We consider the following question: Are there exponents $2<p<q$ such that the
Riesz projection is bounded from $L^q$ to $L^p$ on the infinite polytorus? We
are unable to answer the question, but our counter-example improves a result of
Marzo and Seip by demonstrating that the Riesz projection is unbounded from
$L^\infty$ to $L^p$ if $p\geq 3.31138$. A similar result can be extracted for
any $q>2$. Our approach is based on duality arguments and a detailed study of
linear functions. Some related results are also presented.Comment: This paper has been accepted for publication in Collectanea
  Mathematic

Brevig, Ole Fredrik

English

arXiv

We consider the following question: are there exponents   22 . Our approach is based on duality arguments and a detailed study of linear functions. Some related results are also presented

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Linear functions and duality on the infinite polytorus

We consider the following question: are there exponents   22 . Our approach is based on duality arguments and a detailed study of linear functions. Some related results are also presented.acceptedVersionThis is a post-peer-review, pre-copyedit version of an article published in [Collectanea Mathematica] Locked until 2.2.2020 due to copyright restrictions. The final authenticated version is available online at: https://doi.org/10.1007/s13348-019-00243-

NTNU Open (Norwegian University of Science and Technology)

LINEAR FUNCTIONS AND DUALITYON THE INFINITE POLYTORUSOLE FREDRIK BREVIGAbstract. We consider the following question: Are there exponents 2 < p <q such that the Riesz projection is bounded from Lq to Lp on the infinitepolytorus? We are unable to answer the question, but our counter-exampleimproves a result of Marzo and Seip by demonstrating that the Riesz projectionis unbounded from L∞ to Lp if p ≥ 3.31138. A similar result can be extractedfor any q > 2. Our approach is based on duality arguments and a detailedstudy of linear functions. Some related results are also presented.1. IntroductionLet T∞ = T × T × · · · denote the countably infinite cartesian product of thetorus T = {z ∈ C : |z| = 1}. We equip the T∞ with its Haar measure µ∞, which isequal to the infinite product of the normalized Lebesgue arc measure on T in eachvariable. Let 1 ≤ p ≤ ∞. Every f in Lp(T∞) has a Fourier series expansionf(z) =∑α∈Z∞0cαzαwhere the Fourier coefficients are defined in the standard way and α ∈ Z∞0 meansthat the multi-index α contains only a finite number of non-zero components. TheRiesz projection on T∞ is defined by(1) Pf(z) =∑α∈N∞0cαzα.The initial motivation for the present paper is the following.Question. What is the largest p = p∞ such that the Riesz projection (1) is boundedfrom L∞(T∞) to Lp(T∞)?The Riesz projection is certainly a contraction on the Hilbert space L2(T∞) andsince ‖f‖L2(T∞) ≤ ‖f‖L∞(T∞), we get that p∞ ≥ 2. This question has previouslybeen investigated by Marzo and Seip [8] who demonstrated that p∞ ≤ 3.67632. Wewill obtain the following improvement.Theorem 1. p∞ ≤ p = 3.31138 . . ., where p denotes the unique positive solutionof the equationΓ(1 +p2) 1p=2√π.Date: January 17, 2019.2010 Mathematics Subject Classification. Primary 42B05. Secondary 42B30, 46E30.12 OLE FREDRIK BREVIGFor 2 ≤ p ≤ q ≤ ∞, let ‖P‖q,p denote the norm of the Riesz projection fromLq(T∞) to Lp(T∞). In the case that the Riesz projection is unbounded, we use theconvention ‖P‖q,p = ∞. As explained in [8], for each fixed 2 ≤ q ≤ ∞ there is anumber 2 ≤ pq ≤ q, called the critical exponent, with the property that(2) ‖P‖p,q ={1 if p ≤ pq,∞ if p > pq.The dichotomy (2) is a direct consequence of the fact that we are on the infinitepolytorus. Let f be a function in the unit ball of Lq(T∞) such that ‖Pf‖Lp(T∞) > 1.Consider the functionf2(z) = f(z1, z3, z5, . . .) · f(z2, z4, z6, . . .)which is also in the unit ball of Lq(T∞). The Riesz projection (1) acts independentlyon the variables, so we find thatPf2(z) = Pf(z1, z3, z5, . . .) · Pf(z2, z4, z6, . . .)which implies that ‖Pf2‖Lp(T∞) = ‖Pf‖2Lp(T∞) > ‖Pf‖Lp(T∞). This procedure canbe repeated and so we obtain (2). The example from [8] producing p∞ ≤ 3.67632is a function of only two variables.The present paper is inspired by [3], where linear functions are used as buildingblocks in an similar way to what was just described to construct a counter-examplerelated to Nehari’s theorem for Hankel forms on T∞. The example from [3] improveson an earlier example from [9] by replacing a function of two variables by a linearfunction in an infinite number of variables.Our approach differs from that of [8] (and [2]) in that we do not attempt todirectly construct a counter-example, but instead use duality arguments to inferits existence. This approach leads us to consider the Hardy spaces Hp(T∞), whichare the subspaces of Lp(T∞) consisting of elements such that Pf = f . A standardargument involving the Hahn–Banach theorem (see e.g. [5, Sec. 7.2]) yields that(3) infPψ=ϕ‖ψ‖Lq(T∞) = ‖ϕ‖(Hr(T∞))∗ = supf∈Hr(T∞)|〈f, ϕ〉L2(T∞)|‖f‖Hr(T∞)for 1 ≤ r <∞ and q−1 + r−1 = 1. We will choose ϕ and try to find the optimal fin Hr(T∞) attaining the supremum. This will ensure the existence of ψ in Lq(T∞)attaining the infimum, which be our counter-example through (2).We shall see in Section 3 that if we know the optimal f in the supremum on theright hand side of (3), we can use Hölder’s inequality to construct the element ψin Lq(T∞) of minimal norm such that Pψ = ϕ, thereby attaining the infimum onthe left hand side of (3).As in [3] we will primarily be working with linear functions, which are of theform(4) f(z) =∞∑j=1cjzj .Clearly, ‖f‖2H2(T∞) =∑j≥1 |cj |2 and we easily check that ‖f‖H∞(T∞) =∑j≥1 |cj |.For 1 ≤ p < ∞, optimal norm estimates are given by Khintchine’s inequality.LINEAR FUNCTIONS AND DUALITY ON THE INFINITE POLYTORUS 3Defineap = min(1, Γ(1 +p2) 1p)and bp = max(1, Γ(1 +p2) 1p).If f is a linear function (4) and 1 ≤ p <∞, then we restate a result from [7] as(5) ap‖f‖H2(T∞) ≤ ‖f‖Hp(T∞) ≤ bp‖f‖H2(T∞)and the constants in (5) are optimal. We shall obtain the following companioninequality for dual norms, which might be of independent interest.Theorem 2. Let 1 ≤ p <∞. If f is a linear function (4), then(6) b−1p ‖f‖H2(T∞) ≤ ‖f‖(Hp(T∞))∗ ≤ a−1p ‖f‖H2(T∞).The constants are optimal.Remark. In the case p = ∞, it is easy to deduce by similar considerations (Lemma 4)that ‖f‖(H∞(T∞))∗ = supj≥1 |cj | if f is a linear function (4).Optimality of the constants containing the Gamma function in (5) and (6) botharise from the functionf(z) =z1 + z2 + · · ·+ zd√das d → ∞ through the central limit theorem. In view of (2) and (3), we cantherefore obtain the following general result. Note that Theorem 1 corresponds tothe particular case q = ∞, since Γ(3/2) =√π/2.Theorem 3. Let 2 ≤ p ≤ q ≤ ∞ and set q−1 + r−1 = 1. IfΓ(1 +p2) 1pΓ(1 +r2) 1r> 1,then the Riesz projection is unbounded from Lq(T∞) to Lp(T∞).Remark. Theorem 3 is an improvement on the same statement with requirementp/2·r/2 > 1, which can be deduced from a one-variable example found in [2, Sec. 4].Here is an alternative example to that of [2] obtained by our approach using theHahn–Banach theorem. For w ∈ D, the functional of point evaluation f 7→ f(w)has norm (1− |w|2)−1/r on Hr(T) and the analytic symbol is ϕw(z) = (1−wz)−1.Hence, if w = ε > 0 then ‖ϕε‖(Hr(T))∗ = 1+ r−1ε2 +O(ε4) as ε→ 0. Furthermore,‖ϕε‖Hp(T) =∥∥(1− εz)−p/2∥∥2/pH2(T) = 1 +p4ε2 +O(ε4),so we obtain the desired counter-example as soon as r−1 > p/4 in view of (2). Theoptimal ψw in Lq(T) for this functional can be found in [4, Thm. 6.1], and we notethat it is similar (but not equal to) the counter-example constructed in [2].The present paper is organised into two additional sections. In Section 2 weprove Theorem 2 and Theorem 3. Section 3 is devoted to constructing the elementψ in Lq(T∞) for 1 < q ≤ ∞ of minimal norm such that Pψ(z) = z1 + z2 + · · ·+ zd,thereby realising the infimum (3) in this special case, which is of particular interestdue to the crucial role it plays in the proof of Theorem 2 and Theorem 3.4 OLE FREDRIK BREVIG2. Linear functions on T∞In preparation for the proof of Theorem 2 and Theorem 3, let us recall some basicfacts about linear functions and projections on T∞. The projection Ad obtained byformally setting zj = 0 for j > d has the representationAdf(z1, z2, . . .) =∫T∞f(z1, z2, . . . , zd, zd+1, zd+2, . . .) dµ∞(zd+1, zd+2, . . .).Since Adf is a function the first d variables, we take Lp norm with respect to thesevariables and use the triangle inequality to obtain(7) ‖Adf‖Lp(T∞) ≤ ‖f‖Lp(T∞).Let k ∈ Z. We say that f is k-homogeneous iff(eiθz1, eiθz2, eiθz3, . . .) = ekiθf(z1, z2, z3, . . .).Clearly every f in Lp(T∞) can be decomposed in k-homogeneous parts, say(8) f(z) =∑k∈Zfk(z),where fk is k-homogeneous. The following simple lemma is well-known, but weinclude a short proof for the readers convenience.Lemma 4. Let 1 ≤ p ≤ ∞ and suppose that f in Lp(T∞) is decomposed as in (8).Then ‖fk‖Lp(T∞) ≤ ‖f‖Lp(T∞) for every k ∈ Z.Proof. By the decomposition (8), we find thatfk(z) =∫ π−πf(eiθz1, eiθz2, eiθz3, . . .) e−kiθ dθ2π.By the triangle inequality and interchanging the order of integration, we obtain‖fk‖pLp(T∞) ≤∫ π−π∫T∞∣∣f(eiθz1, eiθz2, eiθz3, . . .)∣∣p dµ∞(z) dθ2π= ‖f‖pLp(T∞),since for each θ the rotation zj 7→ eiθzj does not change the Lp(T∞) norm of f . Let Lin(T∞) denote the space of linear functions (4). Lemma 4 states thatthe projection from Hp(T∞) to Lin(T∞) ∩ Hp(T∞) is contractive. This fact iscrucial to the proof of Theorem 2 and Theorem 3 since it allows us to compute the(Hp(T∞))∗ norm of a linear function ϕ by testing only against functions f fromLin(T∞) ∩Hp(T∞).In view of Khintchine’s inequality (5), the space Lin(T∞) ∩Hp(T∞) consists oflinear functions (4) with square summable coefficients for each 1 ≤ p <∞, althoughthe norms are generally different.Armed with these preliminaries, we will now obtain the key new ingredientneeded in the proofs of Theorem 2 and Theorem 3.Lemma 5. Let 1 ≤ p <∞ and set ϕd(z) = (z1 + · · ·+ zd)/√d. Then‖ϕd‖(Hp(T∞))∗ = ‖ϕd‖−1Hp(T∞).Proof. For the lower bound, we simply note that since ϕd is in Hp(T∞) we obtain(9) ‖ϕd‖(Hp(T∞))∗ = supf∈Hp(T∞)|〈f, ϕd〉H2(T∞)|‖f‖Hp(T∞)≥‖ϕd‖2H2(T∞)‖ϕd‖Hp(T∞)= ‖ϕd‖−1Hp(T∞).LINEAR FUNCTIONS AND DUALITY ON THE INFINITE POLYTORUS 5For the upper bound, we first use (7) and Lemma 4 to the effect that(10) ‖ϕd‖(Hp(T∞))∗ = supf∈Hp(T∞)|〈f, ϕd〉H2(T∞)|‖f‖Hp(T∞)= supf∈Lin(Td)|〈f, ϕd〉H2(T∞)|‖f‖Hp(T∞).Any non-trivial element f in Lin(Td) is of the formf(z) =d∑j=1cjzjwith at least one non-zero coefficient. Define(11) λ = 〈f, ϕd〉H2(T∞) =c1 + · · ·+ cd√d.After rotating each of the variables if necessary, we may assume that cj ≥ 0 for1 ≤ j ≤ d so that λ > 0 whenever f is a non-trivial element in Lin(Td).For 1 ≤ k ≤ d, let fk denote the polynomial obtained by replacing the coefficientsequence (c1, . . . , cd) of f with the shifted sequence(ck, ck+1, . . . , cd, c1, . . . , ck−1).By symmetry, we find that ‖fk‖Hp(T∞) = ‖f‖Hp(T∞). Note also that1dd∑k=1fk(z) =c1 + · · ·+ cddd∑j=1zj = λϕd(z).The triangle inequality therefore allows us to conclude that(12) λ‖ϕd‖Hp(T∞) ≤1dd∑k=1‖fk‖Hp(T∞) = ‖f‖Hp(T∞).Using (10) with (11) and (12), we obtain the upper bound‖ϕd‖(Hp(T∞))∗ = supf∈Hp(T∞)|〈f, ϕd〉H2(T∞)|‖f‖Hp(T∞)≤ λλ‖ϕd‖Hp(T∞)= ‖ϕd‖−1Hp(T∞)which, when combined with the lower bound (9), completes the proof. Another viewpoint is to consider (zj)j≥1 a sequence of independently distributedrandom variables on the torus and f(z) =∑j≥1 cjzj as a weighted random walk inthe plane. The norms ‖f‖Hp(T∞) can now be interpreted as moments of this randomwalk. A simple computation (see Section 3) gives that ‖z1 + z2‖H1(T∞) = 4/π andit is demonstrated in [1] that‖z1 + z2 + z3‖H1(T∞) =31621/3π4Γ6(13)+27422/3π4Γ6(23)= 1.57459 . . .In general it is difficult to compute ‖f‖Hp(T∞) even for simple linear polynomialsf (when p is not an even integer). However, the central limit theorem gives that(13) limd→∞∥∥∥∥z1 + z2 + · · ·+ zd√d∥∥∥∥pHp(T∞)=∫C|Z|pe−|Z|2 dZπ= Γ(1 +p2),since (z1 + z2 + · · ·+ zd)/√d has a limiting complex normal distribution.We are now ready to prove Theorem 2. To conform with the notations of thepresent section and to make the proof clearer, we consider now ϕ in (Hp(T∞))∗and f in Hp(T∞), so ϕ plays the role of f in the statement of the theorem.6 OLE FREDRIK BREVIGProof of Theorem 2. Let ϕ be a linear function in (Hp(T∞))∗. By Lemma 4, theCauchy–Schwarz inequality and Khintchine’s inequality (5), we find that‖ϕ‖(Hp(T∞))∗ = supf∈Lin(T∞)|〈f, ϕ〉H2(T∞)|‖f‖Hp(T∞)≤ supf∈Lin(T∞)‖f‖H2(T∞)‖ϕ‖H2(T∞)‖f‖Hp(T∞)≤‖ϕ‖H2(T∞)ap.Conversely, Khintchine’s inequality (5) also gives that‖ϕ‖(Hp(T∞))∗ = supf∈Hp(T∞)|〈f, ϕ〉H2(T∞)|‖f‖Hp(T∞)≥‖ϕ‖2H2(T∞)‖ϕ‖Hp(T∞)≥‖ϕ‖H2(T∞)bp,since ϕ is in Hp(T∞). To prove optimality of the constants, we appeal to Lemma 5and consider ϕd(z) = (z1 + · · ·+ zd)/√d for d = 1 and as d→ ∞. Theorem 3 also follows easily from Lemma 5 and (13).Proof of Theorem 3. Let 2 ≤ p ≤ q ≤ ∞ and set q−1 + r−1 = 1. Suppose that(14) Γ(1 +p2) 1pΓ(1 +r2) 1r> 1.We want to to prove that the Riesz projection is unbounded from Lq(T∞) toLp(T∞). In view of (2), it is sufficient to find ψ in Lq(T∞) such that‖Pψ‖Lp(T∞)‖ψ‖Lq(T∞)> 1.We pick ψd in Lq(T∞) of minimal norm such that Pψd = ϕd, where ϕd denotes thefunction from Lemma 5. By (3) and Lemma 5, we obtain‖Pψd‖Lp(T∞)‖ψd‖Lq(T∞)= ‖ϕd‖Lp(T∞)‖ϕd‖Lr(T∞).By (13) and our assumption (14), the right hand side is strictly larger than 1 forsome sufficiently large d. 3. Minimal Lq(T∞) normWe will now solve the following problem: For 1 < q ≤ ∞, find the element ψ inLq(T∞) of minimal norm such thatPψ(z) = z1 + z2 + · · ·+ zd = ϕ(z).The strict convexity of Lq(T∞) when 1 < q < ∞ means that the minimizer isunique. Uniqueness of the minimizer holds also for q = ∞, but in this case it is aconsequence of the continuity of ϕ on the polytorus (see e.g. [5, Sec. 8.2]).In view of (3) and (the proof of) Lemma 5, we know that ψ satisfies(15) ‖ψ‖Lq(T∞) =〈ϕ,ψ〉L2(T∞)‖ϕ‖Lp(T∞)=d‖ϕ‖Lp(T∞)with p−1 + q−1 = 1. On the left hand side of (15) we have attained equality inHölder’s inequality, which implies that |ψ| = C|ϕ|p−1 almost everywhere. InsertingLINEAR FUNCTIONS AND DUALITY ON THE INFINITE POLYTORUS 7this into the norm expression ‖ψ‖Lq(T∞) in (15) and using that (p − 1)q = p, wefind that C = d‖ϕ‖−pLp(T∞). From Hölder’s inequality and (15) we also see that〈|ϕ|, |ψ|〉L2(T∞) ≤ ‖ϕ‖Lp(T∞)‖ψ‖Lq(T∞) = 〈ϕ,ψ〉L2(T∞),which is only possible if ϕψ ≥ 0 almost everywhere. Combining these observationsyields thatψ(z) =d‖ϕ‖pLp(T∞)|ϕ(z)|p−2ϕ(z)is the element in Lq(T∞) of minimal norm such that Pψ(z) = z1+z2+· · ·+zd = ϕ(z)for 1 < p ≤ ∞ and p−1 + q−1 = 1. Note that ψ is 1-homogeneous, which we knewin advance by Lemma 4. We can also directly verify that∫T∞ψ(z) zj dm∞(z) =∫T∞ψ(z)z1 + z2 + · · ·+ zdddµ∞(z) = 1,since ψ inherits the symmetry of ϕ.When d = 2, we can actually compute the Fourier series explicitly. We begin byusing the trick z1 + z2 = z2(1 + z1z2) to write ψ(z) = z2Ψ(z1z2), whereΨ(z) =2‖1 + z‖pLp(T)|1 + z|p−2(1 + z).Then we get that‖1 + z‖pLp(T)2=12∫ π−π|1+eiθ|p dθ2π= 2p−1∫ π−πcosp(θ2)dθ2π=2pπ∫ π/20cosp(ϑ) dϑ.Similarly, we compute:∫ π−π|1 + eiθ|p−2(1 + eiθ) e−ikθ dθ2π= 2p−1∫ π−πcosp−1(θ2)e−i(k−1/2)θdθ2π= 2p−1∫ π/2−π/2cosp−1(ϑ) e−i(2k−1)ϑdϑπ=2pπ∫ π/20cosp−1(ϑ) cos((1− 2k)ϑ) dϑThe latter integral, which contains the former as the special case k = 0, 1 is known(see e.g. [6, p. 399]) and we obtain that∫ π−π|1 + eiθ|p−2(1 + eiθ) e−ikθ dθ2π=1pBeta(p+1−2k+12 ,p−1+2k+12)for Beta(x, y) = Γ(x)Γ(y)/Γ(x+ y). Combining everything, we find thatψ(eiθ1 , eiθ2) =∑k∈ZΓ(1 + p/2)Γ(p/2)Γ(1 + p/2− k)Γ(p/2 + k)eikθ1ei(1−k)θ2 .AcknowledgementsThe author would like to extend his gratitude to A. Bondarenko, H. Hedenmalm,E. Saksman and K. Seip for an interesting discussion which culminated in thematerial presented in Section 3 and to the referee for a helpful suggestion.8 OLE FREDRIK BREVIGReferences1. J. M. Borwein, D. Nuyens, A. Straub, and J. Wan, Some arithmetic properties of short randomwalk integrals, Ramanujan J. 26 (2011), no. 1, 109–132.2. O. F. Brevig, J. Ortega-Cerdà, K. Seip, and J. Zhao, Contractive inequalities for Hardy spaces,Funct. Approx. Comment. Math. 59 (2018), no. 1, 41–56.3. O. F. Brevig and K.-M. Perfekt, Failure of Nehari’s theorem for multiplicative Hankel formsin Schatten classes, Studia Math. 228 (2015), no. 2, 101–108.4. B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinite polydiskalgebra, Proc. London Math. Soc. (3) 53 (1986), no. 1, 112–142.5. P. L. Duren, Theory of Hp spaces, Pure and Applied Mathematics, Vol. 38, Academic Press,New York-London, 1970.6. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, eighth ed., Else-vier/Academic Press, Amsterdam, 2015.7. H. König and S. Kwapień, Best Khintchine type inequalities for sums of independent, rotation-ally invariant random vectors, Positivity 5 (2001), no. 2, 115–152.8. J. Marzo and K. Seip, L∞ to Lp constants for Riesz projections, Bull. Sci. Math. 135 (2011),no. 3, 324–331.9. J. Ortega-Cerdà and K. Seip, A lower bound in Nehari’s theorem on the polydisc, J. Anal.Math. 118 (2012), no. 1, 339–342.Department of Mathematical Sciences, Norwegian University of Science and Tech-nology (NTNU), NO-7491 Trondheim, NorwayEmail address: ole.brevig@math.ntnu.no

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Linear functions and duality on the infinite polytorus

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