On the growth of Hardy and Bergman norms of functions in the Dirichlet space

First Advanced Course in Operator Theory and Complex Analysis, University of Seville, June 2004 ON THE GROWTH OF HARDY AND BERGMAN NORMS OF FUNCTIONS IN THE DIRICHLET SPACE DRAGAN VUKOTI´ C Abstract. We review the Chang-Marshall inequality of Moser-Trudinger type for the Dirichlet space. We then use a weaker version of this result to derive a sharp asymptotic estimate for Hardy and Bergman norms of a Dirichlet function for large exponents. Introduction Denote by Hpand Aprespectively the standard Hardy and Bergman spaces of the unit disk D, 0 < p < ∞. The space A∞=H∞consists of all bounded analytic functions in D. Let Ddenote the Dirichlet space of all analytic functions in Dsuch that f0∈A2. It is well known that D ⊂ Hp⊂Apfor all p∈(0,∞). However, D 6⊂ H∞; that is, there exist unbounded functions in D. For any such function fwe obviously have limp→∞ kfkHp=kfkH∞=∞. The main result of this note consists in quantifying this in asymptotic form as follows: We have kfkHp=o(p1/2)as p→ ∞ (and likewise for the Apnorm). The exponent one-half cannot be improved. The proof uses two main tools: an inequality of Chang-Marshall (MoserTrudinger) type and a theorem on the Taylor coefficients of certain logarithmic functions in the Dirichlet space. 2000 Mathematics Subject Classification. 31C25, 30H05. Revised September 10, 2004. The author is supported by MCyT grant BFM2003-07294-C02-01, Spain. 147 148 D. VUKOTI´ C 1. Background We begin by reviewing the basic concepts and collecting the essential facts that will be needed later. 1.1. Hardy spaces. As is customary, we denote by Hpthe standard Hardy space of all functions analytic in the unit disk Dfor which kfkHp= sup 0<r<1µZ2π 0|f(reiθ)|pdθ 2π¶1/p <∞. The functions in any of these spaces have radial limits f(eiθ) almost everywhere on the unit circle T. The space H2admits the well known formula for norm computation: if f∈H2and f(z) = P∞ n=0 anznis its Taylor series in D, then (1) kfk2 H2= ∞ X n=0 |an|2. 1.2. Bergman spaces. Let dA denote the Lebesgue area measure, normalized so that A(D) = 1. If 0 < p < ∞, the Bergman space Apis the set of all analytic functions fin the unit disk Dwith finite Lp(D, dA) norm: kfkp Ap=ZD|f(z)|pdA(z) = 1 πZ1 0Z2π 0|f(reiθ)|pdθ r dr < ∞. Note that kfkApis a true norm if and only if 1 ≤p < ∞and, in this case, Ap is a Banach space. When 0 <p<1, Apis still complete with respect to the metric defined by dp(f, g) = kf−gkp Ap. It follows easily from the formula for Apnorm above and from the fact that the integral means ³R2π 0|f(reiθ)|pdθ 2π´1/p are increasing with rthat kfkAp≤ kfkHpand, therefore, Hp⊂Apfor all p. Formula (1) has its Bergman space analogue: if f∈A2and (an) is the sequence of its Taylor coefficients, then (2) kfk2 A2= ∞ X n=0 |an|2 n+ 1 . 1.3. The Dirichlet space and Beurling’s estimate. The Dirichlet space Dis the set of all analytic functions fin Dwith finite Dirichlet integral. The norm in Dis usually given by (3) kfk2 D=|f(0)|2+ZD|f0(z)|2dA(z) = |a0|2+ ∞ X n=1 n|an|2<∞. HARDY AND BERGMAN NORMS OF DIRICHLET FUNCTIONS 149 When fis a univalent (one-to-one) map, then the Jacobian of the change of variable w=f(z) is precisely |f0(z)|2, so we get kfk2 D=Zf(D) dA(w) = A[f(D)] <∞. In general, f∈ D means that the image Riemann surface f(D) has finite area. It is immediate from (1) and (3) that D ⊂ H2. It is actually a well known fact, although a bit more difficult to prove, that D ⊂ Hpfor all 0 < p < ∞([D], Chapter 6, Exercise 7). In any event, such inclusions are an easy consequence of the (not so easy) inequalities of Moser-Trudinger type that will be discussed here. Obviously, the space Dis not contained in H∞(for example, there are unbounded conformal maps of Donto domains of finite area). It is often convenient to consider the closed subspace D0={f∈ D :f(0) = 0}. Since D ⊂ H2, each function fin Dhas radial limits f(eiθ) almost everywhere. Let Eλ={θ∈[0,2π] : |f(eiθ)|> λ}and let |Eλ|be the normalized arc measure of this set on the unit circle T,i.e., the boundary distribution function of f. In his famous doctoral thesis in 1933, Beurling [Be] obtained the following estimate on this distribution function for the functions in the unit ball of D0. Theorem A.If f∈ D,f(0) = 0, and kfkD≤1then |Eλ| ≤ e−λ2+1. He also showed that this deep result is sharp by using a family of logarithmic functions. It seems that it was observed only much later that Beurling’s estimate implies another important inequality of Moser-Trudinger type. 1.4. The Chang-Marshall inequality. The integrability of exponential expressions of the functions whose derivative has certain integrability properties (in relation to the critical Sobolev index) has been a subject of study for several decades. Take as an example the following variant of the Sobolev imbedding theorem, due to Hardy and Littlewood in the case of analytic functions: whenever 0 < p < 2 and f0∈Ap, we have f∈A2p 2−p. But what can we say about the integrability of fin the critical case f∈ D? The answer clearly cannot be that f∈H∞, as we observed in Subsection 1.3, so it should ideally again be expressed by some integrability condition on f. It turns out that if f∈ D, then it has the following property: ZD e|f(z)|2dA(z)<∞. We can actually get a little more, but not much more! Important results in this respect (in the more general context of real variables) are due to N. Trudinger in the late 1960’s and J. Moser in the early 1970’s, which is why results with this flavor are usually referred to as the 150 D. VUKOTI´ C Moser-Trudinger inequalities. For a detailed bibliography, see Lecture 3 of [Ch], for example. By integrating in polar coordinates, keeping in mind that the integral means ³R2π 0|f(reiθ)|pdθ 2π´1/p are increasing with r, it is easy to see that ZD eα|f(z)|2dA(z)≤1 2πZ2π 0 eα|f(eiθ)|2dθ (α > 0) . It may come as a surprise that even these larger integrals over the unit circle will still be finite when f∈ D. One way of proving this is, as indicated in [CM], by using Beurling’s Theorem A and a nice trick due to Garnett. An alternative and simpler proof via Green’s formula is given in the forthcoming paper [PV]. Theorem B.For every fixed fin Dand for all α > 0we still have Z2π 0 eα|f(eiθ)|2dθ < ∞. Proof. We first prove the statement in the easier case α < 1. Applying Fubini’s theorem to a function g, increasing on [0,∞) and absolutely continuous function on every closed interval of this semi-axis (as in [R], Theorem 8.16), we get Z2π 0 g¡|f(eiθ)|¢dθ −2πg(0) = Z2π 0ÃZ|f(eiθ)| 0 g0(λ)dλ!dθ = 2πZ∞ 0|Eλ|g0(λ)dλ . By choosing g(λ) = eαλ2and taking into account Beurling’s Theorem A, we get (4) Z2π 0 eα|f(eiθ)|2dθ 2π= 1 + 2αZ∞ 0 λeαλ2|Eλ|dλ < ∞ for any α < 1. To prove the statement for arbitrary 0 < α < ∞, we follow the observation due to Garnett from p. 1016 of [CM]. If f(z) = P∞ n=0 anzn, there is obviously a polynomial Pand g∈ D such that f=P+g,g(0) = 0, and k√3αgkD≤1, whence by (4) we have Z2π 0 eα|f(eiθ)|2dθ 2π≤Z2π 0 e2α(|P|2+|g|2)dθ 2π≤e2αkPk2 ∞Z2π 0 e2α|g(eiθ)|2dθ < ∞, which proves the statement. ¤ Even though the integrals considered above are finite for all positive α, they need not be uniformly bounded for all α; in fact, whenever α > 1 they are not (even if we assume that f∈ D0)! This is shown by the same extremal logarithmic functions used by Beurling (see [CM]). In their celebrated paper HARDY AND BERGMAN NORMS OF DIRICHLET FUNCTIONS 151 [CM], Chang and Marshall proved the following impressive result, now usually referred to as the Chang-Marshall inequality: sup ½Z2π 0 e|f(eiθ)|2dθ :kfkD≤1, f(0) = 0¾<∞, thus answering the important open question at that time about the uniform estimate when α= 1. Later on, Marshall [M] simplified the initial (very difficult) proof of this statement. Mathematicians such as Ess´en and Carleson (and many others) have also been working on related problems. We mention the uniform Chang-Marshall inequality with α= 1 primarily as an important historical development but we will not need the full strength of the result. For our purpose, Theorem B (also from [CM]) will suffice. It should also be pointed out that Beurling’s Theorem A alone will not be enough to deduce our main result. 2. Asymptotic formulas for Hardy and Bergman space norms of functions in the Dirichlet space The notation an³bnfor two positive sequences will mean that the finite (nonzero) limit limn→∞ an/bnexists, while an.bnwill mean that an≤C bn for some fixed positive constant Cand all nlarge enough. Similar notation will be used below for positive functions u(p) of a positive real variable pinstead of sequences. It is a standard exercise to check that Hpnorms increase as pincreases and that limp→∞ kfkHp=kfkH∞. In particular, if fis an unbounded function in D, we have limp→∞ kfkHp=∞. This can be quantified as a precise asymptotic relation for the Hardy norms as p→ ∞. Observe that, if kfkD≤1 and f(0) = 0, then the formula for the distribution function used earlier, Beurling’s Theorem A, the change of variable t=λ2, and Stirling’s formula imply Z2π 0|f(eiθ)|pdθ 2π=pZ∞ 0 λp−1|Eλ|dλ ≤pe Z∞ 0 λp−1e−λ2dλ =pe 2Γ³p 2´ ³³p 2e´p+1 2, hence kfkHp.√pas p→ ∞ (and, in particular, f∈Hpfor all p). However, this can be improved to a “little-oh” estimate, as will be shown below. The following auxiliary result will be useful. 152 D. VUKOTI´ C Theorem C.For every real β, the Taylor coefficients anof the function (5) F(z) = µlog 2 1−z¶β have the property that an³n−1(log n)β−1as n→ ∞. Theorem C is stated as Theorem 2.31 and proved on p. 192 of the classical monograph [Z], hence we omit its proof. We are now ready to prove our main result. Theorem 1.(a) If f∈ D, then its Hpnorm enjoys the following asymptotic estimate: (6) kfkHp=o(p1/2)as p→ ∞. The exponent 1/2is best possible; that is, for every ε > 0, there exists a function Fε∈ D such that p−(1/2−ε)kFεkHp→ ∞ as p→ ∞. (b) If f∈ D, then its Apnorm also enjoys the estimate: (7) kfkAp=o(p1/2)as p→ ∞, and the exponent 1/2is best possible in the same sense as in (a). Proof. (a) Let f∈ D. It suffices to prove (6) for p= 2n: the norms kfkHp increase with p, so the general statement will follow from the inequality kfkH2n (2n+ 2)1/2≤kfkHp p1/2≤kfkH2n+2 (2n)1/2, where 2n≤p < 2n+ 2. Now by part (b) of Theorem B, for arbitrary positive αwe have (8) ∞ X n=0 αn n!Z2π 0|f(eiθ)|2ndθ 2π=Z2π 0 eα|f(eiθ)|2dθ 2π<∞. The general term of the series above must eventually be smaller than one, hence kfkH2n (n!)1/(2n)<1 √α,for all n≥Nα. It follows from here by Stirling’s formula that lim sup n→∞ kfkH2n n1/2≤C √α. Since this is true for all positive α, we conclude that lim n→∞ kfkH2n n1/2= 0 , so (6) follows. HARDY AND BERGMAN NORMS OF DIRICHLET FUNCTIONS 153 To see that the exponent one-half is best possible, let ε > 0 be arbitrary and choose β= (1 −ε)/2. Consider the function F=Fεgiven by (5) with εand β as above. Since β < 1/2, by (3) and Theorem C it follows that kFεk2 D= ∞ X n=1 n|an|2³ ∞ X n=1 n−1(log n)2β−2<∞, and so Fε∈ D. Again by Theorem C, the Taylor coefficients an,p of the function Fε(z)p/2=µlog 2 1−z¶pβ/2 behave asymptotically like n−1(log n)(pβ)/2−1. We are allowed to choose βso that pβ > 2. By (1) we have (9) kFεkp Hp=kFp/2 εk2 H2= ∞ X n=1 |an,p|2³ ∞ X n=1 n−2(log n)pβ−2. The latter series is equiconvergent with the integral (10) Z∞ 1 1 x2(log x)pβ−2dx =Z∞ 0 e−ttpβ−2dt = Γ(pβ −1) , which, by Stirling’s formula and for large p, is asymptotically equivalent to (pβ −1)pβ−3/2 epβ−1³µβ e¶pβ ppβ−3/2=app(p−pε−3)/2. When divided by pp/2−pε, this behaves like app(pε−3)/2and hence tends to infinity as p→ ∞. (b) We only have to worry about proving the sharpness, but this is quite similar for the Apspaces too: instead of (9), using (2) one obtains kFεkp Ap³ ∞ X n=1 n−3(log n)pβ−2, and instead of (10): Z∞ 0 e−2ttpβ−2dt = 2−(pβ−1)Γ(pβ −1) . The rest is completely analogous to the end. ¤ The exponent obtained from the apparently crude estimate (on the n-th term of a convergent series) turned out to be the best one. The heuristics behind this is that the remainder of a series of exponential type behaves asymptotically like its general term (and Hp,Apnorms increase with p). Acknowledgments The author would like to thank Miroslav Pavlovi´c for pointing out an inaccuracy in an earlier draft and the referee for discovering several misprints. 154 D. VUKOTI´ C References [Be] Beurling, A. ´ Etudes sur un probl`eme de majoration, Th`ese pour le doctorat, Almquist & Wieksell, Upsalla 1933. [Ch] Chang, S. Y. A. The Moser-Trudinger inequality and applications to some problems in conformal geometry, Nonlinear Partial Differential Equations in Differential Geometry (R. Hardt and M. Wolf, editors), Park City, Utah 1992, 65–125, IAS/Park City Math. Ser., 2, American Mathematical Society, Providence, Rhode Island 1996. [CM] Chang, S. Y. A.; Marshall, D. E. On a sharp inequality concerning the Dirichlet integral, Amer. J. Math. 107 (1985), 1015–1033. [D] Duren, P. L. Theory of HpSpaces, Academic Press, New York-London 1970. Reprint: Dover, Mineola, New York 2000. [M] Marshall, D. E. A new proof of a sharp inequality concerning the Dirichlet integral, Ark. Mat. 27 (1989), 131–137. [PV] Pavlovi´c, M.; Vukoti´c, D. The weak Chang-Marshall inequality via Green’s formula, Rocky Mountain J. Math., to appear. [R] Rudin, W. Real and Complex Analysis, Third Edition, McGraw-Hill, New York 1987. [Z] Zygmund, A. Trigonometric Series, Vol. I, Cambridge University Press, Cambridge 1959. Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain E-mail address:[email protected]