ dγ(B 1 ).
In particular this lemma implies that for all a > 0 there exists a constant d ′ = d ′ a,n such that for all B(x, r) ∈ B a we have The first part of the next lemma, which is taken from [8], says, among other things, that if B(x, r) ∈ B a and |x − y| < br, then B(y, r) ∈ B c for some constant c = c a,b which depends only on a and b.For k ≥ 0 let ∆ k be the set of dyadic cubes at scale k, i.e., Following [8], in the gaussian case we only use cubes whose diameter depends on another parameter l, which keeps track of the distance from the ball to the origin.More precisely, define the layers and define, for k, l ≥ 0, in the first inequality we used that diam(Q) ≤ nm(c Q ) by (2.1), in the second we used that m(c Q ) ≤ 1 and r ≤ am(x) ≤ a, the third follows from the assumption we made, and the final identity follows by noting We denote by α • Q the cube with the same centre as Q and α times its side-length; similar notation is used for balls.Cubes in ∆ γ enjoy the following doubling property: Lemma 2.4.Let α > 0. There exists a constant C α,n , depending only on α and the dimension n, such that for every cube Q ∈ ∆ γ we have Proof: Without loss of generality we may assume that α > 1.Let Q ∈ ∆ γ k,l with center y and side-length 2s.Set B = B(y, s) and note that Using the doubling property for admissible balls from Lemma 2.1 we now obtain Lemma 2.5.Let F ⊆ R n be a non-empty set, let a, b, c > 0 be fixed, and let O a := {x ∈ R n : 0 < d(x, F ) ≤ am(x)}.There exists a sequence (x k ) k≥1 in O a with the following properties: ) with constant depending only on a, b, c, and n.
Remark 2.6.In the above lemma one actually has γ(O 2a ) γ(O a ), but since this fact is not needed below the rather technical proof is omitted.
Proof: Let δ := min{ 1 2 , b} and set O := O a and O ′ := O 2a for brevity.We use a Whitney covering of O ′ by disjoint cubes Q k such that [14,VI.1]).We discard the cubes that do not intersect O and relabel the remaining sequence of cubes as To check that the balls B(c k , diam(Q k )) are admissible, we use the fact that δ ≤ 1 2 to obtain Towards the proof of (ii), we claim that for all x ∈ O, d(x, F ) ≤ 3 max{1, a}d(x, ∁O ′ ).
To prove the claim, we fix x ∈ O and pick an arbitrary y ∈ ∁O ′ .Setting On the other hand, recalling that ε This contradicts the previous inequality and the claim is proved.
Combining the estimate with the claim, we obtain Recalling the inequality |c k − x k | ≤ 1 4 d(x k , F ) proved before, and then using the doubling property in combination with the above inequality, we obtain

Change of aperture for maximal functions
In the proof of Theorem 1.1 we need a change of aperture result for the admissible cone appearing in the definition of non-tangential maximal functions.For this purpose we define, for A, a > 0, the non-tangential maximal function with parameters A, a by , where is the admissible cone with parameters A, a based at the point x ∈ R n .The parameter A is called the aperture of the cone.
In what follows we will fix the dimension n and write L p (γ) := L p (R n ; γ).Theorem 3.1 (Change of aperture).For all A, A ′ , a > 0 there exists a constant D, depending only on A, A ′ , a, and the dimension n, such that for all u ∈ L 1 (γ) and σ > 0 we have and with implied constant independent of u and σ.In particular, The proof of this theorem follows known arguments in the euclidean case [5].We begin with a gaussian weak type (1, 1) estimate from [9].For the convenience of the reader we include an alternative and selfcontained proof.
Then for all τ > 0, with implied constant only depending on a and n.
We denote by c Q the centre of a cube Q.The idea of this proof is that the gaussian density is essentially equal to e − 1 2 |cQ| 2 on an admissible ball B(c Q , r Q ) and the support of To make this precise, consider a cube Q ∈ ∆ γ 0 , and suppose that a ball B(x, r) ∈ B a intersects Q.Then Lemma 2.3 implies that r ≤ 2a(a + n)m(c Q ).As a consequence, for any y ∈ B(x, r), we use the triangle inequality and (2.1) to obtain and thus This inequality implies that with implied constants depending only on a and n.
Using this estimate we obtain where |B(x, r)| denotes the Lebesgue measure of the ball; the constants depend only on a and n.
Next we note that if M * a (1 Q f )(x) > 0, then there exists a ball B ∈ B a such that B ∩ Q = ∅.Using (3.1), we thus have, for all τ > 0 and some C > 0, where M HL denotes the euclidean Hardy-Littlewood maximal operator.
Using the weak type (1, 1) bound for the latter, we get with constants depending only on a and n.
1), and therefore by the second part of Lemma where we denote by Q (l) , l = 1, . . ., N Q , the cubes from ∆ γ 0 that satisfy only depends on a and n.
It follows from the preceding considerations that for x ∈ Q, and thus with implied constant depending only on a and n; the N ′ in the last inequality accounts for the fact that, given Q ′ ∈ ∆ γ 0 , there are at most , where again N ′ depends only on a and n.
Proof of Theorem 3.1: It suffices to prove the inequality for test functions u ∈ C c (R n ).For the rest of the proof we fix u ∈ C c (R n ).Using the doubling property on admissible balls, we fix a constant τ > 0 such that where c a,2A = (1 + 2aA)a is the constant arising from Lemma 2.2(i).For σ > 0 we define b f is defined as in the lemma and b := (A ′ + 2A)a.The scheme of the proof is the following.We first prove (step 1) that, if x ∈ E σ and (y, t) ∈ Γ (2A,a) x (γ), then B(y, A ′ t) ⊆ E σ .We then use this fact (step 2) to prove that for all (y, t) ∈ Γ (2A,a) x (γ) with x ∈ E σ .This eventually gives (step 3) that there exists D = D A,A ′ ,a,n > 0 such that {x ∈ R n ; T * (A,a) u(x) > Dσ} ⊆ E σ .The proof is then concluded using Lemma 3.2 applied to 1 Eσ .In the estimates that follow, the implicit constants are independent of u and σ.
Step 1: We claim that B(y, A ′ t) ⊆ E σ .To prove this, first note that from |x − y| ≤ 2At we have Furthermore, t ≤ am(x), and therefore B(x, (A ′ + 2A)t) ∈ B (A ′ +2A)a = B b .If we now assume that the claim is false, we get where the second inequality uses that B(x, (A ′ + 2A)t) ∈ B b and the last one follows from the definition of the constant τ and the observation that B(y, t) ∈ B ca,2A by Lemma 2.2(i), using that B(x, t) ∈ B a and |x − y| ≤ 2At.This contradicts the fact that x ∈ E σ and the claim is proved.
Step 2: Since B(y, A ′ t) ⊆ E σ , there exists y ∈ B(y, A ′ t) such that y ∈ E σ , that is,

Proof of Theorem 1.1
In this section we follow the method pioneered in [5] for proving square function estimates in Hardy spaces.This method has recently been adapted in a variety of contexts (see [1], [2], [7]).Here, we modify the version given in [7] to avoid using the doubling property on nonadmissible balls, and to take into account differences between the Laplace and the Ornstein-Uhlenbeck operators.As a typical example of the latter phenomenon, we start by proving a gaussian version of the parabolic Cacciopoli inequality.Recall that L is the Ornstein-Uhlenbeck operator, defined for Note that, for all f, g ∈ C 2 b (R n ), one has the integration by parts formula ) for all t > 0, and suppose that with implied constant depending only on the dimension n, C 0 and C 1 .
Considering real and imaginary parts separately, we may assume that all functions are real-valued.Integrating the identity and then using that  For the second term we have, by (4.2), where we used the fact that η 2 satisfies the same assumptions as η and (4.2) was applied to η 2 .To estimate the third term on the right-hand side of (4. 3) we substitute the identity and estimate each of the resulting integrals: |v| 2 dγ dt, |v| 2 dγ dt.
Below we shall apply the lemma with v(x, t) = e −tL u(x), where u is a function in C c (R n ).From the representation e −tL u(x) = R n M t (x, y)u(y) dy where M is the Mehler kernel (see, e.g., [13]), it follows that v satisfies the differentiability and boundedness assumptions of the lemma.
We can now prove the main result of this paper.Recall that .
It will be convenient to define, for ε > 0, .
Proof of Theorem 1.1:As in the proof of Lemma 4.1 it suffices to consider real-valued u ∈ C c (R n ).Throughout the proof we fix a > 0 and set K := c a,1 and K := c 1+2K,2 using the notations of Lemma 2.2.Let F ⊆ R n be an arbitrary closed set and define Note that, since F is closed, F * ⊆ F .For 0 < ε < 1 and 1 < α < 2 put and let ∂R ε α (F * ) be its topological boundary.As in [5, p. 162] and [14, p. 206] we may regularise this set and thus assume it admits a surface measure dσ ε α (y, t).Applying first Green's formula in R n to the section of R ε α (F * ) at level t and using the definition of L (see (4.1)), and subsequently the fundamental theorem of calculus in the t-variable, we obtain the estimate In the above computation, ν / / denotes the projection of the normal vector ν to R ε α onto R n and ν ⊥ the projection of ν in the t direction.In step (i) we used that 1 B(x,t) (y) = 1 B(y,t) (x) and that |x − y| < t and t < am(x) imply t < Km(y) via Lemma 2.2(i); in step (ii) we used that B(y, t) ∩ F * = ∅ implies d(y, F * ) < t.Of course, all implied constants in the above inequalities are independent of F , ε, α, and u.
Fix (z, s) ∈ (F ∩ {τ (z) ≥ 1 2 ε}) × ( 1 16 ε 2 , 19 16 ε 2 ).Then from B(z, 6ε) ⊆ B(z, 24 √ s) ⊆ B(z, 30ε) and the doubling property for the balls where the last step follows from (z, √ s) ∈ Γ (24,4d) z (γ).Combining this with the previous estimate we obtain where the last step follows from the fact that r l = 1 4 ε.We proceed with an estimate for I 3 .For a > 0 let Using Lemma 2.5, we cover G 2 with a sequence of balls B(x k , r k ) with x k ∈ G 2 and r k = In the second inequality we used that y ∈ B(x k , r k ) implies |x k − y| < r k = 1 4 d(x k , F * ), and the third inequality follows from Fubini's theorem and the inequality r