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MULTI-PARAMETER EXTENSIONS OF A THEOREM OF PICHORIDES arXiv:1802.01372v2 [math.CA] 23 Feb 2018 ODYSSEAS BAKAS, SALVADOR RODRÍGUEZ-LÓPEZ, AND ALAN SOLA Abstract. Extending work of Pichorides and Zygmund to the d-dimensional setting, we show that the supremum of Lp -norms of the Littlewood-Paley p square function over the unit ball of the analytic Hardy spaces HA pTd q blows d ´d ` up like pp ´ 1q as p Ñ 1 . Furthermore, we obtain an L log L-estimate for 1 pTd q. Euclidean variants of Pichorides’s theorem are square functions on HA also obtained. 1. Introduction Given a trigonometric polynomial f on T, we define the classical LittlewoodPaley square function ST pf q of f by `ÿ ˘1{2 ST pf qpxq “ |∆k pf qpxq|2 , kPZ where for k P N, we set ∆k pf qpxq “ k 2ÿ ´1 n“2k´1 fppnqei2πnx and ∆´k pf qpxq “ k´1 ´2 ÿ n“´2k `1 fppnqei2πnx and for k “ 0 we take ∆0 pf qpxq “ fpp0q. A classical theorem of J.E. Littlewood and R.E.A.C. Paley asserts that for every 1 ă p ă 8 there exists a constant Bp ą 0 such that }ST pf q}Lp pTq ď Bp }f }LppTq (1) for every trigonometric polynomial f on T, see, e.g., [5] or [19]. The operator ST is not bounded on L1 pTq, and hence, the constant Bp in (1) blows up as p Ñ 1` . In [2], J. Bourgain obtained the sharp estimate Bp „ pp ´ 1q´3{2 as p Ñ 1` . (2) p For certain subspaces of L pTq, however, one might hope for better bounds. In [12], p pTq (1 ă p ď 2), we S. Pichorides showed that for the analytic Hardy spaces HA have (3) sup }ST pf q}Lp pTq „ pp ´ 1q´1 as p Ñ 1` . }f }Lp pTq ď1 p pTq f PHA Higher-dimensional extensions of Bourgain’s result (2) were obtained by the first author in [1]. In particular, given a dimension d P N, if f is a trigonometric polynomial on Td , we define its d-parameter Littlewood-Paley square function by ÿ ` ˘1{2 STd pf qpxq “ |∆k1 ,¨¨¨ ,kd pf qpxq|2 , k1 ,¨¨¨ ,kd PZ 2010 Mathematics Subject Classification. 42B15, 42B25, 42B30. Key words and phrases. Square function, Marcinkiewicz multipliers, Lp estimates. The second author is partially supported by the Spanish Government grant MTM2016-75196-P. 1 2 BAKAS, RODRÍGUEZ-LÓPEZ, AND SOLA where for k1 , ¨ ¨ ¨ , kd P Z we use the notation ∆k1 ,¨¨¨ ,kd pf q “ ∆k1 b ¨ ¨ ¨ b ∆kd pf q, where the direct product notation indicates the operator ∆kj in the j-th position acting on the j-th variable. As in the one-dimensional case, for every 1 ă p ă 8, there is a positive constant Bp pdq such that }STd pf q}Lp pTd q ď Bp pdq}f }Lp pTd q for each trigonometric polynomial f on Td . It is shown in [1] that Bp pdq „d pp ´ 1q´3d{2 as p Ñ 1` . (4) A natural question in this context is whether one has an improvement on the limiting behaviour of Bp pdq as p Ñ 1` when restricting to the analytic Hardy spaces p HA pTd q. In other words, one is led to ask whether the aforementioned theorem of Pichorides can be extended to the polydisc. However, the proof given in [12] relies on factorisation of Hardy spaces, and it is known, see for instance [14, Chapter 5] and [13], that canonical factorisation fails in higher dimensions. In this note we obtain an extension of (3) to the polydisc as a consequence of a more general result involving tensor products of Marcinkiewicz multiplier operators on Td . Recall that, in the periodic setting, a multiplier operator Tm associated to a function m P ℓ8 pZq is said to be a Marcinkiewicz multiplier operator on the torus if m satisfies Bm “ sup kPN “ k`1 2ÿ |mpn ` 1q ´ mpnq| ` n“2k ´1 k ´2ÿ `1 n“´2k`1 ‰ |mpn ` 1q ´ mpnq| ă 8. (5) Our main result in this paper is the following theorem. Theorem 1. Let d P N be a given dimension. If Tmj is a Marcinkiewicz multiplier p operator on T pj “ 1, ¨ ¨ ¨ , dq, then for every f P HA pTd q one has }pTm1 b ¨ ¨ ¨ b Tmd qpf q}Lp pTd q ÀCm1 ,¨¨¨ ,Cmd pp ´ 1q´d }f }LppTd q as p Ñ 1` , where Cmj “ }mj }ℓ8 pZq ` Bmj , Bmj being as in p5q, j “ 1, ¨ ¨ ¨ , d. To prove Theorem 1, we use a theorem of T. Tao and J. Wright [16] on the endpoint mapping properties of Marcinkiewicz multiplier operators on the line, transferred to the periodic setting, with a variant of Marcinkiewicz interpolation for Hardy spaces which is due to S. Kislyakov and Q. Xu [8].ř Since for every choice of signs the randomised version kPZ ˘∆k of ST is a Marcinkiewicz multiplier operator on the torus with corresponding constant Bm ď 2, Theorem 1 and Khintchine’s inequality yield the following d-parameter extension of Pichorides’s theorem (3). Corollary 2. Given d P N, if STd denotes the d-parameter Littlewood-Paley square function, then one has sup }f }Lp pTd q ď1 }STd pf q}Lp pTd q „d pp ´ 1q´d as p Ñ 1` . p f PHA pTd q The present paper is organised as follows: In the next section we set down notation and provide some background, and in Section 3 we prove our main results. Using the methods of Section 3, in Section 4 we extend a well-known inequality due to A. Zygmund [18, Theorem 8] to higher dimensions. In the last section we obtain a Euclidean version of Theorem 1 by using the aforementioned theorem of Tao and Wright [16] combined with a theorem of Peter Jones [7] on a Marcinkiewicz-type decomposition for analytic Hardy spaces over the real line. EXTENSIONS OF A THEOREM OF PICHORIDES 3 2. Preliminaries 2.1. Notation. We denote the set of natural numbers by N, by N0 the set of nonnegative integers and by Z the set of integers. Let f be a function of d-variables. Fixing the first d ´ 1 variables px1 , ¨ ¨ ¨ , xd´1 q, we write f px1 , ¨ ¨ ¨ , xd q “ fpx1 ,¨¨¨ ,xd´1 q pxd q. The sequence of the Fourier coefficients of a function f P L1 pTd q will be denoted by fˆ. Given a function m P L8 pRd q, we denote by Tm the multiplier operator corresponding to m, initially defined on L2 pRd q, by pTm pf qqppξq “ mpξqfppξq, ξ P Rd . Given µ P ℓ8 pZd q, one defines (initially on L2 pTd q) the corresponding periodic multiplier operator Tµ in an analogous way. If λ is a continuous and bounded function on the real line and Tλ is as above, Tλ|Z ř denotes the periodic multiplier operator such that Tλ|Z pf qpxq “ nPZ λpnqfppnqei2πnx (x P T) for every trigonometric polynomial f on T. Given two positive quantities X and Y and a parameter α, we write X Àα Y (or simply X À Y ) whenever there exists a constant Cα ą 0 depending on α so that X ď Cα Y . If X Àα Y and Y Àα X, we write X „α Y (or simply X „ Y ). p 2.2. Hardy spaces and Orlicz spaces. Let d P N. For 0 ă p ă 8, let HA pDd q d denote the space of holomorphic functions F on D , D “ tz P C : |z| ă 1u, such that ż sup }F }pH p pDd q “ |F pr1 ei2πx1 , ¨ ¨ ¨ , rd ei2πxd q|p dx1 ¨ ¨ ¨ dxd ă 8. A 0ďr1 ,¨¨¨ ,rd ă1 Td 8 For p “ 8, HA pDd q denotes the class of bounded holomorphic functions on Dd . It p is well-known that for 1 ď p ď 8, the limit f of F P HA pDd q as we approach the d d distinguished boundary T of D , namely f px1 , ¨ ¨ ¨ , xd q “ lim r1 ,¨¨¨ ,rd Ñ1´ F pr1 ei2πx1 , ¨ ¨ ¨ , rd ei2πxd q exists a.e. in Td and }F }HAp pDd q “ }f }Lp pTd q . For 1 ď p ď 8, we define the analytic p Hardy space HA pTd q on the d-torus as the space of all functions in Lp pTd q that p are boundary values of functions in HA pDd q. Moreover, it is a standard fact that p d p d d p HA pT q “ tf P L pT q : supppf q Ă N0 u. Hardy spaces are discussed in Chapter 7 in [4], where the case d “ 1 is treated, and in Chapter 3 of [14]. If f P L1 pTd q is such that supppfpq is finite, then f is said to be a trigonometric polynomial on Tn , and if moreover supppfpq Ă Nd0 , then f is said to be analytic. It is well-known [4, 14] that for 1 ď p ă 8, the class of trigonometric polynomials on Td is a dense subspace of Lp pTd q and analytic trigonometric polynomials on Td are p dense in HA pTd q. We define the real Hardy space H 1 pTq to be the space of all integrable functions f P L1 pTq such that HT pf q P L1 pTq, where HT pf q denotes the periodic Hilbert 1 pTq transform of f . One sets }f }H 1 pTq “ }f }L1 pTq ` }HT pf q}L1 pTq . Note that HA 1 can be regarded as a proper subspace of H pTq and moreover, }f }H 1 pTq “ 2}f }L1pTq 1 when f P HA pTq. p Given d P N, for 0 ă p ă 8, let HA ppR2` qd q denote the space of holomorphic 2 d 2 functions F on pR` q , where R` “ tx ` iy P C : y ą 0u, such that ż }F }pH p ppR2 qd q “ sup |F px1 ` iy1 , ¨ ¨ ¨ , xd ` iyd q|p dx1 ¨ ¨ ¨ dxd ă 8. A ` y1 ,¨¨¨ ,yd ą0 Rd 8 ppR2` qd q is defined as the space of bounded holomorphic functions For p “ 8, HA p 2 d in pR` q . For 1 ď p ď 8, for every F P HA ppR2` qd q its limit f as we approach the 4 BAKAS, RODRÍGUEZ-LÓPEZ, AND SOLA boundary Rd , namely f px1 , ¨ ¨ ¨ , xd q “ lim y1 ,¨¨¨ ,yd Ñ0` F px1 ` iy1 , ¨ ¨ ¨ , xd ` iyd q, exists for a.e. px1 , ¨ ¨ ¨ , xd q P Rd and, moreover, }F }HAp ppR2` qd q “ }f }Lp pRd q . Hence, as in the periodic setting, for 1 ď p ď 8 we may define the d-parameter analytic p Hardy space HA pRd q to be the space of all functions in Lp pRd q that are boundary p values of functions in HA ppR2` qd q. The real Hardy space H 1 pRq on the real line is defined as the space of all integrable functions f on R such that Hpf q P L1 pRq, where Hpf q is the Hilbert transform of f . Moreover, we set }f }H 1 pRq “ }f }L1 pRq ` }Hpf q}L1 pRq . We shall also consider the standard Orlicz spaces L logr LpTd q. For r ą 0, one may define L logr LpTd q as the space of measurable functions f on Td such that ş r r d Td |f pxq| log p1 ` |f pxq|qdx ă 8. For r ě 1, we may equip L log LpT q with a norm given by ż ( ` |f pxq| |f pxq| ˘ r }f }L log LpTd q “ inf λ ą 0 : logr 1 ` dx ď 1 . λ λ d T For more details on Orlicz spaces, we refer the reader to the books [9] and [19]. 3. Proof of Theorem 1 Recall that a function m P L8 pRq is said to be a Marcinkiewicz multiplier on R if it is differentiable in every dyadic interval ˘r2k , 2k`1 q, k P Z and ż ż “ ‰ 1 Am “ sup |m pξq|dξ ` |m1 pξq|dξ ă 8 (6) kPZ r2k ,2k`1 q p´2k`1 ,´2k s If m P L8 pRq satisfies (6), then thanks to a classical result of J. Marcinkiewicz1, see e.g. [15], the corresponding multiplier operator Tm is bounded on Lp pRq for all 1 ă p ă 8. In [16], Tao and Wright showed that every Marcinkiewicz multiplier operator Tm is bounded from the real Hardy space H 1 pRq to L1,8 pRq, namely }Tm pf q}L1,8 pRq ď Cm }f }H 1 pRq , (7) where the constant Cm depends only on }m}L8 pRq ` Am , with Am as in (6). For the sake of completeness, let us recall that L1,8 pMq stands for the quasi-Banach space of measurable functions on a measure space M endowed with the quasinorm ż dx. }f }L1,8pMq :“ sup t tą0 txPM:|f pxq|ątu Either by adapting the proof of Tao and Wright to the periodic setting or by using a transference argument, see Subsection 3.2, one deduces that every periodic Marcinkiewicz multiplier operator Tm satisfies }Tm pf q}L1,8 pTq ď Dm }f }H 1 pTq , (8) where Dm depends on }m}ℓ8 pZq ` Bm , Bm being as in (5). Therefore, it follows 1 that for every f P HA pTq one has 1 }Tm pf q}L1,8 pTq ď Dm }f }L1pTq , (9) 1 “ 2Dm . We shall prove that for every Marcinkiewicz where one may take Dm multiplier operator Tm on the torus one has sup }Tm pf q}Lp pTq ÀBm pp ´ 1q´1 }f }Lp pTq ď1 p f PHA pTq 1Marcinkiewicz originally proved the theorem in the periodic setting, see [10]. (10) EXTENSIONS OF A THEOREM OF PICHORIDES 5 as p Ñ 1` . To do this, we shall make use of the following lemma due to Kislyakov and Xu [8]. p0 pTq p0 ă p0 ă 8q and Lemma 3 (Kislyakov and Xu, [8] and [11]). If f P HA p0 8 λ ą 0, then there exist functions hλ P HA pTq, gλ P HA pTq and a constant Cp0 ą 0 depending only on p0 such that ´1 |f pxq|, |f pxq|´1 λu for all x P T, ‚ |hλ pxq| ď Cp0 λ mintλ ş ‚ }gλ }pL0p0 pTq ď Cp0 txPT:|f pxq|ąλu |f pxq|p0 dx, and ‚ f “ hλ ` g λ . We remark that by examining the proof of Lemma 3, one deduces that when 1 ď p0 ď 2 the constant Cp0 in the statement of the lemma can be chosen independent of p0 . To prove the desired inequality (10) and hence Theorem 1 in the onedimensional case, we argue as in [11, Theorem 7.4.1]. More precisely, given a 1 ă p ă 2, if f isş a fixed analytic trigonometric polynomial on T, we first write 8 }Tm pf q}pLp pTq “ p 0 λp´1 |tx P T : |Tm pf qpxq| ą λu|dλ and then by using Lemma 3 for p0 “ 1 we obtain }Tm pf q}pLp pTq ď I1 ` I2 , where ż8 I1 “ p λp´1 |tx P T : |Tm pgλ qpxq| ą λ{2u|dλ 0 and I2 “ p ż8 λp´1 |tx P T : |Tm phλ qpxq| ą λ{2u|dλ. 0 1 To handle I1 , we use the boundedness of Tm from HA pTq to L1,8 pTq and Fubini’s theorem to deduce that ż ´1 I1 À pp ´ 1q |f pxq|p dx. T 2 To obtain appropriate bounds for I2 , we use the boundedness of Tm from HA pTq 2 to L pTq and get ż ż8 ż ż8 |f pxq|´2 dxqdλ. λp`1 p |f pxq|2 dxqdλ ` λp´3 p I2 À 0 txPT:|f pxq|ďλu 0 txPT:|f pxq|ąλu Hence, by applying Fubini’s theorem to each term, we obtain ż ż ´1 p ´1 I2 À p2 ´ pq |f pxq| dx ` pp ` 2q |f pxq|p dx. T T Combining the estimates for I1 and I2 and using the density of analytic trigonop pTq, } ¨ }Lp pTq q, (10) follows. metric polynomials in pHA To prove the d-dimensional case, take f to be an analytic trigonometric polynomial on Td and note that if Tmj are periodic Marcinkiewicz multiplier operators (j “ 1, ¨ ¨ ¨ , d), then for fixed px1 , ¨ ¨ ¨ , xd´1 q P Td´1 one can write Tmd pgpx1 ,¨¨¨ ,xd´1 q qpxd q “ Tm1 b ¨ ¨ ¨ b Tmd pf qpx1 , ¨ ¨ ¨ , xd q, where gpx1 ,¨¨¨ ,xd´1 q pxd q “ Tm1 b ¨ ¨ ¨ b Tmd´1 pfpx1 ,¨¨¨ ,xd´1 q qpxd q. Hence, by using (10) in the d-th variable, one deduces that p pp ´ 1q´p }gpx1 ,¨¨¨ ,xd q }pLp pTq }Tmn pgpx1 ,¨¨¨ ,xd´1 q q}pLp pTq ď Cm d where Cmd ą 0 is the implied constant in (10) corresponding to Tmd . By iterating this argument d ´ 1 times, one obtains }Tm1 b ¨ ¨ ¨ b Tmd pf q}pLp pTd q ď rCm1 ¨ ¨ ¨ Cmd sp pp ´ 1q´dp }f }pLppTd q and this completes the proof of Theorem 1. 6 BAKAS, RODRÍGUEZ-LÓPEZ, AND SOLA 3.1. Proof of Corollary 2. We shall use the multi-dimensional version of Khintchine’s inquality: if prk qkPN0 denotes the set of Rademacher functions indexed by N0 over a probability space pΩ, A, Pq, then for every finite collection of complex numbers pak1 ,¨¨¨ ,kd qk1 ,¨¨¨ ,kd PN0 one has › › ÿ ÿ ˘1{2 ` › › |ak1 ,¨¨¨ ,kd |2 (11) ak1 ,¨¨¨ ,kd rk1 b ¨ ¨ ¨ b rkd › p d „p › L pΩ q k1 ,¨¨¨ ,kd PN0 k1 ,¨¨¨ ,kd PN0 for all 0 ă p ă 8. The implied constants do not depend on pak1 ,¨¨¨ ,kd qk1 ,¨¨¨ ,kd PN0 , see e.g. Appendix D in [15], and do not blow up as p Ñ 1. Combining Theorem 1, applied tořd-fold tensor products of periodic Marcinkiewicz multiplier operators of the form kPZ ˘∆k , with the multi-dimensional Khintchine’s inequality as in [1, Section 3] shows that the desired bound holds for analytic polynomials. Since analytic trigonometric polynomials on Td are dense in p pHA pTd q, } ¨ }Lp pTd q q, we deduce that sup }f }Lp pTd q ď1 }STd pf q}Lp pTd q Àd pp ´ 1q´d as p Ñ 1` . p f PHA pTd q It remains to prove the reverse inequality. To do this, for fixed 1 ă p ď 2, choose p an f P HA pTq such that }ST pf q}Lp pTq ě Cpp ´ 1q´1 }f }LppTq , where C ą 0 is an absolute constant. The existence of such functions is shown in p [12]. Hence, if we define g P HA pTd q by gpx1 , ¨ ¨ ¨ , xd q “ f px1 q ¨ ¨ ¨ f pxd q for px1 , ¨ ¨ ¨ , xd q P Td , then }STd pgq}Lp pTd q “ }ST pf q}Lp pTq ¨ ¨ ¨ }ST pf q}Lp pTq ě C d pp ´ 1q´d }f }dLppTq “ C d pp ´ 1q´d }g}LppTd q and this proves the sharpness of Corollary 2. p Remark 4. For the subspace HA,diag pTd q of Lp pTd q consisting of functions of the p form f px1 , . . . , xd q “ F px1 ` ¨ ¨ ¨ ` xd q for some one-variable function F P HA pTq, we have the improved estimate sup }STd pf q}p „ pp ´ 1q´1 , p Ñ 1` . }f }Lp pTd q ď1 p f PHA,diag pTd q This follows from invariance of the Lp -norm and Fubini’s theorem which allow us to reduce to the one-dimensional case. On the other hand, the natural inclusion p p of HA pTk q in HA pTd q yields examples of subspaces with sharp blowup of order ´k pp ´ 1q for any k “ 1, . . . , d ´ 1. Both the original proof of Pichorides’s theorem and the extension in this paper rely on complex-analytic techniques, via canonical factorisation in [12] and conjugate functions in [11]. However, a complex-analytic structure is not necessary in order for an estimate of the form in Corollary 2 to hold. For instance, the same conclusion remains valid for g P H̃0p pTd q, the subset of Lp pTd q consisting of functions with supppfˆq Ă p´Nqd . Moreover, for functions of the form f ` g, where p f P HA pTd q and g P H̃0p pTd q with }f }p “ }g}p ď 1{2, we then have }f ` g}p ď 1 and }STn pf ` gq}p ď }STd pf q}p ` }STd pgq}p À pp ´ 1q´d as p Ñ 1` . EXTENSIONS OF A THEOREM OF PICHORIDES 7 3.2. A transference theorem. In this subsection, we explain how one can transfer the aforementioned result of Tao and Wright on boundedness of Marcinkiewicz multiplier operators from H 1 pRq to L1,8 pRq to the periodic setting. To this end, let us first recall the definition of the local Hardy space h1 pRq introduced by D. Goldberg [6], which can be described as the space of L1 -functions for which the “high-frequency”part belongs to H 1 pRq. Namely, if we take φ to be a smooth function supported in r´1, 1s and such that φ|r´1{2,1{2s ” 1, and set ψ “ 1 ´ φ, one has that f P h1 pRq if, and only if, }f }h1 pRq “ }f }L1 pRq ` }Tψ pf q}H 1 pRq ă `8. The desired transference result is a consequence of D. Chen’s [3, Thm. 29]. Theorem 5. If λ is a continuous and bounded function on R such that }Tλ pf q}L1,8 pRq ď C1 }f }h1 pRq , then }Tλ|Z pgq}L1,8 pTq ď C2 }g}H 1 pTq . Observe that given a Marcinkiewicz multiplier m on the torus, one can construct a Marcinkiewicz multiplier λ on R such that λ|Z “ m. Indeed, it suffices to take λ to be continuous such that λpnq “ mpnq for every n P Z and affine on the intervals of the form pn, n ` 1q, n P Z. In order to use Theorem 5, let ψ be as above and consider the “high-frequency” part λ` of λ given by λ` “ ψλ. Note that for every Schwartz function f we may write Tλ` pf q “ Tλ pfrq, where fr “ Tψ pf q. We thus deduce that }Tλ` pf q}L1,8 pRq “ }Tλ pfrq}L1,8 pRq À }fr}H 1 pRq À }f }h1 pRq , and hence, Theorem 5 yields that Tλ` |Z is bounded from H 1 pTq to L1,8 pTq. Since for every trigonometric polynomial g we can write Tm pgq “ T0 pgq ` Tλ` |Z pgq, where T0 pgq “ mp0qp gp0q, and we have that }T0 pgq}L1,8 pTq “ |mp0q||p gp0q| ď }m}ℓ8 pZq }g}L1pTq ď }m}ℓ8 pZq }g}H 1 pTq , it follows that }Tm pgq}L1,8 pTq À }g}H 1 pTq . 4. A higher-dimensional extension of an inequality due to Zygmund In [18], Zygmund showed that there exists a constant C ą 0 such that for every 1 f P HA pTq, we have }ST pf q}L1 pTq ď C}f }L log LpTq . (12) 1 Note that if one removes the assumption that f P HA pTq, then the Orlicz space L log LpTq must be replaced by the smaller space L log3{2 LpTq, see [1]. p Zygmund’s proof again relies on canonical factorisation in HA pTq, but a higherdimensional extension of (12) can now be obtained from the methods of the previous section. Proposition 6. Given d P N, there exists a constant Cd ą 0 such that for every analytic trigonometric polynomial g on Td one has }STd pgq}L1 pTd q ď Cd }g}L logd LpTd q . The exponent r “ d in the Orlicz space L logd LpTd q cannot be improved. (13) 8 BAKAS, RODRÍGUEZ-LÓPEZ, AND SOLA Proof. By using Lemma 3 and a Marcinkiewicz-type interpolation argument analogous to the one presented in Section 3, one shows that if T is a sublinear operator 1 2 that is bounded from HA pTq to L1,8 pTq and bounded from HA pTq to L2 pTq, then for every r ě 0 one has ż ż r |T pf qpxq| log p1 ` |T pf qpxq|qdx ď Cr r1 ` |f pxq| logr`1 p1 ` |f pxq|qdxs (14) T T for every analytic trigonometric polynomial f on T, where Cr ą 0 is a constant depending only ř on r. If Tωj “ kPZ rk pωj q∆j denotes a randomised version of ST , j “ 1, ¨ ¨ ¨ , d, then 1 2 pTq to L1,8 pTq and HA pTq to L2 pTq and so, by using (14) and iteration, Tωj maps HA one deduces that }pTω1 b ¨ ¨ ¨ b Tωd qpf q}L1 pTd q ď Ad }f }L logd LpTd q (15) for every analytic polynomial f on Td . Hence, the proof of (13) is obtained by using (15) and (11). To prove sharpness for d “ 1, let N be a large positive integer to be chosen later and take V2N “ 2K2N `1 ´ K2N to be the de la Vallée ř Poussin kernel of order 2N , where Kn denotes the Fejér kernel of order n, Kn pxq “ |j|ďn r1 ´ |j|{pn` 1qsei2πjx. Consider the function fN by N `1 fN pxq “ ei2π2 x V2N pxq. ř2N `1 ´1 1 pTq, ∆N `1 pfN qpxq “ k“2N ei2πkx and Then, one can easily check that fN P HA }fN }L logr LpTq À N r . Hence, if we assume that (12) holds for some L logr LpTq, then we see that we must have N À }∆N `1 pfN q}L1 pTq ď }ST pfN q}L1 pTq À }fN }L logr LpTq À N r and so, if N is large enough, it follows that r ě 1, as desired. To prove sharpness in the d-dimensional case, take gN px1 , ¨ ¨ ¨ , xd q “ fN px1 q ¨ ¨ ¨ fN pxd q, fN being as above, and note that N d À }∆N `1 pfN q}dL1 pTq ď }STd pgN q}L1 pTd q À }gN }L logr LpTd q À N r . Hence, by taking N Ñ 8, we deduce that r ě d.  Remark 7. Note that, by using (14) and (11), one can actually show that there exists a constant Bd ą 0, depending only on d, such that }STd pf q}L1,8 pTd q ď Bd }f }L logd´1 LpTd q (16) for every analytic trigonometric polynomial f on Td . Notice that if we remove the assumption that f is analytic, then the Orlicz space L logd´1 LpTd q in (16) must be replaced by L log3d{2´1 LpTd q, see [1]. 5. Euclidean variants of Theorem 1 In this section we obtain an extension of Pichorides’s theorem to the Euclidean setting. Our result will be a consequence of the following variant of Marcinkiewicztype interpolation on Hardy spaces. Proposition 8. Assume that T is a sublinear operator that satisfies: 1 ‚ }T pf q}L1,8pRq ď C}f }L1pRq for all f P HA pRq and 2 ‚ }T pf q}L2pRq ď C}f }L2 pRq for all f P HA pRq, p where C ą 0 is an absolute constant. Then, for every 1 ă p ă 2, T maps HA pRq p to L pRq and moreover, }T }HAp pRqÑLp pRq À rpp ´ 1q´1 ` p2 ´ pq´1 s1{p . EXTENSIONS OF A THEOREM OF PICHORIDES 9 p Proof. Fix 1 ă p ă 2 and take an f P HA pRq. From a classical result due to Peter Jones [7, Theorem 2] it follows that for every λ ą 0 one can write f “ Fλ `fλ , where 1 8 Fλ P HA pRq, fλ P HA pRq and, moreover, there is an absolute constant C0 ą 0 such that ş ş ‚ R |Fλ pxq|dx ď C0 txPR:N pf qpxqąλu N pf qpxqdx and ‚ }fλ }L8 pRq ď C0 λ. p Here, N pf q denotes the non-tangential maximal function of f P HA pRq given by sup |pf ˚ Pt qpx1 q|, N pf qpxq “ |x´x1 |ăt where, for t ą 0, Pt psq “ t{ps2 ` t2 q denotes the Poisson kernel on the real line. Hence, by using the Peter Jones decomposition of f , we have ż8 p }T pf q}LppRq “ pλp´1 |tx P R : |T pf qpxq| ą λ{2u|dλ ď I1 ` I2 , 0 where I1 “ p ż8 λp´1 |tx P R : |T pFλ qpxq| ą λ{2u|dλ ż8 λp´1 |tx P R : |T pfλ qpxq| ą λ{2u|dλ. 0 and I2 “ p 0 We shall treat I1 and I2 separately. To bound I1 , using our assumption on the 1 boundedness of T from HA pRq to L1,8 pRq together with Fubini’s theorem, we deduce that there is an absolute constant C1 ą 0 such that ż ´1 I1 ď C1 pp ´ 1q rN pf qpxqsp dx. (17) R 2 To bound the second term, we first use the boundedness of T from HA pRq to L2 pRq as follows ż8 ż ` ˘ pλp´3 I2 ď C |fλ pxq|2 dx dλ 0 R and then we further decompose the right-hand side of the last inequality as I2,α ` I2,β , where ż8 ż ` ˘ p´3 I2,α “ C pλ |fλ pxq|2 dx dλ txPR:N pf qpxqąλu 0 and I2,β “ C ż8 p´3 pλ 0 ` ż txPR:N pf qpxqďλu ˘ |fλ pxq|2 dx dλ. The first term I2,α can easily be dealt with by using the fact that }fλ }L8 pRq ď C0 λ, ż8 ż I2,α ď C 1 pλp´1 |tx P R : N pf qpxq ą λu|dλ “ C 1 rN pf qpxqsp dx, 0 R 1 where C “ C0 C. To obtain appropriate bounds for I2,β , note that since |fλ |2 “ 1 2 |f ´ Fλ |2 ď 2|f |2 ` 2|Fλ |2 , one has I2,β ď I2,β ` I2,β , where ż8 ż ` ˘ 1 I2,β “ 2C pλp´3 |f pxq|2 dx dλ txPR:N pf qpxqďλu 0 and 2 I2,β “ 2C ż8 0 p´3 pλ ` ż txPR:N pf qpxqďλu ˘ |Fλ pxq|2 dx dλ. 10 BAKAS, RODRÍGUEZ-LÓPEZ, AND SOLA p 1 To handle I2,β , note that since f P HA pRq (1 ă p ă 2) one has |f pxq| ď N pf qpxq for a.e. x P R and hence, by using this fact together with Fubini’s theorem, one obtains ż 1 I2,β ď Cp2 ´ pq´1 rN pf qpxqsp dx. R 2 Finally, for the last term I2,β , we note that for a.e. x in tN pf q ď λu one has |Fλ pxq| ď |f pxq| ` |fλ pxq| ď N pf qpxq ` |fλ pxq| ď p1 ` C0 qλ and hence, ż8 ż ` ˘ 2 I2,β ď C2 λp´2 |Fλ pxq|dx dλ 0 R ż8 ż ż ` ˘ 2 p´2 ďC λ N pf qpxqdx dλ ď C 2 pp ´ 1q´1 rN pf qpxqsp dx, 0 txPR:N pf qpxqąλu R where C 2 “ 4p1 ` C0 q and in the last step we used Fubini’s theorem. Since I2 ď 1 2 I2,α ` I2,β ` I2,β , we conclude that there is a C2 ą 0 such that ż (18) I2 ď C2 rpp ´ 1q´1 ` p2 ´ pq´1 s rN pf qpxqsp dx. R It thus follows from (17) and (18) that }T pf q}LppRq À rpp ´ 1q´1 ` p2 ´ pq´1 s1{p }N pf q}LppRq . To complete the proof of the proposition note that one has p pRqq, }N pf q}LppRq ď Cp }f }Lp pRq pf P HA (19) 1{p where one can take Cp “ A0 , A0 ě 1 being an absolute constant, see e.g. p.278279 in vol.I in [19], where the periodic case is presented. The Euclidean version is completely analogous. Hence, if 1 ă p ă 2, one deduces that the constant Cp in (19) satisfies Cp ď A0 and so, we get the desired result.  Using the above proposition and iteration, we obtain the following Euclidean version of Theorem 1. Theorem 9. Let d P N be a given dimension. If Tmj is a Marcinkiewicz multiplier operator on R pj “ 1, ¨ ¨ ¨ , dq, then }Tm1 b ¨ ¨ ¨ b Tmd }HAp pRd qÑLp pRd q ÀCm1 ,¨¨¨ ,Cmd pp ´ 1q´d as p Ñ 1` , where Cmj “ }mj }L8 pRq ` Amj , Amj is as in p6q, j “ 1, ¨ ¨ ¨ , d. A variant of Pichorides’s theorem on Rd now follows from Theorem 9 and (11). To formulate our result, for k P Z, define the rough Littlewood-Paley projection Pk to be a multiplier operator given by { p P k pf q “ rχr2k ,2k`1 q ` χp´2k`1 ,´2k s sf . For d P N, define the d-parameter rough Littlewood-Paley square function SRd on Rd by ´ ¯1{2 ÿ |Pk1 b ¨ ¨ ¨ b Pkd pf q|2 SRd pf q “ k1 ,¨¨¨ ,kd PZ for f initially belonging to the class of Schwartz functions on Rd . Arguing as in Subsection 3.1, we get a Euclidean version of Corollary 2 as a consequence of Theorem 9. EXTENSIONS OF A THEOREM OF PICHORIDES 11 Corollary 10. For d P N, one has }SRd }HAp pRd qÑLp pRd q „d pp ´ 1q´d as p Ñ 1` . Remark 11. The multiplier operators covered in Theorem 9 are properly contained in the class of general multi-parameter Marcinkiewicz multiplier operators treated in Theorem 61 in Chapter IV of [15]. For a class of smooth multi-parameter Marcinkiewicz multipliers M. Wojciechowski [17] proves that their Lp pRd q Ñ Lp pRd q operator norm is of order pp ´ 1q´d and that they are bounded on the d-parameter Hardy space H p pR ˆ ¨ ¨ ¨ ˆ Rq for all 1 ď p ď 2. Note that the multi-parameter Littlewood-Paley square function is not covered by this result; see also [1] for more refined negative statements. References [1] Bakas, Odysseas. Endpoint Mapping properties of the Littlewood-Paley square function. preprint arXiv:1612.09573 (2016). [2] Bourgain, Jean. On the behavior of the constant in the Littlewood-Paley inequality. In: Geometric Aspects of Functional Analysis (1987-88), pp. 202-208. Lecture notes in math. 1376, Springer Berlin, 1989. [3] Chen, Danin. Multipliers on certain function spaces. PhD thesis, University of Winsconsin-Milwaukee, 1998. [4] Duren, Peter L. Theory of H p spaces. Vol. 38. New York: Academic press, 1970. [5] Edwards, Robert E., and Garth Ian Gaudry. Littlewood-Paley and multiplier theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90. SpringerVerlag Berlin-New York, 1977. [6] Goldberg, David. A local version of real Hardy spaces. PhD thesis, Princeton University, 1978. [7] Jones, Peter Wilcox. L8 estimates for the B̄ problem in a half-plane. Acta Math. 150, no.1-2 (1983): 137–152. [8] Kislyakov, Serguei Vital’evich, and Quanhua Xu. Real interpolation and singular integrals. Algebra i Analiz 8, no. 4 (1996): 75-109. [9] Krasnosel’skii, Mark Aleksandrovich, and Iakov Bronislavovich Rutitskii. Convex functions and Orlicz spaces. P. Noordhoff, Groningen, 1961. [10] Marcinkiewicz, Józef. Sur les multiplicateurs des séries de Fourier. Studia Math. 8 (1939): 78-91. [11] Pavlović, Miroslav. Introduction to function spaces on the disk. Posebna Izdanja [Special editions] 20, Matematicki Institut SANU, Belgrade, 2004. [12] Pichorides, Stylianos K. A remark on the constants of the Littlewood-Paley inequality. Proc. Amer. Math. Soc. 114, no. 3 (1992): 787-789. [13] Rubel, Lee A. and Allen L. Shields. The failure of interior-exterior factorization in the polydisc and the ball. Tohoku Math. J. 24 (1972): 409-413. [14] Rudin, Walter. Function theory in polydiscs. W.A. Benjamin, New York and Amsterdam, 1969. [15] Stein, Elias M. Singular integrals and differentiability properties of functions (PMS-30). Vol. 30. Princeton university press, 2016. [16] Tao, Terence, and James Wright. Endpoint multiplier theorems of Marcinkiewicz type. Rev. Mat. Iberoam. 17, no. 3 (2001): 521-558. [17] Wojciechowski, Michal. A Marcinkiewicz type multiplier theorem for H 1 spaces on product domains. Studia Math. 140, no. 3 (2000): 273-287. [18] Zygmund, Antoni. On the convergence and summability of power series on the circle of convergence (I). Fundamenta Math. 30 (1938): 170-196. 12 BAKAS, RODRÍGUEZ-LÓPEZ, AND SOLA [19] Zygmund, Antoni. Trigonometric series. Vol. I, II. Cambridge University Press, 2002. Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden E-mail address: [email protected], [email protected], [email protected]