Mean growth of Hp functions

Publicacions Matem`atiques, Vol 42 (1998), 301–318. MEAN GROWTH OF HpFUNCTIONS Daniel Girela and Mar´ ıa Auxiliadora M´ arquez Abstract A classical result of Hardy and Littlewood asserts that if 0 <p< q<∞and fis a function which is analytic in the unit disc and belongs to the Hardy space Hp, then, if λ≥pand α=1 p−1 q,we have Z1 0 (1 −r)λα−1µ1 2πZ2π 0 |f(reiθ)|qdθ¶λ/q dr < ∞. We prove that this result is sharp in a very strong sense. Indeed, we prove that if p,q,λand αare as above and ϕis a positive, continuous and increasing function defined in [0,∞) with ϕ(x) xq→∞,asx→∞, then there exists a function f∈Hpsuch that Z1 0 (1 −r)λα−1µZI ϕ¡|f(reiθ)|¢dθ¶λ/q dr =∞, for every non-degenerate interval I⊂[0,2π]. We also prove a result of the same kind concerning functions fsuch that f0∈Hp, 0<p<1. 1. Introduction and statement of results Let ∆ denote the unit disc {z∈C:|z|<1}and Tthe unit circle {ξ∈C:|ξ|=1}. For 0 <r<1 and ganalytic in ∆ we set Mp(r, g)=µ1 2πZ2π 0¯ ¯g(reiθ)¯¯ pdθ¶1/p ,0<p<∞, M ∞ (r, g) = max |z|=r|g(z)|. 1991 Mathematics subject classifications: 30D55. This research has been supported in part by a D.G.I.C.Y.T. grant (PB94-1496) and by a grant from “La Junta de Andaluc´ıa”. 302 D. Girela, M. A. M´ arquez For 0 <p≤∞the Hardy space Hpconsists of those functions g, analytic in ∆, for which kgkHp= sup 0<r<1 Mp(r, g)<∞. Hardy and Littlewood proved in [6] (see also [2, Th. 5.9]) the following. Theorem A. If 0<p<q≤∞and f∈Hp, then (1.1) Mq(r, f)=oÃ1 (1 −r)1 p−1 q!,as r→1. Considering the function f(z)= 1 (1−z) 1 p −εfor small ε>0, we easily see that the exponent 1 p−1 qis best possible. Duren and Taylor proved in [3] (see also [8]) that the Hardy-Littlewood estimate (1.1) is sharp in a stronger sense. Namely, they proved the following result. Theorem B. Let 0<p<q≤∞, and let φ(r)be a positive and non-increasing function on 0≤r<1, with φ(r)→0,asr→1. Then there exists a function f∈Hpsuch that Mq(r, f)6=OÃφ(r) (1 −r)1 p−1 q!,as r→1. Although, as we have said, Theorem A is best possible in a strong sense, Hardy and Littlewood were able to sharpen it in one direction proving the following useful result (see [2, Th. 5.11]). Theorem C. If 0<p<q≤∞,f∈H p ,λ≥p, and α=1 p−1 q, then (1.2) Z1 0 (1 −r)λα−1Mq(r, f)λdr < ∞. The fact that (1.2) implies (1.1) is clear having in mind that Mq(r, f) is an increasing function of r. Let us remark that Flett gave in [4]a proof of Theorem C based on the Marcinkiewicz interpolation theorem. Also, it is worth noticing that if we take q<∞and λ=qthen we obtain the following: If 0 <p<q<∞and f∈Hp, then Z2π 0Z1 0 (1 −r)q p−2|f(reiθ)|qdr dθ < ∞. Our first result in this paper shows that Theorem C is sharp in a very strong sense. Mean growth of Hpfunctions 303 Theorem 1. Let 0<p<q<∞,λ≥p, and α=1 p−1 q.Let ϕ:[0,∞)→[0,∞)be a continuous and increasing function with (1.3) ϕ(x) xq→∞,as x→∞. Then, there exists a function f∈Hpsuch that (1.4) Z1 0 (1 −r)λα−1µZI ϕ¡¯¯f(reiθ)¯¯¢dθ¶λ/q dr =∞, for every non-degenerate interval I⊂[0,2π]. In particular, if 0<p<q<∞and ϕ:[0,∞)→[0,∞)is as above, then there exists a function f∈Hpsuch that ZIZ1 0 (1 −r)q p−2ϕ¡¯¯f(reiθ)¯¯¢dr dθ =∞, for every non-degenerate interval I⊂[0,2π]. According to a classical result of Privalov [2, Th. 3.11], a function f analytic in ∆ has a continuous extension to the closed unit disc ∆ whose boundary values are absolutely continuous on ∂∆ if and only if f0∈H1. In particular, (1.5) f0∈H1⇒f∈H∞. This result has been shown to be sharp. Indeed, Yamashita proved in [9] that there exists a function fanalytic in ∆ with f0∈Hpfor all p∈(0,1) but such that fis not even a normal function, and the first author has recently proved in [5] that no restriction on the growth of M1(r, f0) other than its boundedness is enough to conclude that fis a normal function. We refer to [1] and [7] for the theory of normal functions. On the other hand, Hardy and Littlewood obtained the following generalization of (1.5) (see [2, Th. 5.12]). Theorem D. Let fbe a function which is analytic in ∆.If0<p<1 and f0∈Hpthen f∈Hq, where q=p/(1 −p). Taking f0(z)=(1−z) ε− 1 pfor small ε>0 shows that for each value of p∈(0,1) the index qis best possible. Our next result proves the sharpness of Theorem D in a much stronger sense. 304 D. Girela, M. A. M´ arquez Theorem 2. Let 0<p<1and q=p/(1−p).Letϕ:[0,∞)→[0,∞) be a continuous and increasing function satisfying (1.3). Then, there exists a function fanalytic in ∆with f0∈Hpsuch that (1.6) ZI ϕ¡¯¯f(eiθ)¯¯¢dθ =∞, for every non-degenerate interval I⊂[0,2π]. Let us remark that if pand qare as in Theorem 2 and f0∈Hp, then, by Theorem D, f∈Hqand, hence, fhas a finite non-tangential limit f(eiθ) for almost every θ. Hence, the left hand side of (1.6) makes sense. 2. Proof of the results The proofs of our results will be constructive. Let αand βbe two positive real numbers, and let {δk}∞ k=1 be a sequence of real numbers with (2.1) 0 <δ k<2 −k ,for all k. For k=1,2,..., and j=1,2,... ,2 k, define θk j=2π(2j−1) 2k+1 ,(2.2) Ik j=(θ k j−δ k ,θk j+δ k).(2.3) Notice that, for each k, the intervals Ik j,j=1,2,... ,2 k, are pairwise disjoint. Set (2.4) rk=1−δ k ,k=1,2,... . For k=1,2,..., define (2.5) fk(z)= 2 k X j=1 δα k (1 −rke−iθk jz)β,z∈∆. Let us remark that the functions fkare in fact analytic in the closed unit disc ∆. Actually, the functions fkdepend on α,βand the sequence {δk}, however, we shall not indicate this dependence explicitely. We believe that this will not cause any confusion. Mean growth of Hpfunctions 305 We shall make use of some lemmas to deal with the functions fk. The proofs are elementary and some of them will be omitted. First of all, let us recall that |1−reiθ|≤2|θ|,0<r≤1,1−r≤|θ|≤π,(2.6) |1−reiθ|≥|θ| π,0<r≤1,|θ|≤π,(2.7) |1−eiθ|≥2 |θ| π,|θ|≤π.(2.8) Lemma 1. If l6=m, then |θk l−θk m|≥ π 2 k−1. Lemma 2. If θ∈Ik j, then |eiθk j−rkeiθ|≤2δ k (2.9) and |eiθk l−rkeiθ|≥ 1 2 k−1,for all l6=j.(2.10) Proof: Let θ∈Ik j, then |θ−θk j|<δ k , which, with (2.6), implies |eiθk j−rkeiθ|=|1−rkei(θ−θk j)|≤|1−r k e iδk|≤2δ k . This is (2.9). Now, let l6=jand let ϕk lbe an angle such that eiϕk l=eiθk l and |θ−ϕk l|≤π. Then, using (2.4), (2.8), Lemma 1 and (2.1), we obtain |eiθk l−rkeiθ|=|eiϕk l−rkeiθ|≥|e iϕk l−eiθ|−|e iθ −rkeiθ| =|eiϕk l−eiθ|−δ k≥2|ϕ k l−θ| π−δ k ≥2 π¡|ϕ k l−θ k j|−|θ k j−θ| ¢−δ k≥2 π³π 2 k−1−δ k´−δ k ≥1 2 k−2−2δ k≥1 2 k−1. Hence, (2.10) holds. 306 D. Girela, M. A. M´ arquez Lemma 3. If n<k, then |θn l−θk j|≥ π 2 k,for all l, j. Lemma 4. If θ∈Ik j,n<kand 0<r≤1, then |eiθn l−rnreiθ|≥ 1 2 k+1 ,for all l∈{1,2,... ,2 n}. Proof: Let θ∈Ik jand let ϕn lbe defined as in the proof of Lemma 2. Then, using (2.7), Lemma 3, (2.3) and (2.1), we see that |eiθn l−rnreiθ|=|eiϕn l−rnreiθ|≥|ϕ n l−θ| π ≥1 π¡|ϕ n l−θ k j|−|θ k j−θ| ¢ ≥1 π³π 2 k−δ k´>1 2 k+1 . We shall see that a suitable choice of the numbers α,βand the sequence {δk}will allow us to construct functions fanalytic in ∆ having the properties asserted in Theorems 1 and 2. Precisely, we can prove the following results. Theorem 3. Let 0<p<q<∞,λ≥p, and α=1 p−1 q.Let ϕ:[0,∞)→[0,∞)be a continuous and increasing function satisfying (1.3). Then, there exist two positive numbers αand β, a sequence of real numbers {δk}∞ k=1 which satisfies (2.1), and a sequence of positive numbers {ck}∞ k=1, such that, if fis the function defined by (2.11) f(z)= ∞ X k=1 ckfk(z),z∈∆, then f∈Hpand (1.4) holds for every non-degenerate interval I⊂ [0,2π]. Mean growth of Hpfunctions 307 Theorem 4. Let 0<p<1and q=p/(1−p).Letϕ:[0,∞)→[0,∞) be a continuous and increasing function satisfying (1.3). Then, there exist two positive numbers αand β, a sequence of real numbers {δk}∞ k=1 which satisfies (2.1), and a sequence of positive numbers {ck}∞ k=1, such that, if fis the function defined by (2.12) f(z)= ∞ X k=1 ckfk(z),z∈∆, then fis analytic in ∆,f0∈Hpand (1.6) holds for every non-degenerate interval I⊂[0,2π]. Clearly, Theorem 1 and Theorem 2 follow from Theorem 3 and Theorem 4 respectively. Proof of Theorem 3: Let αbe any positive number, and let (2.13) β=α+1 p. Let {δk}∞ k=1 be a sequence of real numbers which satisfies (2.1) to be specified later. Set ck=2 −k (1 p+2),k=1,2,... . Define the functions fk,k=1,2,..., as in (2.5), let gk=ckfkfor all k and let fbe defined as in (2.11). Hence, f(z)= ∞ X k=1 gk(z),z∈∆. Notice that |gk(z)|=|2−k(1 p+2)fk(z)|≤ 2 −k δ α k (1 −|z|) β≤2 −k (1 −|z|) β for all z∈∆ and, hence, the series P∞ k=1 gk(z) converges uniformly on every compact subset of ∆ and then it defines a function fwhich is analytic in ∆. Now, having in mind the elementary inequality (a1+a2+···+a n) p≤n p(a p 1+a p 2+···+a p n), p>0,a i≥0 for i=1,2,... ,n, 308 D. Girela, M. A. M´ arquez and the fact that for each γ>1 there exists a constant c=cγ>0 such that (2.14) 1 2πZ2π 0 1 |1−reiθ|γdθ ≤c (1 −r)γ−1,0<r<1, and using (2.4) and (2.13), we obtain that kgkkp Hp=kgk(eiθ)kp Lp=1 2πZ2π 0 |gk(eiθ)|pdθ ≤1 2πZ2π 0 cp k2kp δαp k 2k X j=1 1 |1−rke−iθk jeiθ|βp dθ =(2 k c k ) pδ αp k2k1 2πZ2π 0 1 |1−rkeiθ|βp dθ ≤2−kp δαp k c (1 −rk)βp−1=2 −kpc, where cis the positive constant which appears in (2.14) with γ=βp > 1. Thus, we have proved that (2.15) kgkkHp≤2−kc1/p,k=1,2,... , which, clearly, implies that f∈Hp. Next we turn to estimate the value of |f(reiθ)|when θbelongs to one of the intervals Ik jgiven in (2.3), and 0 <r<1, or at least when θis in a suitable subset of Ik jand ris close to 1, say rk<r<1. Suppose that θ∈Ik jand 0 <r<1. Then (2.16) |f(reiθ)|=¯¯¯¯¯ ∞ X n=1 gn(reiθ)¯¯¯¯¯ ≥|g k (reiθ)|− k−1 X n=1 |gn(reiθ)|− ∞ X n=k+1 |gn(reiθ)|. We shall estimate each of these three terms separetely. Mean growth of Hpfunctions 309 First, for θ∈Ik jand 0 <r<1, (2.17) |gk(reiθ)|=ck¯¯¯¯¯¯ 2k X l=1 δα k (1 −rke−iθk lreiθ)β¯¯¯¯¯¯ ≥ck    δα k |1−rke−iθk jreiθ|β− 2k X l=1 l6=j δα k |1−rke−iθk lreiθ|β    . If r>r k , using (2.9) and (2.4), we see that |1−rke−iθk jreiθ|=|eiθk j−rkreiθ|≤|e iθk j−rkeiθ|+|rkeiθ −rkreiθ| ≤2δk+rk(1 −r)≤2δk+(1−r k)=3δ k . If l6=jand r>r k , (2.10), (2.4) and (2.1) give |1−rke−iθk lreiθ|=|eiθk l−rkreiθ|≥|e iθk l−rkeiθ|−|r ke iθ −rkreiθ| ≥1 2k−1−rk(1 −r)≥1 2k−1−(1 −rk)= 1 2 k−1−δ k≥1 2 k. Then, (2.17) implies that (2.18) |gk(reiθ)|≥c kµδ α k (3δk)β−2kδα k¡2k¢β¶ =ckδα kµ1 (3δk)β−2k(1+β)¶,θ∈I k j ,r k<r<1. Let us take the numbers δkso small that (2.19) 2k(1+β)<1 2 1 (3δk)β,k=1,2,... , then, (2.18) and (2.13) give (2.20) |gk(reiθ)|≥ 1 2·3 βc kδ −1/p k,θ∈I k j ,r k<r<1. 316 D. Girela, M. A. M´ arquez So, we have proved that (2.37) |gk(eiθ)|≥ 1 2 β+1 ckδ−1/q k,θ∈I k j . Next, take θ∈Ik jand n<k. Using Lemma 4, we deduce that |gn(eiθ)|≤c n 2 n X l=1 δα n ¯¯1−rne−iθn leiθ¯¯ β =cnδα n 2n X l=1 1 ¯¯eiθn l−rneiθ¯¯ β ≤cnδα n2n¡2k+1¢β=2 n ( 1−2 p) δ α n2 (k+1)β≤2−n2(k+1)β. Hence, (2.38) k−1 X n=1 |gn(eiθ)|≤2 (k+1)β,θ∈I k j . For every positive integer n, define Jn l,l=1,2,... ,2 n, by (2.23), and suppose, as in the proof of Theorem 3, that (2.22) is satisfied. Notice that Lemma 5 holds for every r∈(0,1), and so it also does for r=1. Finally, define the sets Ek j, for k=1,2,..., by (2.24). Then, the same argument used in the proof of Theorem 3 shows that (2.39) ∞ X n=k+1 |gn(eiθ)|≤π β ,θ∈E k j . It follows from (2.36), (2.37), (2.38) and (2.39), that |f(eiθ)|≥ 1 2 β+1 ckδ−1/q k−2(k+1)β−πβ,θ∈E k j , and, taking the numbers δksufficiently small, we have (2.40) |f(eiθ)|≥ 1 2 β+2 ckδ−1/q k,θ∈E k j . For k=1,2,..., let λk=1 2β+2 ck, Mean growth of Hpfunctions 317 and notice that (1.3) implies ϕ(λkx) xq→∞,as x→∞, and so there exists εk>0 such that εϕ³λ kε −1/q´>k, 0<ε≤ε k . We may assume that the numbers δkalso satisfy 0<δ k≤ε k ,k=1,2,... . Therefore, (2.41) δkϕµ1 2β+2 ckδ−1/q k¶>k, k=1,2,... . Also, as in the proof of Theorem 3, we can take the numbers δksmall enough so that (2.31) holds, and then (2.42) |Ek j|≥δ k ,j=1,2,... ,2 k, for all k=1,2,... . From (2.40), the fact that ϕis increasing, (2.41) and (2.42), we conclude that, for each set Ek j,wehave ZE k j ϕ ¡¯ ¯f(e iθ)¯¯¢dθ ≥ZEk j ϕµ1 2β+2 ckδ−1/q k¶dθ =ϕµ1 2β+2 ckδ−1/q k¶|Ek j| ≥kδ −1 kδ k=k. An argument similar to that used at the end of the proof of Theorem 3 shows that this implies that (1.6) holds for every non-degenerate interval I⊂[0,2π]. This finishes the proof. 318 D. Girela, M. A. M´ arquez References 1. J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. 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Departamento de An´alisis Matem´atico Facultad de Ciencias Universidad de M´alaga 29071 M´alaga SPAIN e-mail: [email protected] e-mail: [email protected] Rebut el 24 de febrer de 1997