Marzuq Maher M. H.

University of Toyama Repository

Toyama Math. J.Vol. 33(2010), 43-53Interpolation sequence for the spaces Hq+(φ)(q ≥ 1)Maher M.H. MarzuqAbstract. Let φ be a subadditive increasing real valued functiondefined on [0,∞) and which satisfies φ(x) = 0 if and only if x = 0.For q ≥ 1 we define Hq(φ) to be the set of all functions f which areanalytic in the open unit disc and satisfysup0≤r<1∫ 2pi0[φ(|f(reiθ)|]q dθ <∞.And Hq+(φ) to be the subspace of Hq(φ) of functions which satisfylimr→1∫ 2pi0[φ(|f(reiθ)|)]q dθ = ∫ 2pi0[φ(|f(eiθ)|)]q dθ.In this paper we prove some interpolation theorems for Hq+(φ)1. IntroductionLet us recall some definitions. We call a real-valued function φ definedon [0,∞) a modulus function, if φ is an increasing continuous subadditivefunction that satisfies the condition that φ(x) = 0 if and only if x = 0.Let q ≥ 1. By the class Hq(φ) we mean the collection of all analyticfunctions f defined in the open unit disc ∆ which satisfysup0≤r<1∫ 2pi0[φ(|f(reiθ)|)]q dθ <∞.For q = 1 the spaces were studied in details in [2, 3]. If φ(x) = x, thenHq(φ) becomes the usual Banach space Hq. For φ(x) = xp, 0 < p ≤ 12000 Mathematics Subject Classification. Primary 46B45, 46A45.4344 Maher M.H. Marzuqand q = 1, then Hq(φ) becomes the usual F -spaces Hq. If q = 1 andφ(x) = log(1+xp), then Hq(φ) becomes Np. If q = 1 and φ(x) = log(1+x),then Hq(φ) becomes the class N of functions of bounded characteristic. ForNp = N+ for 0 < p ≤ 1 [8], whereNp = {f ∈ H+(D) ∩ Hp(φ) : limr→1∫ 2pi0log(1 + |f(reiθ)|p) dθ=∫ 2pi0log(1 + |f(eiθ)|p) dθ},andN+ = {f ∈ N : limr→1∫ 2pi0log+(|f(reit)|) dt =∫ 2pi0log+(|f(eit)|) dt}.In general the spaces Hq(φ) are not F -Spaces; see [13, p. 453] for anexample. We shall assume that φ satisfies the additional condition thatφ(et) is a convex function of t, and consequently Hq(φ) ⊆ N .A function f ∈ Hq(φ) is said to belong to the class Hq+(φ) iflimr→1∫ 2pi0[ϕ(|f(reiθ)|)]q dθ = ∫ 2pi0[ϕ(|f(eiθ)|)]q dθ.For the class Hq+(φ), which is a vector space , we define a distance func-tion byρ(f, 0) =[12pi∫ 2pi0[ϕ(|f(eiθ)|)]q dθ] 1q(1.1)For the spaces Hq(p ≥ 1), Hq(0 < p ≤ 1). [5], N+ [13] N+p (p > 0) and(log+H)α(a > 1), [12] become special cases of Hq+(φ).Let α = (αn) = (α1, . . . , αn, . . .), be a sequence of real numbers suchthat αn → 0.Letlq(φ, α) = {(cn) : cn ∈ C and d((cn), 0) =[ ∞∑n=1αn[ϕ(|cn)|)]q] 1q<∞}Let X be the class of analytic functions in ∆ and {zn} is a given sequencein ∆. When a complex sequence {cn} is given, the interpolation problemasks if a function f ∈ X exists such that f{zn} = cn for all n.Interpolation sequence for the spaces Hq+(φ)(q ≥ 1) 45Let Y be a class of complex sequences. If for every sequence (cn) ∈ Ythere is an f ∈ X such that f{zn} = cn, then the sequence {zn} is calleda universal interpolation sequence for the pair (X,Y ), simply written znis u. i. s. for (X,Y ). We are interested in the pair Hq+(φ), lq(φ, α) whereα = (1− |zn|2).A sequence {zn} in ∆ is called uniformly separated sequence (u. s. s.) if∞∑n=1(1− |zn|)<∞, and∞∏n=1m6=n∣∣∣ zn − zn1− z¯mzn∣∣∣ ≥ δ > 0 (m = 1, 2, . . .).L. Carleson [1] showed that {zn} is u. i. s. for (H∞, l∞) if and only if{zn} is u. s. s. For the pair(Hq+(φ), lq(φ, α))we have the following results:for q ≥ 1, φ(x) = x, [11] proved that {zn} is u. i. s. for(Hq+(φ), lq(φ, α))if and only if {zn} is u. s. s. The same result was proved for q = 1 andφ(x) = xp, 0 < p ≤ 1 by [9]. For q = 1, φ(x) = log(1 + x),[14] showed that{zn} is u. s. s., then {zn} is u. i. s. and if {zn} is u. i. s., then(1− |zn|2) log 1|Bn(zn)| → 0 as n→∞,whereBn(z) =∞∏m=1m6=n|zm|zmzm − z1− z¯mz .In this paper we obtain results which generalize the above mentionedresults.In section 2 we will prove that Hq+(φ) and lq(φ, α) are F -spaces in thesense of Banach, [4].In section 3 we prove that if {zn} is u. s. s. then {zn} is u. i. s. for(Hq+(φ), lq(φ, α)), we also proved that if φ satisfies limx→∞ϕ−1(ax)ϕ−1( 1x) <∞ for all real a and Tφ,q(Hq+(φ))= lq(φ, α) then {zn} is u. s. s. whereTφ,q(f(z)) =((φ−1(1− |zn|2)) 1q−1f(zn)).2. The spaces Hq+(φ), lq(φ,α).In this section we will show that the spaces Hq+(φ) and lq(φ, α) are F -spaces.46 Maher M.H. MarzuqTheorem 2.1. Hq+(φ) is an F -space in the sense of Banach,[4]. Thatisi. Let fn be functions in Hq+(φ) such that ρ(fn, 0)→ 0 as n→∞. Thenfor any α ∈ C, ρ(αfn, 0)→ 0 as n→∞.ii. Let αn ∈ C be such that αn → 0. Then for each f ∈ Hq+(φ),ρ(αnf, 0)→ 0 as n→∞.iii. Hq+(φ) is complete with respect to the metric (1.1).Proof. i. Suppose {fn} is a sequence in Hq+(φ), ρ(fn, 0) → 0 andβ ∈ C. Nowρ(βfn, 0) =(12pi∫ 2pi0[ϕ(|βfn(θ)|q dθ) 1q≤ [|β|+ 1]ρ(fn, 0)→ 0,where[|β|] is the greatest integer in |β|.ii. Suppose βn ∈ C, βn → 0 and f ∈ Hq+(φ), without loss of generalitywe may assume |βn| ≤ 1, so[ϕ(|βnf |)]q ≤ [φ(|f |)]q and ϕ(|βnf(θ)|)→ 0 a. e.Hence by Lebesgue convergence theorem we getlimn→∞12pi∫ 2pi0[φ(|βnf(θ)|]q dθ = 0.Thus ρ(βnf, 0)→ 0 as n→∞.iii. Suppose {fn} is a Cauchy sequence in Hq+(φ). By Lemma 3 in [2] ap-plied to[φ(|f |)]q which is subharmonic for q ≥ 1, from [6, Lemma 5.1],we get|f(z)| ≤ ϕ−1 cρ(f, 0)((1− |z|) 1q) for z ∈ ∆.Therefore|fn(z)− fm(z)| ≤ ϕ−1 cρ(fn, fm)((1− r) 1q) < εInterpolation sequence for the spaces Hq+(φ)(q ≥ 1) 47for n,m > N(ε) and for all z ∈ {w : |w| ≤ r < 1}. Hence {fn(z)}is a Cauchy sequence in C. Since {fn(z)} converges uniformly oncompact subsets of ∆ and fn(z) is analytic, then {fn} converges toan analytic function f . Clearly, {φ(|fn|)} converges uniformly oncompact subsets to φ(|f |). Therefore∫ 2pi0[ϕ(|f(reiθ)|]q dθ = limn→∞∫ 2pi0[ϕ(|fn(reiθ)|)]q dθ≤ limn→∞∫ 2pi0[ϕ(|fn(θ)|)]q dθ ≤ M,hence, f ∈ Hq(φ). But Hq(φ) ⊆ N , so limr→1f(reiθ) = f(θ) a. e. andfn(θ) → f(θ) in measure. Now choose a subsequence fnj such thatfnj (θ)→ f(θ) a. e., thenρ(f, fn) =(12pi∫ 2pi0[ϕ(|f(θ)− fn(θ|)]q dθ) 1q≤ limj→∞(12pi∫ 2pi0[ϕ(|fnj (θ)− fn(θ)|]q dθ) 1q≤ ρ(fnj , fn) + ε for large j.Thus if nj and n are sufficiently large, we haveρ(f, fn)→ 0, as n→∞.It remains to show that f ∈ Hq+(φ). Since fnj (θ) → f(θ) a. e., thenthere exists a compact set E ⊂ [0, 2pi] such that m(E) > 2pi − ε andfnj → f uniformly on E, hence(12pi∫ 2pi0[ϕ(|fnj (θ)|)]q dθ) 1q≤(12pi∫E[φ(|f(θ)|)]q dθ) 1q+ε , for large j.48 Maher M.H. MarzuqAlso,(∫Ec[ϕ(|fnj (θ)|)]q dθ) 1q≤(∫Ec[ϕ(|fnj (θ)− f(θ)|)]q dθ) 1q+(∫Ec[ϕ(|f(θ)|)]q dθ) 1q≤ ρ(fnj , f) +(∫Ec[ϕ(|f(θ)|)]q dθ) 1q≤ ε+(∫Ec[ϕ(|f(θ)|)]q dθ) 1q,but fnj ∈ Hq+(ϕ), so(∫ 2pi0[ϕ(|fnj (reiθ)|)]q dθ) 1q≤(∫ 2pi0[ϕ(|f(θ)|)]q dθ) 1q+ ε,letting j →∞,(∫ 2pi0[ϕ(|f(reiθ)|)]q dθ) 1q≤(∫ 2pi0[ϕ(|f(θ)|)]q dθ) 1q+ ε,for any ε > 0. Hencelimr→1(∫ 2pi0[ϕ(|f(reiθ)|)]q dθ) 1q=(∫ 2pi0[ϕ(|f(θ)|)]q dθ) 1q,this proves that f ∈ Hq+(φ).Theorem 2.2. lq(ϕ, α) is an F -space.Proof. Parts (i), (ii) are exactly the same as in Theorem 2.1. For com-pleteness, suppose {xn} is a Cauchy sequence, let ε > 0 be given, thenthere exists N such thatd(xk, xm) < ε for all k,m > N(ε).Hence ∞∑n=1αn(ϕ(|xkn − xmn |))q< εq, (2.1)Interpolation sequence for the spaces Hq+(φ)(q ≥ 1) 49where xk = (xkn), xm = (xmn ). Therefore|xkn − xmn | < ϕ−1[εα1qn].Thus {xkn} is a Cauchy sequence in C for each n. So it must converge tosome xn ∈ C.Let x = {xn}, then Minkowski’s inequality gives( ∞∑n=1αn (ϕ(|xn|))q) 1q≤( ∞∑n=1αn(ϕ(|xn − xkn|))q) 1q+( ∞∑n=1αn(ϕ(|xkn|))q) 1q,but {xn} is a Cauchy sequence, hence the second term of the right side isbounded by M which does not depend on k, also∑αn(ϕ(|xn − xkn|))q= limm→∞∑αn(ϕ(|xmn − xkn|))q< εqby (2.1). Thus xn → x and x ∈ lq(φ, α).3. Interpolation TheoremsWe will assume an additional condition on f , for each g ∈ H1 the functionlog[ϕ−1(|g(z)|) 1q + 1] is integrable on the unit circle.Now we prove a theorem which generalizes theorem 1 in [14].Theorem 3.1. If {zn} is u. s. s. then {zn} is u. i. s. for(Hq+(φ), lq(φα)),where α = (1− |zn|2).Proof. Let c = (cn) ∈ lq(φ, α) and letg(z) =∞∑n=1(1− |zn|2)2[ϕ|cn|)]q Bn(z)Bn(zn) · 1(1− z¯nz)2 .clearly g is analytic in ∆ since∞∑n=1(1−|zn|2)[ϕ(|cn|)]q <∞. Also g ∈ H1,because12pi∫ 2pi0|g(reiθ)| dθ ≤∞∑n=1(1− |zn|2)2[(ϕ|cn|)]q 1|Bn(zn)| 12pi∫ 2pi0dθ|1− z¯nz|≤ 1δ∞∑n=1(1− |zn|2)[ϕ(|cn|)]q<∞.50 Maher M.H. MarzuqSince log[ϕ−1(|g(z)| 1q ) + 1] is integrable, hence by [7, p. 53] there existsf1 ∈ H1 such that |f1(z)| = φ−1(|g(z)|1q )+1. Since Bk(zn) = 0 for all k 6= n,g (zn) = [φ (|cn|)]q. Hence, |f1(zn)| = |cn| + 1 and f1(zn) = (|cn| + 1)eiθn ,where cn = |cn|eiαn and f3(zn) = eiαn , then (f1f2 − f3)(zn) = cn. Sinceϕq(|f1(z)|) ≤ |g(z)|+ ϕq (1), we have f1 ∈ Hq(φ) and so is f1f2 − f3.By Carleson’s Theorem there exists functions f2, f3 in H∞ such thatf2(zn) = ei(αn−θn).The following theorem generalizes the one given by [11].Theorem 3.2. Suppose φ satisfieslimx→∞ϕ−1(ax)ϕ−1(1x)<∞,for all real a and Tφ,q(Hq+(φ))= lq(φ, α), then {zn} is u. s. s.We need the following lemma:Lemma. The sequence E = {en} is bounded in lq(φ, α) in the senseof topological vector space W. Rudin [10], where en = (0, 0, . . . , 0, 1, 0, . . .)and 1 appears in the nth place.Proof. Let s > 0 be given and let Bs = {x ∈ lq(φ, α)‖x‖ < s}. We needto show that there exists r0 such that E ⊂ rBs for all r > r0. Let r0 besuch that (φq)−1(s) = 1r0 , then for r > r0∥∥∥enr∥∥∥ = ϕq(1r)< ϕq( 1r0)= s,this implies enr ∈ Bs. Therefore E ⊂ rBs.Proof of Theorem 3.2. LetN qϕ = {f ∈ Hq+(ϕ) : f(zn) = 0 for all n}.The quotient space Hq+(ϕ)/Nqϕ is an F -space and Tφ,q induces a one-to-onebounded linear functional ∆ϕ,q (since Tϕ,q is bounded) from Hq+(ϕ)/Nqϕonto lq(φ, α). Hence, the inverse is bounded which implies that ∆−1φ,q(E)Interpolation sequence for the spaces Hq+(φ)(q ≥ 1) 51is bounded on Hq+(ϕ), i. e. there exists M > 0 such that ∆−1φ,q(E) ⊂ BM,which means that for all n, there exists fn ∈ Hq+(ϕ) such that‖fn‖ ≤ M, (3.1)andfn(zk) =ϕ−1(1− |zk|2)“1q”−1n = k,0 n 6= k.For n > k, letFnk(z) = fk(z)∏j=1j 6=k1− z¯jzz − zj ,then‖Fnk‖ ≤ ‖fk‖ and Fnk ∈ Hq+(ϕ),so by [2]|Fnk(z)| ≤ ϕ−1(c‖Fnk‖(1− |z|2) 1q), (3.2)but,|Fnk(z)| =∣∣∣∣ n∏j=1j 6=k1− z¯jzkzk − zj∣∣∣∣∣∣∣∣fk(zk)∣∣∣∣,thus|Fnk(zk)| =[ϕ−1(1− |zk|2)( 1q)−1] n∏j=1j 6=k∣∣∣1− z¯jzkzk − zj∣∣∣,and by using (3.1) and (3.2) we get∣∣∣∣ n∏j=1j 6=k1− z¯jzkzk − zj∣∣∣∣ ≤ ϕ−1(αk)ϕ−1(cMαk)= ϕ−1(xk)ϕ−1( 1xk)< ∆ <∞,for xk large, where xk = αkcM and ∆ is a constant. Thus {zk} is u. s. s.52 Maher M.H. MarzuqReferences[1] L. Carleson, An interpolation problem for bounded analytic functions,Amer. J. of Math., 80 (1958), 921–930.[2] W. Deeb and M. Marzuq, H(φ) spaces, Cand. Math. Bull., Vol. 29(3),(1986), 295–301.[3] W. Deeb, R. Khalil and Marzuq, Isometric multiplication of Hardy-Orlicz spaces, Bull. Austral. Math. Soc. 34 (1986), 177–189.[4] N. Dunford and J.T., Shwartz Linear operators, 1: General theory,Pure and Appl. Math., Vol. 7, Interscience, N.Y. 1958.[5] P. L. Duren, Theory of Hq spaces, Academic Press, N.Y. 1970.[6] M. Hasumi and S. Kataoka, Remarks on Hardy-Orlicz classes, Arch.Math. 51(1988), 455–463.[7] K. Hoffman, Banach spaces of analytic functions, Prentice Hall Inc.,N. J., (1962).[8] Z. Jianzhong, A note on Hardy-Orlicz spaces, Cand. Math. Bull. Vol.33(1), (1990), 29–33.[9] V. Kabaila, Interpolation sequences for Hq classes in the case p < 1,Litov. Mat. Sb., 3 (1963), #1, 141–147 (Russian).[10] W. Rudin, Functional Analysis, McGraw-Hill, N.Y. (1973).[11] H. S. Shapiro and A. L. Shields, On some interpolation problems foranalytic functions, Amer. J. Math., 83(1961), 513–532.[12] M. Stoll, Mean growth and Taylor coefficients of some topological al-gebras of analytic functions, Ann. Polon. Math. 35(1977), 139–158.[13] N. Yanagihara, Multipliers and linear functional for the class N+,Trans. Amer. Math. Soc. 180(1973), 449–461.[14] N. Yanagihara, Interpolation theorems for the class N+, III. J. Math.18(1974), 427–435.Interpolation sequence for the spaces Hq+(φ)(q ≥ 1) 53Maher M.H. Marzuq84 Raymond RoadPlymouth, MA 02360(Received February 4, 2011)

Interpolation sequence for the spaces H^<q>_<+>(φ)(q≥1)

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Interpolation sequence for the spaces H^<q>_<+>(φ)(q≥1)

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