A multiplier inclusion theorem on product domains

In this note it is shown that the class of all multipliers from the $d$-parameter Hardy space $H^1_{\mathrm{prod}} (\mathbb{T}^d)$ to $L^2 (\mathbb{T}^d)$ is properly contained in the class of all multipliers from $L \log^{d/2} L (\mathbb{T}^d)$ to $L^2(\mathbb{T}^d)$.


Introduction
Let d be a positive integer. If X is a subspace of L 1 pT d q, then we denote by M XÑL 2 pT d q the class of all multipliers from X to L 2 pT d q, namely the class M XÑL 2 pT d q consists of all functions m : Z d Ñ C such that for every f P X one has ř pk1,¨¨¨,k d qPZ d |mpk 1 ,¨¨¨, k d q p f pk 1 ,¨¨¨, k d q| 2 ă 8. In [1], it was shown that the class of all multipliers from the (real) Hardy space H 1 pTq to L 2 pTq is properly contained in the class of all multipliers from L log 1{2 LpTq to L 2 pTq. Our goal in this note is to extend this result to the multiparameter setting. First of all, note that if H 1 prod pT d q denotes the d-parameter (real) Hardy space over the d-torus, then L log d LpT d q Ă H 1 prod pT d q and hence, one automatically has M H 1 prod pT d qÑL 2 pT d q Ă M L log d LpT d qÑL 2 pT d q . On the other hand, by adapting the argument given in [1] to the multi-parameter case, one deduces that the best we can expect is that M H 1 prod pT d qÑL 2 pT d q Ă M L log d{2 LpT d qÑL 2 pT d q . In this note we prove that this is indeed the case, namely we strengthen the trivial exponent r " d in L log r LpT d q to the optimal one, r " d{2. In particular, our main result in this note is the following theorem. Moreover, the above inclusion is proper and it is sharp, in the sense that the exponent r " d{2 in L log d{2 LpT d q cannot be improved.
The multiplier inclusion (1.1) is obtained by a series of reductions. First, arguing as in [1] and by using D. Oberlin's characterisation of the class M H 1 prod pT d qÑL 2 pT d q given in [7], it follows that the proof of (1.1) is reduced to showing the following higher-dimensional version of an inequality due to Zygmund (see Theorem 7.6 in Chapter XII of [14]), a result of independent interest. To state this version of Zygmund's inequality on T d , let J denote the set of all "intervals" of integers of the form˘t2 n´1 ,¨¨¨, 2 n`1´2 u, n P N 0 , in other words, J consists of all the sets in Z of the form t2 k´1 ,¨¨¨, 2 k`1´2 u, k P N 0 and t´2 l`1`2 ,¨¨¨,´2 l`1 u, l P N 0 . Proposition 2. Let J be as above.
If E Ă Z d is a non-empty set satisfying the condition In turn, (1.3) will be a corollary of a higher-dimensional extension of a result due to Seeger and Trebels [12] concerning sharp bounds of sums involving "smooth" Littlewood-Paley projections on T d . To state this result, fix a Schwartz function η supported in p´2, 2q such that η| r´1,1s " 1 and consider φpξq " ηpξq´ηp2ξq. For k P N, set φ k pξq " φp2´kξq and for k " 0, set φ 0 " η. One can easily see that ř kPN0 φ k pξq " 1 for every ξ P R. Then, for k P N 0 , the corresponding "smooth" Littlewood-Paley projection in the periodic setting is defined by for any, say, trigonometric polynomial f on T. On the d-torus we put initially defined over trigonometric polynomials f on T d . Then, Proposition 2 is a consequence of the following result.
Proposition 3. There exists a constant C d ą 0, depending only on the dimension d and our choice of φ, such that the following inequality holds for every trigonometric polynomial f on T d and for each p ą 2.
The proof of Proposition 3 is an adaptation of the work of Seeger and Trebels [12] to the higher-dimensional setting combined with a well-known inequality on multiple martingales, see section 2.2. At this point, it should be mentioned that, in fact, we expect that which, of course, implies (1.4). However, as our primary goal is to establish Theorem 1 and since (1.4) is enough for that purpose, we shall not pursue this in the present note. The paper is organised as follows. In section 2 we give some notation and background and in section 3 we show how the proof of our multiplier inclusion theorem follows from Proposition 2. In section 4, we prove that Proposition 3 implies Proposition 2 and then, in section 5 we give a proof of Proposition 3. In the last section we briefly present some further applications of our work.

Notation and background
We denote by Z the set of integers, by N the set of positive integers, and by N 0 the set of non-negative integers.
The cardinality of a finite set A is denoted by #tAu. If X and Y are positive quantities such that X ď CY , where C ą 0 is a constant, then we write X À Y . To specify the dependence of this constant on some additional parameters α 1 ,¨¨¨, α n we write X À α1,¨¨¨,αn Y . If X À Y and Y À X, we write X " Y .
In this note, we identify T with r0, 1q in the standard way.
2.1. Product Hardy spaces and the class M H 1 prod pT d qÑL 2 pT d q . For 0 ă r ă 1, let P r denote the Poisson kernel on T given by P r pxq " p1´r 2 q{p1´2r cos x`r 2 q, x P T. For x P T, let Γpxq " tz P D : |z´e i2πx | ď 2p1´|z|qu, where D denotes the unit disc in the complex plane. Then, the d-parameter (real) Hardy space H 1 prod pT d q consists of all integrable functions f on the d-torus such that f˚P L 1 pT d q, where for px 1 ,¨¨¨, x d q P T d one has It follows by the work of D. Oberlin [7] that m : Z d Ñ C belongs to the class where I is the unique dyadic interval in T of the form I " rs2´m, ps`1q2´mq, s " 0, 1,¨¨¨, 2 m´1 such that x P I. For m P N, let D m " E m´Em´1 denote the martingale differences acting on functions defined on T. For m " 0, we set D 0 " E 0 .
For a given d-tuple pm 1 ,¨¨¨, m d q of non-negative integers, we define to be the corresponding operators acting on functions on the d-torus.
In [3], Chang, Wilson, and Wolff obtained the" good-λ" inequalityˇ which holds for all λ ą 0 and ǫ ą 0, where C 0 ą 0 is an absolute constant. In particular, this estimate implies that there exists a constant C ą 0 such that for all p ą 2. By using (2.2), Chang, Wilson, and Wolff obtained in [3] an inequality analogous to (2.2) involving Lusin area integrals. In [8], Pipher extended (2.2) and its analogous version on Lusin area integrals to the two-parameter setting and in [5], R. Fefferman and Pipher extended the aforementioned inequality of Chang, Wilson, and Wolff involving Lusin area integrals to ℓ 2 -valued functions. The argument of R. Fefferman and Pipher can be easily adapted to obtain an ℓ 2 -valued extension of (2.2), see [4]. By using this ℓ 2 -valued extension of (2.2) together with induction on d, one deduces that there exists a constant C d ą 0, depending only on the dimension d P N, such that for every p ą 2, see also, e.g., [4,Proposition 4.5] and [2].

Thin sets in Harmonic
Analysis. Let G be a compact abelian group and let Λ be a non-empty set in its dual p G. In this note, we shall only consider the case for every Λ-polynomial f . The smallest constant App, Λq such that the above inequality holds is called the Λppq constant of Λ.
for every Λ-polynomial. It follows by the work of Rudin [11] and Pisier [9] that a spectral set Λ is Sidon if and only if, it is a Λppq set for any p ą 2 and its Λppq constant grows like p 1{2 as p Ñ 8.
for every Λ-polynomial. Here, we use the notation r|f |s " where pr γ q γ denotes the set of Rademacher functions. It is well-known that if Λ is a Λppq set for all p ą 2 with Λppq constant growing like p k{2 , k P N, then Λ is a q-Rider set with q " 2k{pk`1q, see [10, Théorème 6.3].

Proposition 2 implies Theorem 1
To prove that Proposition 2 implies Theorem 1, we adapt the argument given in [1] to the multi-parameter setting by using the characterisation of M H 1 prod pT d qÑL 2 pT d q . To be more specific, assume that Proposition 2 holds and take an arbitrary m in the class M H 1 prod pT d qÑL 2 pT d q . Then, by definition, we need to show that for every Towards this aim, fix an f P L log d{2 LpT d q and note that the sum ÿ pk1,¨¨¨,k d qPZ d where J is as in the introduction and the statement of Proposition 2. Hence, by where E f is a set in Z d defined as follows. Given Then, having chosen a set of d-tuples p r k 1 ,¨¨¨, r k d q as above, we define Notice that as the choice of d-tuples p r k 1 ,¨¨¨, r k d q is not necessarily unique, there might be several choices of sets E f . We just choose one of them to write ÿ Note that any such set 3.1. Sharpness of (1.1). We remark that, in fact, the above argument shows that if m P M H 1 prod pT d qÑL 2 pT d q , then there is a constant C m ą 0, depending only on m, such that ÿ pk1,¨¨¨,k d qPZ d To see that the exponent r " d{2 in L log d{2 LpT n q in (1.1) cannot be improved, we argue as in [1]. More specifically, assume that for some r ą 0 every multiplier from H 1 prod pT d q to L 2 pT d q is a multiplier from L log r LpT d q to L 2 pT d q. We shall prove that r ě d{2. To this end, for a large positive integer N , take f to be a trigonometric polynomial on T d given by f " denotes the de la Vallée Poussin kernel of order 2 N and K n is the Fejér kernel on T of order n P N. Since }K n } L 1 pTq " 1 and }K n } L 8 pTq À n, we deduce that ż So, if we take M " pmpk 1 ,¨¨¨, k d qq k1,¨¨¨,k d PZ with mpk 1 ,¨¨¨, k d q " 1{ ? k 1¨¨¨kd for k 1 ą 0,¨¨¨, k d ą 0 and mpk 1 ,¨¨¨, k d q " 0 otherwise, namely when at least one of the coordinates is less or equal than 0, then M P M H 1 prod pT d qÑL 2 pT d q and hence, ÿ we see that, by choosing N to be large enough, we must have r ě d{2.

Remark 4.
A similar argument shows that the Orlicz space L log d{2 LpT d q in p1.3q cannot be improved. Indeed, if E is a set satisfying p1.2q, then by making use of the argument presented above, we see that the exponent r " d{2 in L log d{2 LpT d q in the right-hand side of higher-dimensional Zygmund's inequality p1.3q is sharp.
To show that the inclusion p1.1q is proper, take Λ to be a Sidon set in Z that cannot be written as a finite union of lacunary sequences, see [11,Remark 2.5(3)]. Then M " χ Λˆ¨¨¨ˆΛ belongs to the class M L log d{2 LpT d qÑL 2 pT d q , see, e.g., [1,Proposition 4]. However, it can be easily checked that M " χ Λˆ¨¨¨ˆΛ does not satisfy (2.1) and hence, we deduce that χ Λˆ¨¨¨ˆΛ P M L log d{2 LpT d qÑL 2 pT d q zM H 1 prod pT d qÑL 2 pT d q .

Proposition 3 implies Proposition 2
Our goal in this section is to prove that Proposition 3 implies Proposition 2. Towards this aim, take E Ă Z d to be a set satisfying the assumption of Proposition 2, i.e. condition (1.2). Assume first that E satisfies (1.2) with D E " 1. By duality, to prove (1.3), it suffices to show that E is a Λppq set in Z d for every p ą 2 with Λppq constant growing like p d{2 as p Ñ 8. In other words, it is enough to show that for every E-polynomial f one has for every p ą 2, where A E is an absolute constant, independent of p and f . As we will see momentarily, if D E " 1, then, in fact, A E depends only on d and in particular, it can be taken to be independent of E.
To prove (4.1), fix an E-polynomial f and note that for every pk 1 ,¨¨¨, k d q P N d 0 one has by the triangle inequality Observe that the quantity on the right-hand side of the last inequality equals to }f } L 2 pT n q , as suppp p f q Ă E. Hence, (4.1) follows from (1.4) and (4.2) in the case where D E " 1. Moreover, note that, in the case where D E " 1, the implied constant in (4.2) depends only on the dimension d and on our choice of φ and, in particular, it is independent of E.
In the case where D E ą 1, write f " Then, by using the triangle inequality and the previous step we have since, by our construction and the L 2 -theory, }f i } L 2 pT d q ď }f } L 2 pT d q for all i " 1,¨¨¨, D E .

Proof of Proposition 3
To prove Proposition 3, note that, as p ą 2, it follows by Minkowski's inequality that .
This last inequality follows from the next lemma which is a d-dimensional analogue of [12, Lemma 2.3].
Lemma 5. Let δ be a Schwartz function that is even, supported in p´4, 4q and such that δ| r´2,2s " 1.
Define ψpξq " δpξq´δp2ξq. For k P N, put ψ k pξq " ψp2´kξq and for k " 0, put ψ 0 " δ. Consider the operator acting on functions defined over the torus. For k 1 ,¨¨¨, k d P N 0 we use the notation Ψ k1,¨¨¨,k d " Ψ k1 b¨¨¨b Ψ k d . There exists a constant C d ą 0, depending only on the dimension d and on ψ, such that for all d-tuples of non-negative integers pm 1 ,¨¨¨, m d q and pk 1 ,¨¨¨, k d q one has }E m1,¨¨¨,m d Ψ k1,¨¨¨,

2)
where A " j P t1,¨¨¨, du : m j ă k j ( and In p5.2q we make the convention that if A " H, then ś jPA 2 mj´kj " 1. The proof of Lemma 5 will be given in the next subsection. By using the above lemma and in particular estimate (5.3) one can easily complete the proof of Proposition 3. Towards this aim, we argue as in the proof of [ .
Since we have , the proof of Proposition 3 will be complete once we prove Lemma 5. This will be done in the following subsection.

5.1.
Proof of Lemma 5. The proof of this Lemma is a straightforward adaptation of [12,Lemma 2.3] to the multi-parameter setting. For the sake of simplicity, we shall only present the proof of the two-dimensional case. A similar argument establishes the higher-dimensional case.
Let ψ be as in the statement of Lemma 5. Following [12], we use the notation ψ psq pξq " pi2πξq s ψpξq, s P t´1, 0, 1u and for k P N 0 we put For s " 0 we write ψ p0q " ψ and Ψ p0q k " Ψ k . Notice that we may write Ψ psq Our assumption on the support of ψ implies that K psq k is in fact a trigonometric polynomial on T. By using the Poisson summation formula, see, e.g., Corollary 2.6 in Chapter VII of [13], it is straightforward to see that }K psq k } L 1 pTq À ψ 1. Therefore, it follows that }Ψ for all s 1 , s 2 P t´1, 0, 1u and we thus deduce that ÿ s1,s2Pt´1,0,1u for all k 1 , k 2 P N 0 , where the summation is taken with respect to all possible choices of s 1 , s 2 P t´1, 0, 1u.
If we assume, without loss of generality, that x pαq 1 ă x pβq 1 , then by the mean value theorem, Hence, by using the definition of E m2 and E m2´1 , it follows by the mean value theorem for integrals that there are x pαq 2 P I 2 and x pβq 2 Without loss of generality we may assume that x pαq 2 ă x pβq 2 . Hence, by applying the mean value theorem, we deduce that It only remains to consider the subcase where m 1 ě k 1 and m 2 ă k 2 , the other one (m 1 ă k 1 and m 2 ě k 2 ) being symmetric. We need to show that }D m1,m2 Ψ k1,k2 } L 8 pT 2 qÑL 8 pT 2 q À 2 k1´m1 2 m2´k2 .
Hence, the proof of (5.3) for n " 2 is complete.
6. Some Further remarks and applications 6.1. Applications in thin sets. Proposition 2 gives examples of Λppq sets in Z d whose corresponding Λppq constant grows like p d{2 as p Ñ 8 and they cannot be written as products of Sidon sets. Moreover, those sets, namely the class of the sets E Ă Z d that cannot be written as d-fold products of sets in Z and satisfy the condition sup I1,¨¨¨,I d PJ # E X pI 1ˆ¨¨¨ˆId q ( ă 8, are examples of 2d{pd`1q-Rider sets in Z d that cannot be written as products of Sidon sets in Z. Note that if Λ 1 ,¨¨¨, Λ d are lacunary sequences in Z, then Λ 1ˆ¨¨¨ˆΛd satisfies (1.2) and we thus recover the well-known fact that Λ 1ˆ¨¨¨ˆΛd is a Λppq set in Z d whose constant grows like p d{2 as p Ñ 8. However, Proposition 2 cannot handle spectral sets of the form Λ 1ˆ¨¨¨ˆΛd , where Λ j is a Sidon set that is not a finite union of lacunary sequences (j " 1,¨¨¨, d).

6.2.
A version of (1.4) for "rough" projections. For k P N consider the classical Littlewood-Paley projections For k " 0, set ∆ 0 pf qpxq " p f p0q. For k 1 ,¨¨¨, k d P N 0 we write ∆ k1,¨¨¨,k d " ∆ k1 b¨¨¨b ∆ k d .