On a problem of Pichorides

Let $S^{(\Lambda)}$ denote the classical Littlewood-Paley square function formed with respect to a lacunary sequence $\Lambda$ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of $S^{(\Lambda)}$ from the analytic Hardy space $H^p_A (\mathbb{T})$ to $L^p (\mathbb{T})$ and of the behaviour of the $L^p (\mathbb{T}) \rightarrow L^p (\mathbb{T})$ operator norm of $S^{(\Lambda)}$ ($1<p<2$) in terms of the ratio of the lacunary sequence $\Lambda$. Namely, if $\rho_{\Lambda}$ denotes the ratio of $\Lambda$, then we prove that $$ \sup_{\substack{ \| f \|_{L^p (\mathbb{T})} = 1 \\ f \in H^p_A (\mathbb{T}) } } \big\| S^{(\Lambda)} (f) \big\|_{L^p (\mathbb{T})} \lesssim \frac{1}{p-1} (\rho_{\Lambda} - 1 )^{-1/2} \quad (1<p<2)$$ and $$ \big\| S^{(\Lambda)} \big\|_{L^p (\mathbb{T}) \rightarrow L^p (\mathbb{T})} \lesssim \frac{1}{(p-1)^{3/2}} (\rho_{\Lambda} - 1 )^{-1/2} \quad (1<p<2)$$ and that the exponents $r=1/2$ in $(\rho_{\Lambda} - 1 )^{-1/2} $ cannot be improved in general. Variants in higher dimensions and in the Euclidean setting are also obtained.

and that the exponents r " 1{2 in pρ Λ´1 q´1 {2 cannot be improved in general. Variants in higher dimensions and in the Euclidean setting are also obtained.

Introduction
Given a strictly increasing sequence Λ " pλ j q jPN0 of positive integers, consider the corresponding Littlewood-Paley projections`∆ It is well-known that in the case where Λ " pλ j q jPN0 is a lacunary sequence in N, namely the ratio ρ Λ :" inf jPN0 pλ j`1 {λ j q is greater than 1, the Littlewood-Paley square function S pΛq can be extended as a sublinear L p pTq bounded operator for 1 ă p ă 8; see [15] or [38] for the periodic case and [33] for the Euclidean case.
Other proofs of the aforementioned theorem of Bourgain were obtained by the author in [1] and by Lerner in [24].
In 1992, in [29], Pichorides showed that if we restrict ourselves to the analytic Hardy spaces, then one has the improved behaviour pp´1q´1 as p Ñ 1`. More specifically, Pichorides proved in [29] that if Λ " pλ j q jPN0 is a lacunary sequence of positive integers, then one has (1.2) sup where the implied constants in (1.2) depend only on the lacunary sequence Λ and not on p. As remarked by Pichorides, see Remark (i) in [29,Section 3], if Λ " pλ j q jPN0 is a lacunary sequence in N with ratio ρ Λ P p1, 2q, then the argument in [29] yields that for fixed 1 ă p ă 2 the implied constant in the upper estimate in (1.2) is Oppρ λ´1 q´2q.
Our main purpose in this paper is to solve the aforementioned problem implicitly posed by Pichorides in [29] and more specifically, our goal is to improve the exponent r " 2 in pρ Λ´1 q´2 obtained in [29] to the following optimal estimate. Theorem 1. There exists an absolute constant C 0 ą 0 such that for every 1 ă p ă 2 and for every lacunary sequence Λ " pλ j q jPN0 in N with ratio ρ Λ P p1, 2q one has The exponent r " 1{2 in pρ Λ´1 q´1 {2 in (1.3) is optimal in the sense that for every given λ "close" to 1`, one can construct a lacunary sequence Λ in N with ratio ρ Λ " λ and choose a p close to 1`such that (1.4) sup Note also that for each fixed 1 ă p ă 2, Theorem 1 implies that for every lacunary sequence Λ in N with ratio ρ Λ P p1, pq one has and this estimate, as remarked in [29], is best possible in general; see also Remark 6 below. Furthermore, we also establish the sharp behaviour of the L p pTq Ñ L p pTq operator norm of S pΛq in terms of the ratio ρ Λ of Λ. In other words, the following non-dyadic version of the aforementioned result of Bourgain (1.1) is obtained in this paper. Theorem 2. There exists an absolute constant C 0 ą 0 such that for every 1 ă p ă 2 and for every lacunary sequence Λ " pλ j q jPN0 in N with ratio ρ Λ P p1, 2q one has Moreover, as in Theorem 1, the exponent r " 1{2 in pρ Λ´1 q´1 {2 in (1.5) cannot be improved in general.
The proofs of (1.3) and (1.5) are based on the observation that it suffices to show where for the case of Theorem 1 one takes X to be the real Hardy space H 1 pTq and for the case of Theorem 2, one takes X to be the Orlicz space L log 1{2 LpTq. Indeed, regarding the proof of Theorem 1, having established the aforementioned weak-type inequality, one then argues as in [4]. More precisely, (1.3) is obtained by using (1.6) for X " H 1 pTq, the trivial estimate › › S pΛq › › L 2 pTqÑL 2 pTq " 1 and a result of Kislyakov and Xu on Marcinkiewicz-type interpolation between analytic Hardy spaces [21]. Regarding Theorem 2, having shown (1.6) for X " L log 1{2 LpTq, the proof of (1.5) is obtained by arguing as in [1], namely by first interpolating between (1.6) for X " L log 1{2 LpTq and › › S pΛq › › L 2 pTqÑL 2 pTq " 1 and then using Tao's converse extrapolation theorem [36].
The proof of (1.6) for X " H 1 pTq and X " L log 1{2 LpTq can be obtained by using the arguments of Tao and Wright [37] that establish the endpoint mapping properies of general Marcinkiewicz multiplier operators "near" L 1 pRq. However, we remark that for the case of Theorem 1, namely to prove (1.6) for X " H 1 pTq, one can just use a version of Stein's classical multiplier theorem on Hardy spaces [31,32] obtained by Coifman and Weiss in [12].
Furthermore, an adaptation of the aforementioned argument to the Euclidean setting, where one uses the work of Tao and Wright [37] combined with a theorem of Peter Jones [20] on Marcinkiewicz-type decomposition for functions in analytic Hardy spaces on the real line (instead of the theorem of Kislyakov and Xu mentioned above), gives a Euclidean version of Theorem 1. Moreover, by using the work of Lerner [24], one obtains a variant of Theorem 2 to the Euclidean setting as well as an alternative proof of Theorem 2.
At this point, it is worth noting that, given any strictly increasing sequence Λ " pλ j q jPN0 of positive integers, if 2 ă p ă 8 then S pΛq has an L p pTq Ñ L p pTq operator norm that is independent of Λ. More specifically, Bourgain, by using duality and his extension [7] of Rubio de Francia's theorem [30], proved in [10] that for every strictly increasing sequence Λ of positive integers the L p pTq Ñ L p pTq operator norm of S pΛq (2 ă p ă 8) behaves like (1.7) › › S pΛq › › L p pTqÑL p pTq " p pp Ñ 8q and the implied constants in (1.7) do not depend on Λ. In particular, if Λ is a a lacunary sequence in N, then the implied constants in (1.7) are independent of ρ Λ . Moreover, it is well-known that, in general, Littlewood-Paley square functions formed with respect to arbitrary strictly increasing sequences might not be bounded on L p for 1 ď p ă 2; see e.g. [11]. For more details on Littlewood-Paley square functions of Rubio de Francia type, see [23] and the references therein.
The present paper is organised as follows. In Section 2, we give some notation and background. In Section 3, we give a proof of Theorem 1 and present its optimality in the sense explained above. In Section 4 we prove Theorem 2 and in Section 5 we extend our results to Littlewood-Paley square functions formed with respect to finite unions of lacunary sequences. In Section 6 we extend (1.3) and (1.5) to higher dimensions and in the last section of this paper we obtain variants of (1. 3) and (1.5) to the Euclidean setting.

Notation and Background
2.1. Notation. We denote the set of integers by Z, the set of natural numbers by N and the set of non-negative integers by N 0 . The real line is denoted by R and the complex plane by C. We identify strictly increasing sequences of positive integers with subsets of N in a standard way.
If x P R, then rxs stands for the unique integer such that rxs´1 ă x ď rxs and txu denotes the integer part of x namely, txu is the unique integer satisfying txu ď x ă txu`1.
The notation |¨| is used for either the one-dimensional Lebesgue measure of a Lebesgue measurable set A Ď R or for the modulus of a complex number a P C.
The logarithm of x ą 0 to the base λ ą 0 is denoted by log λ pxq. If λ " e, we write log x.
If A is a finite set, then #pAq denotes the number of its elements. Given a, b P Z with a ă b, ta,¨¨¨, bu denotes the set ra, bs X Z and will occasionally be referred to as an "interval" in Z. Moreover, a function m : Z Ñ R is said to be "affine" in ta,¨¨¨, bu if, and only if, there exists a function µ : R Ñ R such that µpnq " mpnq for all n P ta,¨¨¨, bu and µ is affine in ra, bs.
If there exists an absolute constant C ą 0 such that A ď CB, we shall write A À B or B Á A and say that A is OpBq. If C ą 0 depends on a given parameter η, we shall write A À η B. If A À B and B À A, we write A " B. Similarly, if A À η B and B À η A, we write A " η B.
If J is an arc of the torus T :" R{p2πZq with |J| ă π, then 2J denotes the arc that is concentric to J with length equal to 2|J|.
We identify functions on the torus T with 2π-periodic functions defined over the real line. The Fourier coefficient of a function f in L 1 pT d q at pn 1 ,¨¨¨, n d q P Z d is given by then we say that f is an analytic trigonometric polynomial on T d . Similarly, if f is a Schwartz function on R d then its Fourier transform at pξ 1 ,¨¨¨, ξ d q P R d is given by Vector-valued functions on T are denoted by F . Moreover, for 0 ă p ă 8, we use the notation If pX, A, µq is a measure space, we use the standard notation }f } L 1,8 pXq :" sup λą0 λ¨µ`tx P X : |f pxq| ą λu˘(.

2.2.
Multipliers. If m P ℓ 8 pZ d q, then T is said to be a multiplier operator with associated multiplier m if, and only if, for every trigonometric polynomial f on T d one has the representation for every px 1 ,¨¨¨, x d q P T d . In this case, we also say that m is the symbol of T and we write T " T m . For j P t1,¨¨¨, du, if T j is the multiplier operator acting on functions over T with symbol m j P ℓ 8 pZq, then T 1 b¨¨¨b T d denotes the multiplier operator acting on functions over T d whose associated symbol is given by mpn 1 ,¨¨¨, n d q " m 1 pn 1 q¨¨¨m d pn d q for all pn 1 ,¨¨¨, n d q P Z d .
Multiplier operators acting on functions over Euclidean spaces are defined similarly. Given a bounded function m on R, we denote by T m| Z the multiplier operator acting on functions defined over the torus with associated symbol given by µ " m| Z , i.e. µ : Z Ñ C and µpnq " mpnq for all n P Z.
2.3. Function spaces. The real Hardy space H 1 pTq is defined to be the class of all functions f P L 1 pTq such that Hpf q P L 1 pTq, where H denotes the periodic Hilbert transform. For f P H 1 pTq, we set }f } H 1 pTq :" }f } L 1 pTq`} Hpf q} L 1 pTq . One defines the real Hardy space H 1 pRq on the real line in an analogous way.
It is well-known that H 1 pTq admits an atomic decomposition. More specifically, following [12], a function a is said to be an atom in H 1 pTq if it is either the constant function a " χ T or there exists an arc I in T such that supppaq Ď I, ş I apxqdx " 0, and }a} L 2 pTq ď |I|´1 {2 . The characterisation of H 1 pTq in terms of atoms asserts that f P H 1 pTq if, and only if, there exists a sequence of atoms pa k q kPN in H 1 pTq and a sequence pµ k q kPN P ℓ 1 pNq such that where the convergence is in the H 1 pTq-norm and moreover, if we define then there exist absolute constants c 1 1 , c 1 ą 0 such that (2.1) c 1 1 }f } H 1 at pTq ď }f } H 1 pTq ď c 1 }f } H 1 at pTq . For more details on real Hardy spaces, see [12], [17], [18] so that f equals a.e. to the limit of r f as one approaches the boundary R d of pR 2 q d , where R 2 :" tx`iy : x P R, y ą 0u. The Hardy space H 8 A pR d q is the class of all functions f P L 8 pR d q that are boundary values of bounded analytic functions r f on pR 2 q d . One defines H p A pT d q in an analogous way (0 ă p ď 8). Also, it is well-known that for 1 ď p ď 8 the Hardy space H p A pT d q coincides with the class of all functions f P L p pT d q such that suppp p f q Ď N d 0 . It thus follows that, in the one-dimensional case, H 1 A pTq Ď H 1 pTq and }f } H 1 pTq " 2}f } L 1 pTq for all f P H 1 A pTq. Moreover, the class of analytic trigonometric polynomials on T d is dense in pH p A pT d q, }¨} L p pT d q q for all 0 ă p ă 8. For more details on one-dimensional analytic Hardy spaces, we refer the reader to the book [14].
For r ą 0, L log r LpTq denotes the class of all measurable functions f on the torus satisfying ż T |f pxq| log r pe`|f pxq|qdx ă 8.
If we equip L log r LpTq with the norm where Φ r ptq :" tr1`logp1`tqs r (t ě 0), then pL log r LpTq, }¨} L log r LpTq q becomes a Banach space. For more details on Orlicz spaces, see [22]. It is well-known that M c is of weak-type p1, 1q and bounded on L p pTq for every 1 ă p ď 8; see Chapter I in [33] or Section 13 in Chapter I in [38]. Analogous bounds hold for M d ; see [34].

2.5.
Khintchine's inequality. In several parts of this paper, we pass from square functions of the form p ř j |s j | 2 q 1{2 to corresponding families of functions ř j˘s j and vice versa by using a standard randomisation argument involving Khintchine's inequality for powers p P r1, 2q. Recall that given a probability space pΩ, A, Pq and a countable set of indices F , a sequence pr n q nPF of independent random variables on pΩ, A, Pq satisfying Ppr n " 1q " Ppr n "´1q " 1{2, n P F , is said to be a sequence of Rademacher functions on Ω indexed by F . Then, Khintchine's inequality asserts that for every finitely supported complex-valued sequence pa j1,¨¨¨,j d q j1,¨¨¨,j d PF one has for all 0 ă p ă 8, where the implied constants do not depend on pa j1,¨¨¨,j d q j1,¨¨¨,j d PF .
In the special case where p is "close" to 1`, for instance when p P r1, 2q, the implied constants in (2.2) can be taken to be independent of p P r1, 2q; see Appendix D in [33].

Proof of Theorem 1
Let Λ " pλ j q jPN0 be a lacunary sequence in N with ratio ρ Λ P p1, 2q. To prove Theorem 1, we shall first establish the weak-type inequality (1.6) for X " H 1 pTq and then use a Marcinkiewicz-type interpolation argument for analytic Hardy spaces on the torus.
As mentioned in the introduction, the proof of (1.6) for X " H 1 pTq can be obtained by using either the work of Tao and Wright [37] or a classical result of Coifman and Weiss [12] and more specifically, by using the argument of Coifman and Weiss that establishes [12,Theorem (1.20)]. As the former approach will be used in the proof of Theorem 2 in Section 4, we shall present here a proof of the desired weak-type inequality that uses arguments of [12].
The following lemma is a consequence of the argument of Coifman and Weiss establishing [12,Theorem (1.20)]. Lemma 3. Let Λ " pλ j q jPN0 be a lacunary sequence in N with ratio ρ Λ P p1, 2q.
If`r ∆ pΛq j˘j PN0 is defined as above and pΩ, A, Pq is a probability space, for ω P Ω, consider the operator r T pΛq ω given by where pr j q jPN0 denotes the set of Rademacher functions on Ω indexed by N 0 . Then, there exists an absolute constant A 0 ą 0 such that where m pΛq j are as above, j P N 0 . Our first task is to show that m pΛq ω satisfies the following "Mikhlin-type" condition where C 0 ą 0 is an absolute constant, independent of ω and Λ. The verification of (3.2) is elementary. Indeed, to show (3.2), note that for every n P Z there exist at most 3 non-zero terms in the sum m pΛq ω pnq " ÿ jPN0 r j pωqm pΛq j pnq and so, it suffices to show that for every j P N 0 one has for all n P Zzt0u, where the implied constant is independent of n and Λ. To show (3.3), fix a j P N 0 and take an n P supp`m pΛq j˘. We may assume that j P N, as the case j " 0 is treated similarly and gives the same bounds. Suppose first that n P t´λ j`1 ,¨¨¨,´λ j´1 u Y tλ j´1 ,¨¨¨, λ j`1 u. Observe that in the subcase where λ j´1 ď |n| ă λ j one has δ j pnq " 0 and so, (3.3) trivially holds. If we now assume that λ j ď |n| ď λ j`1 , then one has as desired. The case where n P t´λ j´1 ,¨¨¨,´λ j´2 u Y tλ j´2 ,¨¨¨, λ j´1 u is handled similarly and also gives |n|δ j pnq ď 2pρ Λ´1 q´1. Therefore, (3.3) holds and so, (3.2) is valid with C 0 " 6. Consider now an arbitrary non-constant atom a in H 1 pTq. Then, the proof of [12,Theorem (1.20)], together with the estimate (3.2), yields that where the implied constant does not depend on r T pΛq ω and a. Indeed, notice that it follows from [12, (1.16)] that ż Tˇr T pΛq ω paqpxqˇˇdx ď 3π˜ż and hence, by using (3.3) and arguing exactly as on pp. 578-579 in [12], one deduces that ż and so, (3.4) follows. Note that since › › m pΛq ω › › ℓ 8 pZq ď 3, we deduce from (3.4) that there exists an absolute constant D 0 ą 0 such that for any non-constant atom a in H 1 pTq. Moreover, observe that the constant atom a 0 " χ T trivially satisfies (3.5), since 0 :" maxtD 0 , 3u. Therefore, (3.5) holds for all atoms in H 1 pTq and hence, by arguing as e.g. on pp. 129-130 in [18], we deduce that (3.5) holds in the whole of H 1 pTq with A 0 " c 1 D 1 0 , c 1 being the constant in (2.1). Having established Lemma 3, we are now ready to prove (1.6) for X " H 1 pTq. Note that by using (1.6) for X " H 1 pTq combined with the fact that }g} H 1 pTq " 2}g} L 1 pTq for g P H 1 A pTq and the density of analytic trigonometric polynomials in pH 1 A pTq, }¨} L 1 pTq q, one deduces that there exists an absolute constant M 0 ą 0 such that Now, in order to prove (1.6) for X " H 1 pTq, fix an arbitrary trigonometric polynomial f and define ph j q jPN0 by h j :" r ∆ pΛq j pf q, where r ∆ pΛq j are the "smoothed-out" versions of ∆ pΛq j , introduced above. It follows from Corollary 2.13 on p. 488 in [17] and the definition of ph j q jPN0 that where the implied constant does not depend on f and Λ. If we fix a probability space pΩ, A, Pq, observe that, by using the definition of ph j q jPN0 together with Khintchine's inequality (2.2) and Fubini's theorem, one has where the implied constant is independent of f and Λ. Here, r T pΛq ω denotes the multiplier operator in the statement of Lemma 3. Hence, by using the last estimate, (3.1), and (3.7), we obtain (1.6) for X " H 1 pTq.
To complete the proof of Theorem 1, one argues as in [4]. More precisely, fix a p P p1, 2q and take an arbitrary f P H p A pTq with }f } L p pTq " 1. Assume first that p is "close" to 1`, for instance, suppose that p P p1, 3{2s.
)ˇˇˇd α and I 2 :" p2πq´1p To handle I 1 , we use (3.6), the properties of g α P H 1 A pTq, Fubini's theorem, and the assumption that }f } L p pTq " 1 as follows, where C 0 " 4M 0 C with M 0 and C being the constants in (3.6) and in Lemma 4, respectively. To handle I 2 , we first use the fact that }S pΛq } L 2 pTqÑL 2 pTq " 1 and get and then, arguing as e.g. in the proof of [28, Theorem 7.4.1], we write I 2 ď I 1 with C ą 0 being the constant in Lemma 4. Hence, by using Fubini's theorem and the assumption that }f } L p pTq " 1, we have where C 1 ą 0 is an absolute constant. Putting all the estimates together, we obtain where the implied constant does not depend on f , p, and Λ. We remark that (1.3) also holds for p P p3{2, 2q. To see this, observe that [10, Theorem 2] implies that, e.g., where C ą 0 does not depend on the lacunary sequence Λ. Hence, arguing as in the case where p P p1, 3{2s, namely by using Lemma 4 and, in particular, by interpolating between (3.6) and the L 3 pTq Ñ L 3 pTq bound mentioned above, we deduce that (1.3) is also valid for p P p3{2, 2q. Therefore, the proof of Theorem 1 is complete.
3.1. Optimality of Theorem 1. In this subsection, it is shown that the exponent is optimal in the sense that for every λ "close" to 1`one can exhibit a lacunary sequence Λ with ratio ρ Λ P rλ, λ 3 q and choose a p " ppλq "close" to 1`such that (1.4) holds. For this, the idea is to consider lacunary sequences Λ whose terms essentially behave like λ jqj pλ´1q´1 for all j P N 0 , where q j " 1, j P N 0 . To be more precise, we need the following elementary construction.
Proof. Given a λ ą 1 with λ 3 ă 2, fix a r λ P pλ 3{2 , λ 2 q and then consider the auxiliary sequence pα j q jPN0 given by α j :" r r λ j s, j P N 0 . Note that it follows from the definition of pα j q jPN0 that Note also that, since r λ ă λ 2 , the left-hand side of (3.11) implies that for all j ě j 0 .
To prove (3.9), note that for all j P N one has where we used the fact that the map t Þ Ñ tpt`1q´1 is increasing on r0, 8q and (3.11). This completes the proof of the left-hand side of (3.9). To prove the right-hand side of (3.9), note that by the definition of Λ one has Since r λ`λ´1 ă λ 2`λ´1 and the map t Þ Ñ t 3´t2´t`1 is increasing on r1, 8q, the proof of the right-hand side of (3.9) follows from the last estimate.
Given a λ ą 1 with λ 3 ă 2, construct a lacunary sequence Λ in N as in Lemma 5 and then consider the corresponding square function S pΛq .
Let N P N be such that and observe that if we choose p " λ, then Consider now the de la Vallée Poussin kernel V N of order N , namely K n being the Fejér kernel of order n; K n pxq :" ř |j|ďn`1´| j|{pn`1q˘e ijx , x P T. As in [4], consider the analytic trigonometric polynomial f N given by Using (3.15), one has (3.16) }f N } L p pTq À 1.
Indeed, arguing as on p. 424 in [26], observe that }f N } L 1 pTq À 1 and }f N } L 8 pTq À N and hence, interpolation implies that }f N } L p pTq À N 1´1{p . Therefore, (3.16) follows from (3.15). Next, we claim that the set A pΛq ( is non-empty and has cardinality (3.17) #pA To prove that A pΛq N is non-empty, observe that it follows from (3.8) and (3.14) that and hence, there exists a k 0 P N such that Using now the right-hand side of (3.9), we obtain It thus follows that k 0 P A pΛq N and so, A pΛq N is a non-empty set of indices. In order to prove (3.17), note that if we set k :" #pA pΛq N q and k 0 P N is as above, then the definition of A pΛq N and the left-hand side of (3.9) give 2N ě λ k0`k´1 ě λ k´1 λ k0 ě λ k´1 N and so, λ k´1 ď 2. Hence, k ď 1`log λ 2 " rlog λs´1 and since rlog λs´1 " pλ´1q´1, we get the upper estimate By the definitions of A pΛq N , k 0 , k, and the right-hand side of (3.9), we obtain 2N ď λ k0`k`1 ď λ 3pk`1q λ k0´1 ă λ 3pk`1q N and so, pλ´1q´1 " log λ 2 ă 3pk`1q. Hence, we also get the lower estimate and therefore, the proof of (3.17) is complete in view of (3.18) and (3.19).
Going now back to the proof of (1.4), observe that, thanks to the definition of f N , one has for all j P A pΛq N . Therefore, by using (3.20) and (3.14), one deduces that for all j P A pΛq N with j ą k 0`1 , k 0 being as above. Hence, arguing again as in [10], by using Minkowski's inequality together with the last estimate and (3.17), we get It thus follows from the last estimate combined with (3.16) that and this implies the desired bound (1.4), as (3.15) gives log N " pp´1q´1.
Remark 6. Observe that for any fixed p P p1, 2q, if one sets λ :" p 1{3 and defines Λ " pλ j q jPN0 , N , and f N as above, then one has λ ď ρ Λ ă λ 3 " p, log N " pp´1q´1 " pλ´1q´1 and the previous argument shows that which is the lower estimate mentioned in Remark (i) in [29,Section 3]. Notice that the aforementioned lower bound also shows that the estimate in Theorem 1 cannot be improved in general.

A classical inequality of Paley.
A classical theorem of Paley [27] asserts that if Λ " pλ j q jPN0 is a lacunary sequence in N, then for every f P H 1 A pTq the sequence p p f pλ j qq jPN0 is square summable. Moreover, Paley's argument in [27] yields that if Λ is a lacunary sequence in N with ratio ρ Λ P p1, 2q then Paley's inequality was extended by D. Oberlin's in [25]; see Corollary on p. 45 in [25]. We remark that by using Lemma 3, iteration and multi-dimensional Khintchine's inequality (2.2), one recovers D. Oberlin's extension of Paley's inequality, namely if Λ n " pλ pnq jn q jnPN0 is a lacunary sequence in N with ratio ρ Λn P p1, 2q for n P t1,¨¨¨, du, then Furthermore, an adaptation of the argument presented in the previous subsection shows that the exponents r " 1{2 in ś d n"1 pρ Λn´1 q´1 {2 in (3.22) cannot be improved in general.

Proof of Theorem 2
As mentioned in the introduction, the first step in the proof of Theorem 2 is to establish the weak-type inequality (1.6) for X " L log 1{2 LpTq and this inequality will be obtained by using the work of Tao and Wright on the endpoint behaviour of Marcinkiewicz multiplier operators acting on functions defined over the real line [37].
To be more precise, let η be a fixed Schwartz function that is even, supported in p´8,´1{8q Y p1{8, 8q and such that η| r1{4,4s " 1. For j P N, set η j pξq :" ηp2´jξq for ξ P R and denote by r ∆ j the periodic multiplier operator whose symbol is given by η j | Z , i.e. r ∆ j " T ηj | Z . We shall also consider the sequence of functions pφ j q jPN given by By arguing exactly as on pp. 547-549 in [37] one deduces that [37, Proposition 9.1] implies that for every f P L log 1{2 LpTq there exists a sequence pF j q jPN of non-negative functions such that for every j P N one has We omit the details. Notice that (4.2) can be regarded as a periodic analogue of [37,Proposition 4.1]. Fix now a lacunary sequence Λ " pλ k q kPN0 of positive integers with ρ Λ P p1, 2q. Note that it follows from the mapping properties of the periodic Hilbert transform that for every a, b P R such that a ă b, the multiplier operator T χ ta,¨¨¨,bu is of weaktype p1, 1q and bounded on L p pTq for all 1 ă p ă 8 with corresponding operator norm bounds that are independent of a, b. We may thus assume, without loss of generality, that λ 0 ě 8.
For technical reasons, in order to suitably adapt the relevant arguments of [37] to the periodic setting and prove that S pΛq satisfies (1.6) for X " L log 1{2 LpTq, it would be more convenient to work with S p r Λq , where r Λ is a strictly increasing sequence in N, which is associated to Λ and satisfies the properties of the following lemma.
There exists a Λ 1 Ď N such that if we regard r Λ :" Λ Y Λ 1 Y p2 j`3 q jPN0 as a strictly increasing sequence in N and write r Λ " p r λ k q kPN0 , then r λ 0 " 8 and moreover, r Λ has the following properties: (1) For every k P N there exists a positive integer j k ě 4 such that Proof. The desired construction is elementary. First of all, note that if we regard the set Λ d :" Λ Y p2 j`3 q jPN0 as a strictly increasing sequence in N and write Λ d " pλ pdq k q kPN0 , then λ pdq 0 " mintλ 0 , 8u " 8 and for every k P N there exists a unique positive integer j k ě 4 such that I be as above.
k s, and ra p8q k , λ pdq k q have lengths that are less or equal than 2 j k´3 . Here, we make the standard convention that ra, as " tau for a P R. For such a choice of a p1q k ,¨¨¨, a p8q k , we define To verify property p2q, observe that by the definition of Λ 1 one has Hence, the proof of the lemma is complete.
Note that if r Λ is as in Lemma 7, then one has (4.3) S pΛq pf qpxq À S p r Λq pf qpxq px P Tq for every trigonometric polynomial f on T. Hence, it is enough to work with the operator S p r Λq instead of S pΛq in view of (4.3). We now fix a trigonometric polynomial f on T. For every given k P N, we consider j k P N as in Lemma 7, namely j k is the positive integer satisfying the property r r λ k´1 , r λ k q Ď r2 j k´1 , 2 j k q and we then set where pF j q jPN is the sequence of non-negative functions associated to f such that (4. where the implied constant does not depend on Λ and ω. As some of the technicalities in the periodic setting are slightly more involved than the ones in the Euclidean case, for the convenience of the reader, in Subsection 4.1 we present how (4.5) can be obtained by adapting the arguments of Tao and Wright [37] to our case. Assuming now that (4.5) holds, note that it easily follows from the definition (4.4) of`r F k˘k PN0 and the second property of r Λ in Lemma 7 that there exists an absolute constant A ą 0 such that for all x P T.
To justify the last step, notice that if T is a linear, translation-invariant operator acting on functions defined over the torus and such that }T } L p 0 pTqÑL p 0 pTq ď 1 for some p 0 ą 1 and }T } L log r LpTqÑL 1 pTq ď D 0 for some r ą 0, then by carefully examining the proof of [36, Theorem 1.1], one deduces that (4.11) }T } L p pTqÑL p pTq ď D 0 M r,p0 pp´1q´r pp Ñ 1`q, where M r,p0 ą 0 is a constant depending only on r and p 0 , but not on D 0 . Indeed, if T is above, then note that in the proof of [36, Theorem 1.1], }T } L p pTqÑL p pTq is estimated by the sum of the quantities on the right-hand sides of [36, (13)] and [36, (14)]. In the proof of [36, (13)] only the fact that T is bounded on L p0 pTq is used in [36]. Moreover, it follows from the argument in [36, Section 3] that the implicit constant on the right-hand side of [36, (14)] depends linearly on the implicit constant in [36, (1)]. In turn, the proof of [36, Lemma 2.1] yields that the implicit constant in [36, (1)] depends linearly on }T } L log r LpTqÑL 1 pTq and one can thus conclude that (4.11) holds. To complete the justification of (4.10), observe that in our case we have T " ř kPN0 r k pωq∆ p r Λq k and so, by employing (4.8) (i.e. p 0 " 2) and (4.9), we may take r " 3{2 and D 0 " A 0 pρ Λ´1 q´1 {2 . We thus see that (4.10) indeed holds, in view of (4.11).
Therefore, the proof of (1.5) is now obtained by using (4.10), Khintchine's inequality (2.2), and (4.3).  if one defines F j :" | r ∆ j pf q| then it follows from the work of Tao and Wright adapted to the torus (see also the next subsection) that for every choice of ω P Ω one has (4.12) Hence, by using the following Littlewood-Paley inequality (4.13) together with (4.12), (2.2), and (4.3), one obtains (1.6) for X " H 1 pTq. Note that (4.13) is a consequence of e.g. [12,Theorem 1.20] or the work of Stein [31,32].

4.1.
Proof of (4.5). In this subsection, we show how one can adapt the argument on pp. 535-540 in [37] to the periodic setting and establish (4.5).
For this, fix a trigonometric polynomial f on T and let r F :" p r F k q kPN be as above. We shall prove that for every α ą 0 one has (4.14)ˇˇ!x P T :ˇˇˇˇÿ where C 0 ą 0 is an absolute constant, independent of ω, Λ, α, and f . Towards this aim, fix an α ą 0 and consider the set where M c denotes the centred Hardy-Littlewood maximal function acting on functions defined over T. Then one has By using a Whitney-type covering lemma; see e.g. pp. 167-168 in [33], it follows that there exists a countable collection of arcs pJq JPJ in T whose interiors are mutually disjoint, satisfy |J| ď 1{8 for all J P J as well as the properties where c 0 ą 0 is an absolute constant. We thus have where the absolute constant N 0 ą 0 is as in (4.15 where the implied constant in (4.20) is independent of F and α. Define G " pg k q kPN by g k pxq :" Using (4.19), (4.20) as well as Minkowski's integral inequality one deduces that where the implied constant in (4.21) does not depend on F and α. We also define B " pb k q kPN by b k :" r F k´gk , k P N. Then it easily follows from (4.20) that where the implied constant in (4.22) is independent of F and α. Notice that we also have that ş J b k pxqdx " 0 for all k P N, but we will not exploit this property here.
Next, we write r Λ " p r λ k q kPN0 and for k P N define the intervals I k :" r r λ k´1 , r λ k q and I 1 k :" p´r λ k ,´r λ k´1 s.
. Hence, to prove (4.14), it suffices to show that where C 0 is the constant in (4.14). We shall only focus on the proof of (4.23), as the proof of (4.24) is completely analogous. To prove (4.23), for k P N consider the functions and for a fixed Schwartz function ψ that is even, supported in r´2, 2s and such that ψ| r´1,1s " 1, let ψ k pξq :" ψ`|I k |´1 " ξ´ξ I k ‰˘, ξ P R, where ξ I k denotes the centre of I k . Consider now the multiplier T ψ k | Z and note that it follows from the definition of ψ k and Lemma 7 that where j k P N is such that I k Ď r2 j k´1 , 2 j k q and r ∆ j k " T ηj k | Z with η j being as in the previous section. Hence, it follows from (4.1) and the smoothness of ψ that there exists an absolute constant M 0 ą 0 such that (4.25) |T ψ k | Z pf qpxq| ď M 0`r F k˚φI k˚φ j k˘p xq for all x P T, where j k P N is as above, i.e. I k Ď r2 j k´1 , 2 j k q and φ j k is as in the previous section, namely φ j k pxq " 2 j k r1`2 2j k sin 2 px{2qs´3 {4 , x P T. For k P N, if r F k is not identically zero, define the function a k by Otherwise, define a k pxq :" 0, x P T. Hence, the definition of a k and (4.25) imply that }a k } L 8 pTq ď M 0 for all k P N. For each k P N, we thus have that nd so, to prove (4.23), it suffices to show thaťˇˇ!
x P T :ˇˇˇˇÿ where C 1 0 ą 0 is an absolute constant that does not depend on ω, Λ, α, and f . Since r F k " g k`bk , it is enough to prove thaťˇˇ!
for some absolute constant C 2 0 ą 0. To establish the first bound involving G " pg k q kPN , note that by using Chebyshev's inequality, Parseval's identity twice as well as the fact that }a k } L 8 pTq ď M 0 for all k P N, one obtainšˇˇ! x P T :ˇˇˇˇÿ kPN r k pωq∆ p r Λ,`q k`a k pg k˚φI k˚φ j k q˘pxqˇˇˇˇą where the implied constant is independent of ω, Λ, α, and f . Using now Young's inequality twice combined with the fact that there exists an absolute constant C ą 0 such that }φ I k } L 1 pTq ď C and }φ j k } L 1 pTq ď C for all k P N, we deduce thaťˇˇ!
where the implied constant does not depend on ω, Λ, α, and f . Hence, the desired inequality for G " pg k q kPN follows from the last estimate combined with (4.21).
To prove the desired weak-type inequality involving B " pb k q kPN , for J P J we write b k,J :" χ J b k , k P N and, as in [37], we havěˇˇ! We shall prove separately that (4.26) and (4.27) where the implied constants in (4.26) and (4.27) do not depend on ω, Λ, α, and f . 4.1.1. Proof of (4.26). To prove the desired bound for I 1 , observe that by using Chebyshev's inequality, Parseval's identity twice, the bound }a k } L 8 pTq ď M 0 and then Young's inequality together with the fact that there exists an absolute constant C ą 0 such that }φ j k } L 1 pTq ď C, one has (4.28) where the implied constant is independent of ω, Λ, α, and f . To get an appropriate bound for the right-hand side of (4.28), we shall use the following lemma.
Lemma 10. Let b k,J and φ I k be as above.
Proof. We may suppose, without loss of generality, that 2J can be regarded as an interval in r´π, πq. To prove (4.29), take an x P r´π, πq and consider two cases; x P 2J and x R 2J. Assume first that x R 2J. In the subcase where |x´y| ď 3π{2 for all y P J (with π ď y ă π) note that for all y, y 1 P J we have |x´y| ě |x´y 1 |{3. Hence, if y J denotes the centre of J, we get where the implied constants in the above chain of inequalities do not depend on k, J and we used the fact that for all x P r´3π{2, 3π{2s, which is a direct consequence of the elementary inequalities y{8 ď sinpy{2q for y P r0, 3π{2s and sinpyq ď y for all y ě 0. Notice that the inequality on the left-hand side of (4.30) holds for all x P R. If we now consider the subcase where |x´y| ą 3π{2 for some y P J (with´π ď y ă π), then |x´y| " |x| " |y| for all y P J (noting that |J| ď 1{8) and so, by using (4.30), one has |b k,J˚φI k pxq| À ż r´π,πq If x P 2J, observe that |I k ||x´y| ď 2|I k ||J| ď 2 for all y P J and hence, by using the trivial estimate |b k,J˚φI k pxq| ď }b k,J } L 1 pTq |I k | and (4.30), we get where the implied constant is independent of k, J.

4.1.2.
Proof of (4.27). We shall now estimate the remaining term I 2 . As in [37], for k P N 0 and J P J with |I k ||J| ą 1, we write b k,J˚φI k˚φ j k " χ Tzp2Jq pxqpb k,J˚φI k˚φ j k qpxq`χ 2J pxqpb k,J˚φI k˚φ j k qpxq for x P T and hence, we have the inequality :"ˇˇ!x P T :ˇˇˇˇÿ kPN,JPJ : We shall first handle the term I piq 2 . For this, we need the following lemma. Lemma 11. Assume that |I k ||J| ą 1.
There exists an absolute constant C 0 ą 0 such thaťˇχ for all x P T.
Proof. To prove the desired estimate, note that there exists an absolute constant C ą 0 such that for all x P r´π, πq. To see this, consider first the case where x P r´π{2, π{2s and write φ I k˚φ j k pxq " I 1 pxq`I 2 pxq`I 3 pxq, where I 1 pxq :" p2πq´1 ż´π ďyăπ: |x|ą2|y| φ j k pyqφ I k px´yqdy, Observe that in this subcase one has |x´y| ď 3π{2 for all y P r´π, πq and so, by using (4.30) twice, one gets Since ş T φ j k pyqdy À 1, one deduces that I 1 pxq is Opφ I k pxqq. Similarly, one shows that I 2 pxq is Opφ I k pxqq and I 3 pxq is Opφ j k pxqq. Therefore, by putting the above estimates together, it follows that (4.35) holds when x P r´π{2, π{2s. If we now assume that x P r´π, πq and |x| ą π{2, we write The first integral is handled as in the first case. For the second integral, we have |y| " |x| for all y P r´π, πq with |x´y| ą 3π{2 and so, by using a version of (4.30) for φ j k as well as the fact that ş T φ I k pyqdy À 1, we get ż´π ďyăπ: |x´y|ą3π{2 φ j k pyqφ I k px´yqdy À ż´π ďyăπ: |x´y|ą3π{2 Therefore, by using (4.35), one has (4.36) |b k,J˚φI k˚φ j k pxq| ď C " |b k,J |˚φ I k pxq`|b k,J |˚φ j k pxq ‰ px P Tq.
We may suppose that 2J can be regarded as an interval in r´π, πq. To estimate the first term on the right-hand side of (4.36), take an x P r´π, πqz2J and consider first the case where |x´y| ď 3π{2 for all y P J. Note that, by using (4.30), one has where the constants in the above chain of inequalities do not depend of k, J and we used the assumption that |I k ||J| ą 1 as well as the fact that M c pχ J qpxq " |J|rdistpx, Jqs´1 when x R 2J. In the case where |x´y| ą 3π{2 for some y P J, one has |x| " |y| " |x´y| " 1 for all y P J and hence, Since we have 2 j k |J| ě |I k ||J| ą 1, a similar argument shows that |b k,J |˚φ j k pxq is Op}b k,J } L 1 pTq |J|´1rM c pχ J qpxqs 3{2 q for all x R 2J and hence, the proof of the lemma is complete.
Having established Lemma 11, observe that it follows from Chebyshev's inequality and Parseval's identity that where C ą 0 is an absolute constant. Since we may write an application of a periodic version of the Fefferman-Stein maximal theorem [16, Theorem 1 (1)] (for r " 3{2 and p " 3) gives us that where in the last step we used the fact that the arcs pJq JPJ have mutually disjoint interiors. We thus deduce that there exists an absolute constant A 0 ą 0 such that Hence, by using (4.33), (4.37), and (4.18), it follows that where the implied constant does not depend on ω, Λ, α, and f .
We shall now prove that To this end, for |I k ||J| ą 1, we decompose the projection ∆ where the operators T P k,J , T Q k,J , T r P k,J are defined as follows. The operators T P k,J and T r P k,J are of convolution-type and their kernels p k,J and r p k,J are trigonometric polynomials given by p k,J pxq :" e i r λ k´1 x K t|J|´1u pxq px P Tq and r p k,J pxq :" e ip r λ k´1 qx K t|J|´1u pxq px P Tq respectively, where K N denotes the Fejér kernel of order N . Observe that T P k,J and T r P k,J are multiplier operators and their symbols P k,J and r P k,J are supported in r λ k´1´t |J|´1u,¨¨¨, r λ k´1`t |J|´1u ( and r λ k´1´t |J|´1u,¨¨¨, r λ k´1`t |J|´1u ( , respectively. The symbol of the multiplier operator T Q k,J is given by Hence, if we set c k,J pxq :" a k pxqχ 2J pxqpb k,J˚φI k˚φ j k qpxq px P Tq, we have :"ˇˇ!x P T :ˇˇˇˇÿ kPN,JPJ : To estimate A piiq 2 , note that by using Chebyshev's inequality and Parseval's identity twice, as well as the fact that the for every given n P Z there are at most 3 non-zero terms pT P k,J pc k,J qq p pnq for |I k ||J| ą 1, one gets for some absolute constant C ą 0. The right-hand side of (4.41) will essentially be estimated as in the previous case. More precisely, we have the following pointwise bound.
Lemma 12. If T P k,J and c k,J are as above, then one has (4.42) |T P k,J pc k,J qpxq| ď C 0 }b k,J } L 1 pTq |J|´1rM c pχ J qpxqs 2 for all x P T, where C 0 ą 0 is an absolute constant.
Proof. First of all, note that since c k,J is supported in 2J, it follows from the definition of T P k,J that we may write (4.43) |T P k,J pc k,J qpxq| ď p2πq´1 ż 2J |c k,J pyq|K t|J|´1u px´yqdy.
To prove (4.42), we shall consider two cases; x P 2J and x R 2J. If x P 2J then we use the trivial bound }K t|J|´1u } L 8 pTq À |J|´1 and so, (4.43) gives |T P k,J pc k,J qpxq| À |J|´1}c k,J } L 1 pTq . Hence, by first using the bound }a k } L 8 pTq À 1 and then Young's inequality combined with the facts that }φ I k } L 1 pTq À 1 and }φ j k } L 1 pTq À 1, one gets |T P k,J pc k,J qpxq| À |J|´1}b k,J } L 1 pTq .
Since M c pχ J qpxq " 1 when x P 2J, the desired inequality (4.42) follows from the last estimate.
In the case where x R 2J, by using (4.43) and the definition of the Fejér kernel, then one deduces, by arguing as in the proof of Lemma 11, that (4.44) |T P k,J pc k,J qpxq| À }c k,J } L 1 pTq t|J|´1u´1 " distpx, Jq ‰´2 .
By using (4.42), (4.41) gives and so, by arguing as in the previous case, namely by using a periodic version of [16, Theorem 1 (i)] (this time for r " 2 and p " 4) as well as (4.33), we deduce that It thus follows from the last etimate combined with (4.18) that To obtain an appropriate estimate for B piiq 2 , observe that by using Chebyshev's inequality, the triangle inequality as well as by exploiting the translation-invariance of the operator ÿ kPN: where r J denotes the arc r´|J|, |J|s in T (noting that by our construction |J| ď 1{8) and r c k,J is a translated copy of c k,J such that supppr c k,J q Ď r J. We thus see that, in view of (4.18), it suffices to handle the second term on the right-hand side of (4.47). To this end, fix a Schwartz function s supported in r´1, 1s with sp0q " 1 and set s J pxq :" sp|J|´1xq. Hence, suppps J q Ď r J " r´|J|, |J|s and s J p0q " 1. So, if σ J denotes the periodisation of s J , then supppσ J q Ď r J and moreover, the Fourier coefficients of σ J satisfy the properties (4.48) x σ J pnq " |J|p sp|J|nq for all n P Z and (4.49) ÿ nPZ x σ J pnq " 1.
Consider now the multiplier operator T R k,J whose symbol R k,J is given by R k,J pnq :" Q k,J pnq´`Q k,J˚x σ J˘p nq, n P Z.
Notice that if q k,J denotes the kernel of T Q k,J , then one has for all x P Tzp2 r Jq and so, it is enough to show that for every J P J one has (4.50) where D 0 ą 0 is an absolute constant. Towards this aim, consider the trigonometric polynomial γpxq :" e ix´1 px P Tq and observe that, by using the Cauchy-Schwarz inequality, the left-hand side of (4.50) is majorised by Hence, by using Parseval's theorem, one has (4.51) .
We shall prove that there exists an absolute constant C 0 ą 0 such that (4.52) B J ď C 0 |J| 3{2 α for every J P J . To show (4.52), given a k P N such that |I k ||J| ą 1, consider the following subsets of I k , L k,J :" r λ k´1 ,¨¨¨, r λ k´1`t |J|´1u ( and L 1 k,J :" r λ k´1´t |J|´1u,¨¨¨, r λ k´1 ( . The desired estimate (4.52) will easily be obtained by using the following lemma.
Lemma 13. Let k P N and J P J be such that |I k ||J| ą 1. If L k,J , L 1 k,J are as above, then for every n P Z one has where the implied constants do not depend on I k , J. Here, M d denotes the centred discrete maximal function.
Proof. We shall establish (4.53) first. To this end, fix an arbitrary n P Z and consider the following cases; n R I k and n P I k . Case 1. If n R I k , then either n ă r λ k´1 or n ě r λ k . Assume first that n ă r λ k´1 . Since Q k,J is supported in I k , one has Moreover, note that since s is a Schwartz function, we deduce from (4.48) that there exists an absolute constant A 0 ą 0 such that (4.55) |x σ J plq| ď A 0 |J| p1`|J| 2 l 2 q 2 for every l P Z.
Hence, by using the fact }Q k,J } ℓ 8 pZq " 1, it follows from (4.55) that where A 1 0 ą 0 is an absolute constant and in the last step we used the fact if A is an "interval" in Z and n R A then one has M d pAqpnq " #pAqpdistpn, Aqq´1. Hence, by using the last estimate, together with the elementary bound it follows that |R k,J pnq| is OprM d pχ L k,J qpnqs 2 q and so, (4.53) holds when n ă λ k´1 . If n ě λ k , then a similar argument shows that |R k,J pnq| is OprM d pχ L 1 k,J qpnqs 2 q and hence, (4.53) holds when n R I k .
Case 2. We shall now prove (4.53) in the case where n P I k . Towards this aim, observe that if n P L k,J Y L 1 k,J , then where we used (4.55) and the fact that if A is an "interval" in Z and n P A then one has M d pχ A qpnq " 1.
If we now assume that I k z`L k,J Y L 1 k,J˘‰ H and n P I k z`L k,J Y L 1 k,J˘, then Q k,J pnq " 1 and so, (4.49) gives To bound the second term, note that by using (4.55) and the fact that }Q k,J } ℓ 8 pZq " 1, one has ÿ where the implied constants are independent of k and J. The third term is handled similarly and is OrM d pχ L 1 k,J qpnqs 2 . For the first term, notice that by using (4.55) one has ÿ lă r λ k´1 One shows that the fourth term is OprM d pχ L 1 k,J qpnqs 2 q in a completely analogous way. Hence, by putting all the estimates together, we deduce that (4.53) also holds in this subcase.
We shall now prove (4.54). Towards this aim, we argue as in the proof of (4.53) and, for a given n P Z, consider the cases where n R I k and n P I k .
Case 1'. If n R I k , then Q k,J pn`1q " Q k,J pnq " 0 and so, one has |R k,J pn`1q´R k,J pnq| "ˇˇˇˇÿ Notice that, since for all k P Z one has x σ J pkq " |J|p sp|J|kq and s is a Schwartz function, it follows from the last estimate and the mean value theorem that where A ą 0 is an absolute constant. Therefore, by arguing as in case 1, if n ă r λ k´1 , the quantity on the right-hand side of (4.56) is Op|J|rM d pχ L k,J qpnqs 2 q, whereas if n ě r λ k , the quantity on the right-hand side of (4.56) is Op|J|rM d pχ L 1 k,J qpnqs 2 q. We thus deduce that (4.54) holds when n R I k .
Case 2'. If n R I k , then either n P L k,J Y L 1 k,J or n P I k zpL k,J Y L 1 k,J q (assuming that I k z`L k,J Y L 1 k,J˘‰ H).
If n P L k,J Y L 1 k,J , we have rM d pχ L k,J qpnqs 2`r M d pχ L 1 k,J qpnqs 2 " 1 and hence, (4.57) |Q k,J pn`1q´Q k,J pnq| À |J| " |J|`"M d pχ L k,J qpnq where the implied constants do not depend on k, J. As in case 1', by using the definition of σ J and the mean value theorem, one has ÿ We thus deduce that the right-hand side of the last inequality is Op|J|q, which combined with (4.57), gives (4.54) in the subcase where n P L k,J Y L 1 k,J . It remains to treat the subcase where n P I k zpL k,J Y L 1 k,J q. To this end, note that, in view of (4.49), we may write For the first term, we have |x σ J pn`1´lq´x σ J pn´lq| and, by arguing as above, we see that both terms on the right-hand side of the last estimate are Op|J|rM d pχ L k,J qpnqs 2 q. Similarly, one shows that |T k,J pnq| is Op|J|rM d pχ L 1 k,J qpnqs 2 q. For the last term, observe that if n P I k zpL k,J Y L 1 k,J q, then Q k,J pn`1q " Q k,J pnq " 1 and so, Hence, by arguing as above, one deduces that where the implied constants do not depend on k, J. Therefore, (4.54) also holds in this subcase and hence, the proof of the lemma is complete.
Going now back to the proof of (4.52), observe that for every n P Z one hašˇp r c k,J q p pn`1q´pr c k,J q p pnqˇˇď ż r J |e ix´1 ||r c k,J pxq|dx ď 2|J|}c k,J } L 1 pTq À |J|}b k,J } L 1 pTq and |pr c k,J q p pnq| À }b k,J } L 1 pTq . It thus follows from Lemma 13 that there exists an absolute constant C 0 ą 0 such that By using a version of the Fefferman-Stein theorem [16] for the discrete maximal operator M d ; see e.g. [19,Theorem 1.2] as well as the fact that the "intervals" L k,J are mutually disjoint and have cardinality Op|J|´1q, we deduce that where in the above chain of inequalities the implied constants are independent of k and J. Similarly, one shows that and we thus have where D 0 ą 0 is an absolute constant. Hence, by using the last estimate combined with (4.33), one obtains (4.52).

Variants and remarks in the one-dimensional case
The argument of Tao and Wright [37] adapted to the torus, as presented in the previous section, can actually be used to handle Littlewood-Paley square functions formed with respect to finite unions of lacunary sequences in N. To be more specific, observe that an increasing sequence Λ of positive integers can be written as a finite union of lacunary sequences in N if, and only if, the quantity s finite. Hence, given a finite union of lacunary sequences Λ, if f is an arbitrary trigonometric polynomial on T and pF j q jPN and p r F k q kPN are as in the previous section, it follows that there exists an absolute constant A ą 0 such that for all x P T.
Note that (5.2) can be regarded as the analogue of (4.6) in this case. Therefore, by arguing exactly as in Subsection 4.1, one shows that Hence, by using interpolation (where in this case, one uses (5.3) instead of (4.7)) and Tao's converse extrapolation theorem [36], one deduces that }S pΛq } L p pTqÑL p pTq is Oppp´1q´3 {2 σ 1{2 Λ q for p P p1, 2q. Similarly, by arguing as in Remark 9, one shows that the H p A pTq Ñ L p pTq operator norm of S pΛq is Oppp´1q´1σ 1{2 Λ q for p P p1, 2q. We thus obtain the following theorem.

Theorem 14.
There exists an absolute constant C 0 ą 0 such that for every strictly increasing sequence Λ " pλ j q jPN0 in N with σ Λ ă 8 one has If Λ is a lacunary sequence in N with ratio ρ Λ P p1, 2q then one can easily check that σ Λ À pρ Λ´1 q´1 and hence, (1.3) and (1.5) are direct corollaries of (5.4) and (5.5), respectively.
Next, as in Subsection 3.1, consider the analytic trigonometric polynomial f M given by f M pxq :" e ip2M`1qx V M pxq, x P T and note that, by our choice of M and p, one has }f M } L p pTq " 1. Hence, by arguing as in Subsection 3.1, one has where in the last step (5.8) was used. This completes the proof of (5.6). The proof of (5.7) is completely analogous.

5.1.
Zygmund's classical inequality revisited. Let Λ " pλ j q jPN0 be an increasing sequence that can be written as a finite union of lacunary sequences in N or, equivalently, σ Λ ă 8 with σ Λ P N being as in (5.1). Notice that (4.1), (4.2), (5.2), and the argument on pp. 530-531 in [37] yield the following version of a classical inequality due to Zygmund (see Theorem 7.6 in Chapter XII of [38]) where A ą 0 is an absolute constant. Therefore, by using (5.9) and duality; see e.g. Remarque on pp. 350-351 in [5], one deduces that for every trigonometric polynomial g such that supppgq Ď Λ one has (5.10) }g} L p pTq ď Cσ where C ą 0 is an absolute constant. We remark that the exponent r " 1{2 in σ 1{2 Λ in (5.10) cannot be improved and therefore, the exponent r " 1{2 in σ 1{2 Λ on the right-hand side of (5.9) is sharp. Indeed, this can be shown by using the work of Bourgain [8]. To be more specific, since we assume that σ Λ ă 8, we may write Λ as a disjoint union of lacunary sequences Λ 1 ,¨¨¨, Λ N of positive integers with ρ Λj ě 2 for all j " 1,¨¨¨, N and N " σ Λ . Since lacunary sequences with ratio greater or equal than 2 are quasiindependent sets in Z; see [9,Definition 4], it follows that we may decompose Λ as a disjoint union of N " σ Λ quasi-independent sets. We thus conclude that for every finite subset A of Λ there exists a quasi-independent subset B of A such that #pBq ě N´1#pAq " σ´1 Λ #pAq and hence, (5.10) as well as its sharpness (in terms of the dependence with respect to σ Λ ) follows from [8, Lemma 1], see also pp. 101-102 in [9].
For variants of the aforementioned result of Zygmund in higher dimensions, see [2], [3] and the references therein.

Higher-dimensional versions of Theorems 1 and 2
In this section we extend Theorem 1 to the d-torus. To state our result, fix a d P N and consider strictly increasing sequences Λ n " pλ pnq jn q jnPN0 of positive integers, n P t1,¨¨¨, du. Then, the d-parameter Littlewood-Paley square function formed with respect to Λ n (n P t1,¨¨¨, du) is given by and is initially defined over the class of all trigonometric polynomials f on T d .
Theorem 15. Given a d P N, there exists an absolute constant C d ą 0 such that for every 1 ă p ă 2 and for all lacunary sequences Λ n " pλ pnq jn q jnPN0 in N with ratio ρ Λn P p1, 2q, n P t1,¨¨¨, du, one has Proof. We shall prove (6.1) first. To this end, the idea is to fix a probability space pΩ, A, Pq and show that there exists an absolute constant C ą 0 such that for every choice of ω P Ω one has for every analytic trigonometric polynomial g on T. Then the proof of (6.1) is obtained by iterating (6.3).
To prove (6.3), note that by using estimate p3.2q in Bourgain's paper [10], which follows from Bourgain's extension [7] of Rubio de Francia's theorem [30], one deduces that there exists an absolute constant M ą 0 such that for every q P r1, 2s one has (6.4) }h} L q pTq ď M › › S pΛq phq › › L q pTq for every analytic trigonometric polynomial h on T. Hence, in order to establish (6.3), fix a trigonometric polynomial g and consider the trigonometric polynomial h ω given by Observe that (6.4) applied to h ω implies that for all p P p1, 2q and for every analytic trigonometric polynomial g on T. Hence, the desired estimate (6.3) follows from the last inequality combined with (1.3). Alternatively, (6.3) can be obtained by using (4.12) and Marcinkiewicz interpolation for analytic Hardy spaces for the operator T pΛq ω :" ř jPN0 r j pωq∆ pΛq j , as explained in Section 3.
To complete the proof of (6.1), fix a p P p1, 2q and take an analytic trigonometric polynomial f on T d . By iterating (6.3), we get where C ą 0 is the constant in (6.3). Hence, by using the last estimate together with multi-dimensional Khintchine's inequality (2.2) and the density of analytic trigonometric polynomials on T d in pH p A pT d q, }¨} L p pT d q q, (6.1) follows. The proof of (6.2) is similar. Indeed, by iterating (4.10), one gets for every trigonometric polynomial f on T d , where r Λ n is a strictly increasing sequence in N associated to Λ n and is constructed as in Lemma 7, n P t1,¨¨¨, du. Hence, the proof of (6.2) is obtained by using the last estimate, (2.2), the fact that for every trigonometric polynomial f on T d one has S pΛ1,¨¨¨,Λ d q pf qpxq À d S p r Λ1,¨¨¨, r Λ d q pf qpxq px P T d q, and the density of trigonometric polynomials on T d in pL p pT d q, }¨} L p pT d q q. An adaptation of the argument presented in Subsection 3.1 to higher dimensions shows that (6.1) is optimal in the following sense. For every choice of λ "close" to 1`, there exists a lacunary sequence Λ "`λ j˘j PN0 in N with ratio ρ Λ P rλ, λ 3 q and a p "close" to 1`such that Similarly, one deduces that (6.2) is also optimal.
Remark 17. By arguing as in the previous section, one shows that if Λ n " pλ pnq jn q jnPN0 are finite unions of lacunary sequences (n " 1,¨¨¨, d), then and that the above estimates cannot be improved in general.

Euclidean Variants
Let Λ " pλ j q jPZ be a countable collection of positive real numbers indexed by Z such that λ j`1 ą λ j for all j P Z as well as lim jÑ´8 λ j " 0 and lim jÑ`8 λ j "`8.
For j P Z define the j-th "rough" Littlewood-Paley projection P pΛq j to be the multiplier with symbol χ Ij YI 1 j , where I j :" rλ j´1 , λ j q and I 1 j :" p´λ j ,´λ j´1 s. Given a d P N and sequences Λ 1 ,¨¨¨, Λ d as above, define the d-parameter "rough" Littlewood-Paley square function initially over the class of Schwartz functions on R d .
A Euclidean version of (6.1) is given by the following theorem.
Theorem 18. Given a d P N, consider sequences Λ 1 ,¨¨¨, Λ d that are as above and moreover, satisfy σ Λn :" sup jPZ #`Λ n X r2 j´1 , 2 j q˘ă 8 for all n P t1,¨¨¨, du. Then, there exists an absolute constant C d ą 0, depending only on d, such that Similarly, a Euclidean version of (6.2) is the content of the following theorem.
Then, there exists an absolute constant C d ą 0, depending only on d, such that 7.1. Proof of Theorem 18. Let n P t1,¨¨¨, du be a fixed index. Note that if we consider a probability space pΩ, A, Pq and Λ n " pλ pnq jn q jnPZ is as in the hypothesis of the theorem, then it follows from the work of Tao and Wright [37], see also Remark 9, that there exists an absolute constant C 0 ą 0 such that for every choice of ω n P Ω.
We can now argue as in [4,Section 5]. Assume first that p P p1, 3{2s and note that by using ( where A 0 ą 0 is an absolute constant; see the proof of [4,Proposition 8]. To show that (7.4) also holds for p P p3{2, 2q, one can use, e.g., (7.4) for p " 3{2 and duality so that one gets an appropriate H 3 A pRq to H 3 A pRq bound and then, as in the previous case, interpolate between this bound and (7.3).
Remark 20. By adapting the argument of Section 5 to the Euclidean setting, where one uses a variant of the construction in the proof of [4, Corollary 10], one can show that the exponents r " 1{2 in σ 1{2 Λn in (7.1) are optimal. 7.2. Proof of Theorem 19. The proof of (7.2) for d " 1 is obtained by carefully examining Lerner's argument that establishes [24, Theorem 1.1]. To be more specific, the proof of (7.2) will be a consequence of the following weighted estimate pn P t1,¨¨¨, duq combined with an extrapolation result, which is due to Duoandikoetxea; see [13,Theorem 3.1]. Recall that a non-negative locally integrable function on R is said to be an A 2 weight if, and only if, rws A2 :" sup JĎR: J interval xwy J xw´1y J is finite. Here, we use the notation xσy J :" |J|´1 ş J σpyqdy. For a fixed index n P t1,¨¨¨, du, set r Λ n :" Λ Y p2 j q jPZ and observe that for every Schwartz function f one has S pΛnq R pf qpxq À S p r Λnq R pf qpxq for all x P R. Hence, in order to establish (7.5), it suffices to prove that for every A 2 weight w one has pn P t1,¨¨¨, duq for each fixed index n P t1,¨¨¨, du. To prove (7.6), note that by arguing exactly as in [24], namely by using duality and [24,Theorem 2.7], it suffices to show that there exists an absolute constant C 2 ą 0 such that (7.7) for every A 2 weight w and for every countable collection of Schwartz functions pψ j q jPZ , see also [24,Remark 4.1]. Here, as in [24], given a dyadic lattice I in R; see [24, Definition 2.1], one sets S φ,I pf qpxq :"˜ÿ kPZ ÿ JPI: |J|"2´k where φ is a Schwartz function satisfying suppp p φq Ď r1{2, 2s and for k P Z we use the notation φ k pxq :" 2 k φp2 k xq, x P R.
By using [24,Lemma 3.2], for every k P Z, one gets › › ›P p r Λnq j k pψ j k q˚φ k › › › L 2 pw k,I q ď C 0 rws A2 }ψ j k } L 2 pwq for all j k P Z such that r I pnq j k Ď r2 k´1 , 2 k`1 q, where C 0 ą 0 is a constant depending only on the function φ. Hence, (7.7) is obtained by combining the last two estimates. Therefore, we deduce that (7.5) holds.
Hence, by arguing as in [ for each index n P t1,¨¨¨, du.
To obtain the higher-dimensional case, given a probability space pΩ, A, Pq, observe that (7.8) combined with a Euclidean version of [10, (3.2)] implies that for every ω n P Ω and for all indices n P t1,¨¨¨, du. Therefore, (7.2) is obtained by first iterating (7.9) and then using multi-dimensional Khintchine's inequality (2.2).
Remark 21. We remark that either by adapting the modification of Lerner's argument [37] presented above to the periodic setting or by using an appropriate variant of (7.9) and transference; see Theorem 3.8 in Chapter VII in [35], one obtains an alternative proof of (6.2) and in particular, of Theorem 2 as well as of the bound (5.5) in Theorem 14.
Remark 22. By adapting the argument of Section 5 to the Euclidean setting, where one uses a variant of the construction in the proof of [1, Proposition 6.1], one can show that the exponents r " 1{2 in σ 1{2 Λn in (7.2) cannot be improved in general. Remark 23. Assume that Λ " pλ j q jPZ is a countable collection of non-negative real numbers satisfying the assumptions of Theorem 7.2. Then, it follows from the work of Tao  for every compact subset K in R. Hence, by using (7.10), interpolation, and a version of Tao's converse extrapolation theorem for operators restricted to compact sets in R; see p. 2 in [36], it follows that for every compact subset K of the real line. Notice that (7.11) is weaker than (7.2) (for d " 1).