Polynomials on Polydiscs - Matematikcentrum · 2013. 7. 22. · Polynomials on Polydiscs We will be...

31
Polynomials on Polydiscs Kristian Seip Norwegian University of Science and Technology June 12, 2013

Transcript of Polynomials on Polydiscs - Matematikcentrum · 2013. 7. 22. · Polynomials on Polydiscs We will be...

  • Polynomials on Polydiscs

    Kristian Seip

    Norwegian University of Science and Technology

    June 12, 2013

  • Polynomials on Polydiscs

    We will be interested in polynomials F in d complex variables:

    F (z) =∑α

    aαzα;

    here z = (z1, ..., zd ), α = (α1, ..., αd ) is a multi-index with αjnonnegative integers, and

    zα = zα11 · · · zαdd .

    We will link such polynomials to analysis on polydiscs byconsidering Lp norms (and more general Lp integrals) withrespect to normalized Lebesgue measure σd on the unitpolycircle Td . In particular, we set

    ‖F‖p :=

    {(∫Td |F (z)|

    pdσd (z))1/p

    , 1 ≤ p

  • Elementary facts

    We should bear in mind that‖F‖22 =

    ∑α |aα|2 (by orthonormality of zα)

    ‖F‖p ≤ ‖F‖q when p ≤ q (by Hölder’s inequality)‖F‖∞ = supz∈Dd |F (z)| (by maximum principle).

    The polycircle Td is referred to as the distinguished boundaryof the polydisc Dd .

  • Hardy spaces on Td

    The three main results to be presented in this talk are all aboutpolynomials on polydiscs. We will, however, also need toconsider the closely related Hardy spaces Hp(Td ); for1 ≤ p

  • First perspective: From d = 1 to d > 1

    Rudin’s approach (from the preface of his 1969 “FunctionTheory in Polydiscs”): “Briefly, the object is to see how much ofour extremely detailed knowledge about holomorphic functionsin the unit disc [...] can be carried over to an analogoussituation in several variables, namely to polydiscs.”

    This suggests both purely function theoretic problems (studiedin depth by Rudin) and how the extensive interplay betweenfunction theory and operator theory on the unit disc carries overto the unit polydisc.

  • Second perspective: From d

  • Third perspective: Independent Steinhaus variables

    We may interpret the interaction between the independentvariables z1, z2, ..., zd on Td in probabilistic terms, now viewingthe measure σd as a probability measure. A random variablewith a uniform distribution on the unit circle T is known as aSteinhaus variable; our polynomial F is in this context known asa polynomial chaos with respect to d independent Steinhausvariables.

    In the simplest (linear) case, whenF (z) = a1z1 + · · ·+ adzd , Khinchin’s inequality says that‖F‖p ' ‖F‖2 with constants that do not depend on d (bestconstants obtained by Sawa (1985) and König (2013))Lp norms could potentially give interesting informationabout the distribution of polynomial chaosPolynomial chaos arising from Dirichlet polynomials viaBohr’s lift seems interesting but is not easy to handle.

  • First result: Factorization

    A simple and important fact in the case d = 1 is that every f inH1(T) can be factored as

    f = gh, where g,h ∈ H2(T) and ‖g‖22 = ‖h‖22 = ‖f‖1.

    Reason: we may factor out the zeros of f such that f = BΦ,where B is a so-called Blaschke product and Φ is without zeros.Then ‖f‖1 = ‖Φ‖1, and it suffices to set g = BΦ1/2 andh = Φ1/2.

  • Nehari’s theorem

    In general, functions in several variables do not factor in thisway, but operator theory helps us see what is the rightanalogue. Namely, the most important application of thefactorization of H1 functions is to establish Nehari’s theorem(1957), which, in the case d = 1, characterizes boundedHankel forms Hψ : H2(Td )× H2(Td )→ C, defined as

    Hψ(f ,g) = 〈fg, ψ〉L2(Td ),

    where ψ ∈ H2(Td ) is the symbol of the form.

    Theorem (Nehari)

    Hψ is bounded on H2(T)× H2(T) if and only if ψ defines abounded linear functional on H1(T).

    Proof: The sufficiency is obvious; the necessity followsimmediately from the factorization of functions in H1(T).

  • Weak factorization

    To establish Nehari’s theorem for d > 1, it would be enough tohave f =

    ∑∞j=1 gjhj with

    ∞∑j=1

    ‖gj‖2‖hj‖2 ≤ C‖f‖1

    for a constant C independent of f . If this holds, we say thatH1(Td ) admits weak factorization and set

    ‖f‖w ,1 := inf∑j gj hj=f

    ∞∑j=1

    ‖gj‖2‖hj‖2.

    Theorem (Ferguson–Lacey (2002), Lacey–Terwilleger (2009))

    H1(Td ) admits weak factorization for every d > 1.

  • Interpretation of Nehari’s theorem

    Nehari’s theorem (for d

  • From d = 1 to d =∞: Multiplicative Hankel forms

    If ψ ∈ H2(T), then the corresponding Hankel form can bewritten in discrete form as

    Hψ(a,b) =∞∑

    j,k=0

    ajbkρj+k

    where ψ(z) =∑∞

    n=0 ρnzn. It is bounded whenever

    |Hψ(a,b)| ≤ M‖a‖2‖b‖2. Nehari’s theorem says that everybounded Hankel form is such that ρn = ϕ̂(n),n ≥ 0, whereϕ ∈ L∞(T). The infinite dimensional version of a Hankel form isequivalent to

    Hψ(a,b) =∞∑

    j,k=1

    ajbkρj·k .

    For each n ∈ N, one considers the decompositionn = pα11 · · · p

    αdd in prime numbers and we associate with n as

    before the multi-index α = (α1, . . . , αd ). Then ρ(n) = ψ̂(α).

  • A multiplicative version of Nehari’s theorem?

    Clearly, if ψ ∈ L∞(T∞) and ρ(n) = ψ̂(α1, . . . , αd ), then themultiplicative bilinear Hankel form is bounded. The questionasked by Helson (2005) is whether the converse is true, i.e.,whether for every bounded multiplicative bilinear form there is abounded function on T∞ that defines the form.

    Helson proved that if the bilinear form is Hilbert-Schmidt, i.e.∑ρ(n)2d(n)

  • The Schur test

    Helson, in his last paper published posthumously in 2010, usedthe Schur test to estimate the norm of Hψ. In our case, all weneed is a sequence of positive numbers cj such that∑

    k

    ρjkck ≤ Bcj .

    Then Hψ will be bounded on H2 × H2 with norm ≤ B.

    A key point is this: If we view Hψ as a linear functional acting onH1 and equip H1 with the norm ‖ · ‖w ,1, then this functional willhave the same bound B on its norm.

    Helson used the following simple and natural form: ρn = 1 forn ≤ N and ρn = 0 otherwise.

  • Helson’s conjecture

    By his simple argument involving the Schur test, Helsonobtained a large class of Dirichlet polynomials Q for which‖Q‖2 ≤ 2‖Q‖w ,1, namely those with unimodular coefficients. Togive a negative answer to the question of whether weakfactorization extends to the infinite-dimensional polydisc, whichwas his main object, he “only” needed a sequence ofpolynomials QN for which ‖QN‖1 = o(‖QN‖2) when N →∞. Hemade the following plausible conjecture:

    ‖N∑

    n=1

    n−s‖1 = o(√

    N).

    The conjecture is a very interesting but difficult numbertheoretic/probabilistic problem. (I think it is best understood inprobabilistic terms as a particular instance of polynomialchaos.)

  • Solution to Helson’s problem

    Although the discrete representation of multiplicative Hankelforms suggests a strong link to number theory, we can avoidthe difficulties encountered by Helson, while still using theSchur test in the same way. Namely, consider instead thehomogeneous polynomial

    ψ(z) = (z1 + z2) · (z3 + z4) · · · (z2k−1 + z2k ).

    Theorem (Ortega-Cerdà–Seip (2012))

    supf 6=0,f∈H1(D2k ) ‖f‖w ,1/‖f‖1 ≥ (π2/8)k/2.

    The proof shows that this particular ψ has optimal weakfactorization ψ · 1 (a function that “factors” badly!) and thus‖ψ‖w ,1 = ‖ψ‖2. Our conclusion is that Nehari’s theorem doesnot extend to d =∞ or, equivalently, it does not extend tomultiplicative Hankel forms.

  • Second result: The Bohnenblust–Hille inequality

    The Bohnenblust–Hille inequality (1931) estimates the size ofthe coefficients (measured in terms of a suitable `p norm) of apolynomial F in terms of ‖F‖∞, with a constant that does notdepend on the dimension d .

    Keep in mind that for a general polynomial F (z) =∑

    aαzα, wetrivially have (∑

    α

    |aα|2) 1

    2 ≤ ‖F‖∞.

  • The Bohnenblust–Hille inequality

    We now restrict ourselves to m-homogeneous polynomialsF (z) =

    ∑|α|=m aαz

    α and ask: Is it possible to have

    ( ∑‖α‖=m

    |aα|p) 1

    p ≤ C‖F‖∞

    for some p < 2 with C depending on m but not on d?

    Bohnenblust–HilleYES, and 2m/(m + 1) is the smallest possible p.

    It is of basic interest also to know the asymptotic behavior of Cwhen p = 2m/(m + 1) and m→∞; Bohnenblust and Hilleobtained a super-exponential bound roughly as mm/2.

  • Polarization: the crux of the matter

    The profound insight of the work of Bohnenblust–Hille is thatthere is a one-to-one correspondence (called polarization)between m-homogeneous polynomials and symmetric m-linearforms that essentially preserves L∞ norms. (Polarization haslater found many important applications.)It is easier to do estimates with forms; in fact, thetransformation from homogeneous polynomials to symmetricmultilinear forms seems to be an indispensable tool for provingthe Bohnenblust–Hille inequality. (The ideas for dealing withmultilinear forms come from earlier work of Littlewood (1930)on corresponding estimates for bilinear forms.)

  • The polynomial BH-inequality is hypercontractive

    The following is a recent improvement of the Bohnenblust–Hilleinequality, based on a new way of connecting estimates forsymmetric multilinear forms and homogeneous polynomials:

    Theorem (Defant, Frerick, Ortega-Cerdà, Ounaı̈es, Seip 2011)Let m and d be positive integers larger than 1. Then we have( ∑

    |α|=m

    |aα|2m

    m+1)m+1

    2m ≤ e√

    m(√

    2)m−1 supz∈Dd

    ∣∣∣ ∑|α|=m

    aαzα∣∣∣

    for every m-homogeneous polynomial∑|α|=m aαz

    α on Cd .

    Diniz–Muñoz-Fernández–Pellegrino–Seoane-Sepúlveda (2012)and others have later studied this and other similar constants. Itremains unknown whether the constant grows sub-exponentially, but for the applications that I am aware of,exponential growth is good enough.

  • Third result: GCD sums from Poisson integrals

    Recall the Poisson kernel on D:

    P1(w , z) =1− |w |2

    |1− wz|2.

    The Poisson integral

    u(w) =∫T

    f (z)P1(w , z)dσ1(z)

    gives a harmonic function whose radial limits on T coincide a.e.with f if say f is integrable. Choose in particular f (z) = zn andw = r > 0. Then

    rn =∫T

    f (z)P1(r , z)dσ1(z) =∫T

    f (z)P1(r , z)dσ1(z).

  • GCD sums from Poisson integrals ctd.

    An even more elaborate way of writing this, is:

    r |m−n| =∫T

    zmznP1(r , z)dσ1(z).

    If we lift this to the polydisc, we get something interesting!

  • GCD sums from Poisson integral ctd.

    We write

    Pd (w , z) =d∏

    k=1

    1− |wk |2

    |1− wkzk |2.

    It is convenient in this definition to allow w to be a point in theinfinite-dimensional polydisc D∞. Let β = (β(1), ..., β(d),0,0, ...)and µ = (µ(1), ..., µ(d),0,0, ...) be multi-indices, and set|β − µ| = (|β(1) − µ(1)|, ..., |β(d) − µ(d)|,0,0, ...). Then ifr = (r1, r2, ...) is a sequence of positive numbers in D∞, we get

    r |β−µ| =∫Td

    zβzµPd (r , z)dσd (z).

    It leads to the following lemma.

  • Lemma on Poisson integrals

    Lemma

    For a positive sequence r in D∞, arbitrary multi-indicesβ1 = β1, ..., βN with d = max

    ⋃j suppβj and complex numbers

    c1, ..., cN , we have

    N∑k ,`=1

    r |βk−β`|ckc` =∫Td

    ∣∣∣ N∑j=1

    cjzβj∣∣∣2Pd (r , z)dσd (z).

  • Integers in multi-index notation

    If we let p = (p1,p2, ...) denote the sequence of primes writtenin ascending order, then we may associate with every integer na unique multi-index β and write n = pβ; setting nk = pβk andn` = pβ` and taking into account that

    |a− b| = a + b − 2 min(a,b),

    then we get

    (p−1j )|βk−β`| =

    (gcd(nk ,n`))2

    nkn`.

    More generally, if r = p−α := (p−α1 ,p−α2 , ...) for some α > 0,

    then

    r |βk−β`| =(gcd(nk ,n`))2α

    (nkn`)α.

  • Lemma on GCD sums

    Setting cj ≡ 1 in the lemma on Poisson integrals, we get

    Lemma

    For arbitrary positive integers n1,n2, ...,nN written as nj = pβj ,we have

    N∑k ,`=1

    (gcd(nk ,n`))2α

    (nkn`)α=

    ∫Td

    ∣∣∣ N∑j=1

    zβj∣∣∣2Pd (p−α, z)dσd (z).

    We are particularly interested in estimating how large suchsums can be when N is fixed and the positive integers aredistinct; to this end, set

    Γα(N) = sup1N

    N∑k ,`=1

    (gcd(nk ,n`))2α

    (nkn`)α,

    where the supremum is taken over all distinct n1, ...,nN .

  • Estimates of Γα(N)

    The study of such GCD sums was initiated by Koksma (1930s)who observed that they show up in uniform distribution theory.Erdős (1940) raised the problem of estimating Γ1(N); Gál(1949) solved this problem and showed that

    Γ1(N) ' (log log N)2.

    Dyer and Harman (1986), motivated by applications in metricDiophantine approximation, proved that

    Γ1/2(N)� exp(

    c log Nlog log N

    )and also that

    Γα(N)� exp(

    (log N)(4−4α)/(3−2α))

    for 1/2 < α < 1.

  • A sharp result for 0 < α < 1

    Theorem (Aistleitner–Berkes–Seip 2013)

    Γα(N) ≤

    exp

    (c(α)(log N)1−α(log log N)−α

    ), 1/2 < α < 1

    exp(c(log N log log N)1/2

    ), α = 1/2

    N(1+o(1))(1−2α), 0 < α < 1/2

    when N →∞.

    These estimates are best possible up to the precise value ofc(α), with a possible exception for the case α = 1/2.

  • About the proof of this theorem

    Summarized in two sentences: The body of the proof is acombination of (1) a combinatorial reasoning (coming fromGál’s work) regarding the left-hand side of the identity

    N∑k ,`=1

    r |βk−β`|ckc` =∫Td

    ∣∣∣ N∑j=1

    cjzβj∣∣∣2Pd (r , z)dσd (z).

    when ck ≡ 1 and (2) an analytic argument applied to theright-hand side of the same identity. The point of thecombinatorial argument is to prepare for the analytic argumentby reducing our consideration to multi-indices of small support(of order log N).

    By our lemma on Poisson integrals, we may as well view ourtheorem as a function theoretic result. In this perspective, I findthe application of Gál’s argument a surprising contribution ofcombinatorics to function theory in polydiscs.

  • Application: A Carleson–Hunt-type inequality

    Theorem (Aistleitner–Berkes–Seip 2013)For every 1-periodic real-valued function f of mean 0 andbounded variation, there exists a constant c such that thefollowing holds. For every finite and strictly increasingsequence of positive integers (nk )1≤k≤N and every associatedfinite sequence of real numbers (ck )1≤k≤N , we have

    ∫ 10

    (max

    1≤M≤N

    ∣∣∣∣∣M∑

    k=1

    ck f (nkx)

    ∣∣∣∣∣)2

    dx ≤ c (log log N)4N∑

    k=1

    c2k .

    Thus: The “price” we pay when replacing cos(2πx) (classicalCarleson–Hunt) by a function f of bounded variation is a factor(log log N)4. (Here the best possible power of log log N isknown to lie in the interval [2,4].)

  • Remark on the Riemann zeta-function

    Observation: Our estimates for Γα(N) are of the same order ofmagnitude as the conjectured maximal order of ζ(α + it) forN < t < 2N.

    This looks as a mere coincidence and could be so too, but, onthe other hand, a concrete link has been found in a closelyrelated situation: Hilberdink (2009) has shown that the largesteigenvalue of the matrix(

    (gcd(n,m))2α

    (nm)α

    )Nn,m=1

    can be used to estimate from below the maximum of ζ(α + it)for N < t < 2N, but his bound is a little smaller than the bestknown lower bound for this maximum.