PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 85(99) (2009), 55–61 DOI:10.2298/PIM0999055J

ON THE SOLID HULL OF THE HARDY–LORENTZ SPACE

Miroljub Jevtić and Miroslav Pavlović

Communicated by Žarko Mijajlović

Abstract. The solid hulls of the Hardy–Lorentz spaces H^{p,q} , 0 < p < 1, 0 < q ⩽ ∞ and H_{0}^{p,∞} , 0 < p < 1, as well as of the mixed norm space H_{0}^{p,∞,α} , 0 < p ⩽ 1, 0 < α < ∞, are determined.

Introduction

In [JP1 ] the solid hull of the Hardy space H^{p}, 0 < p < 1, is determined. In this article we determine the solid hulls of the Hardy–Lorentz spaces H^{p,q}, 0 < p < 1, 0 < q ⩽ ∞ and H_{0}^{p,∞} , 0 < p < 1, as well as of the mixed norm space H_{0}^{p,∞,α} , 0 < p ⩽ 1, 0 < α < ∞. Since H^{p,p} = H^{p} our results generalize [JP1 , Theorem 1].

Recall, the Hardy space H^{p}, 0 < p ⩽ ∞, is the space of all functions f holomorphic in the unit disk U, (f ∈ H(U)), for which ∥f∥_{p} = lim_{r→1} M_{p}(r, f) < ∞, where, as usual,

M_{p}(r, f) = ( \frac{1}{2π} ∫_{0}^{2π} |f(re^{it})|^{p}dt) ^{1/p}, 0 < p < ∞,
M_{∞} (r, f) = sup_{0⩽t<2π} |f(re^{it})|.

Now we introduce a generalization and refinement of the spaces H^{p}; the Hardy– Lorentz spaces H^{p,q}, 0 < p < ∞, 0 < q ⩽ ∞.

Let σ denotes normalized Lebesgue measure on T = ∂U and let L^{0}(σ) be the space of complex-valued Lebesgue measurable functions on T. For f ∈ L^{0}(σ) and s ⩾ 0 we write

λ_{f} (s) = σ({ξ ∈ T : |f(ξ)| > s})

for the distribution function and

f^{⋆}(s) = inf ({t ⩾ 0 : λ_{f} (t) ⩽ s})

for the decreasing rearrangement of |f| each taken with respect to σ.

2000 Mathematics Subject Classification: Primary 30D55; Secondary 42A45.

Research supported by the grant ON144010 from MNS, Serbia.

The Lorentz functional ∥ · ∥_{p,q} is defined at f ∈ L^{0}(σ) by
∥f∥_{p,q} = ( ∫_{0}^{1} (f^{⋆}(s)s^{1/p})^{q} \frac{ds}{s}) ^{1/q} for 0 < q < ∞,
∥f∥_{p,∞} = sup{f^{⋆}(s)s^{1/p} : s ⩾ 0}.

The corresponding Lorentz space is L^{p,q}(σ) = {f ∈ L^{0}(σ) : ∥f∥_{p,q} < ∞}. The space L^{p,q}(σ) is separable if and only if q ≠ ∞. The class of functions f ∈ L^{0}(σ) satisfying lim_{s→0} (f^{⋆}(s)s^{1/p}) = 0 is a separable closed subspace of L^{p,∞} (σ), which is denoted by L_{0}^{p,∞} (σ).

The Nevanlinna class N is the subclass of functions f ∈ H(U) for which

sup_{0<r<1} ∫_{T} log^{+} |f(rξ)| dσ(ξ) < ∞.

Functions in N are known to have non-tangential limits σ-a.e. on T. Consequently every f ∈ N determines a boundary value function which we also denote by f. Thus

f(ξ) = lim_{r→1} f(rξ) σ-a.e. ξ ∈ T.

The Smirnov class N^{+} is the subclass of N consisting of those functions f for which

lim_{r→1} ∫_{T} log^{+} |f(rξ)| dσ(ξ) = ∫_{T} log^{+} |f(ξ)| dσ(ξ).

We define the Hardy–Lorentz space H^{p,q}, 0 < p < ∞, 0 < q ⩽ ∞, to be the space of functions f ∈ N^{+} with boundary value function in L^{p,q}(σ) and we put ∥f∥_{H^{p,q}} = ∥f∥_{p,q}. The functions in H^{p,∞} with a boundary value function in L_{0}^{p,∞} (σ) form a closed subspace of H^{p,∞} , which is denoted by H_{0}^{p,∞} . The cases of major interest are of course p = q and q = ∞; indeed H^{p,p} is nothing but H^{p}, and H^{p,∞} is the weak-H^{p}.

The mixed norm space H^{p,q,α} , 0 < p ⩽ ∞, 0 < q, α < ∞, consists of all f ∈ H(U) for which

∥f∥_{H^{p,q,α}} = ∥f∥_{p,q,α} = ( ∫_{0}^{1}(1 − r)^{qα−1} M_{p}(r, f)^{q}dr) ^{1/q} < ∞.

H^{p,q,α} can also be defined when q = ∞, in which case it is sometimes known as the weighted Hardy space H^{p,∞,α} , and consists of all f ∈ H(U) for which

∥f∥_{p,∞,α} = sup_{0<r<1} (1 − r)^{α} M_{p}(r, f) < ∞.

The functions in H^{p,∞,α} 0 < p ⩽ ∞ for which lim_{r→1} (1 − r)^{α} M_{p}(r, f) = 0 form a closed subspace which is denoted by H_{0}^{p,∞,α} .

Throughout this paper, we identify the holomorphic function f(z) = ∑ _{k=0}^{∞} f̂(k)z^{k} with its sequence of Taylor coefficients {f̂(k)}_{k=0}^{∞} .

If f(z) = ∑ _{k=0}^{∞} f̂(k)z^{k} belongs to H^{p,q}, then

(1)  f̂(k) = O((k + 1)^{(1/p)−1} ), if 0 < p < 1 and 0 < q ⩽ ∞.

(See [Al] and [Co].)

In this paper we find the strongest condition that the moduli of an H^{p,q}, 0 < p < 1, 0 < q ⩽ ∞, satisfy. Our result shows that the estimate (1) is optimal only if q = ∞.

To state our results in a form of theorems we need to introduce some more notations

A sequence space X is solid if {b_{n}} ∈ X whenever {a_{n}} ∈ X and |b_{n}| ⩽ |a_{n}|. More generally, we define S(X), the solid hull of X. Explicitly,

S(X) = {{λ_{n}} : there exists {a_{n}} ∈ X such that |λ_{n}| ⩽ |a_{n}|}.
A complex sequence {a_{n}} is of class l(p, q), 0 < p, q ⩽ ∞, if

∥{a_{n}}∥_{p,q}^{q} = ∥{a_{n}}∥_{l(p,q)}^{q} = ∑_{n=0}^{∞} ( ∑_{k∈I_{n}} |a_{k}|^{p}) ^{q/p} < ∞,

where I_{0} = {0}, I_{n} = {k ∈ N : 2^{n−1} ⩽ k < 2^{n}}, n = 1, 2, … In the case where p or q is infinite, replace the corresponding sum by a supremum. Note that l(p, p) = l^{p}.

For t ∈ R we write D^{t} for the sequence {(n + 1)^{t}}, for all n ⩾ 0. If λ = {λ_{n}} is a sequence and X a sequence space, we write λX = {{λ_{n}x_{n}} : {x_{n}} ∈ X}; thus, for example, {a_{n}} ∈ D^{t}l^{∞} if and only if |a_{n}| = O(n^{t}).

We are now ready to state our first result.

Theorem 1. If 0 < p < 1 and 0 < q ⩽ ∞, then S(H^{p,q}) = D^{(1/p)−1} l(∞, q).

In particular, S(H^{p}) = D^{(1/p)−1} l(∞, p), 0 < p < 1. This was proved in [JP1 ]. Also, S(H^{p,∞} ) = D^{(1/p)−1} l^{∞} means that the estimate (1) valid for the Taylor coefficients of an H^{p,∞} , 0 < p < 1, function is sharp.

Our second result is as follows:

Theorem 2. If 0 < p < 1, then S(H_{0}^{p,∞} ) = D^{(1/p)−1} c_{0}, where c_{0} is the space of all null sequences.

Our method of proving Theorem 1 and Theorem 2 depend upon nested embedding [Le, Theorem 4.1] for Hardy–Lorentz spaces. Thus, the strategy is to trap H^{p,q} between a pair of mixed norm spaces and then deduce the results for H^{p,q} from the corresponding results for the mixed norm spaces. Our Theorem 1 will follow from the following two theorems:

Theorem L. [Le] Let 0 < p_{0} < p < s ⩽ ∞, 0 < q ⩽ t ⩽ ∞ and β >

(1/p_{0}) − (1/p). Then

(2)  D^{−β} H^{p_{0} ,q,β+(1/p)−(1/p_{0} )} ⊂ H^{p,q} ⊂ H^{s,q,(1/p)−(1/s)} ,
(3)  D^{−β} H_{0}^{p_{0} ,∞,β+(1/p)−(1/p_{0} )}  ⊂ H_{0}^{p,∞} ⊂ H_{0}^{s,∞,(1/p)−(1/s)} .
Theorem JP 1. [JP1 ] If 0 < p ⩽ 1, 0 < q ⩽ ∞ and 0 < α < ∞, then

S(H^{p,q,α} ) = D^{α+(1/p)−1} l(∞, q).

To prove Theorem 2 we first determine the solid hull of the space H_{0}^{p,∞,α} , 0 < p ⩽ 1, 0 < α < ∞. More precisely, we prove

Theorem 3. If 0 < p ⩽ 1 and 0 < α < ∞, then S(H_{0}^{p,∞,α} ) = D^{α+(1/p)−1} c_{0}.

Given two vector spaces X, Y of sequences we denote by (X, Y) the space of multipliers from X to Y. More precisely,

(X, Y) = {λ = {λ_{n}} : {λ_{n}a_{n}} ∈ Y, for every {a_{n}} ∈ X}.

As an application of our results we calculate multipliers (H^{p,q}, l(u, v)), 0<p<1, 0 < q ⩽ ∞, (H_{0}^{p,∞} , l(u, v)), 0 < p < 1, and (H^{p,∞} , X), 0 < p < 1, where X is a solid space. These results extend some of the results obtained by Lengfield [Le, Section 5].

1. The solid hull of the Hardy–Lorentz space H^{p,q} , 0 < p < 1, 0 < q ⩽ ⩽ ⩽ ∞

Proof of Theorem 1. Let 0 < p < 1. Choose p_{0} and s so that p_{0} < p < s ⩽ 1 and a real number β so that β + (1/p) − (1/p_{0}) > 0. As an easy consequence of Theorem JP we have

S(D^{−β} H^{p_{0} ,q,β+(1/p)−(1/p_{0} )} ) = D^{(1/p)−1} l(∞, q).

Also, by Theorem JP,

S(H^{s,q,(1/p)−(1/s)} ) = D^{(1/p)−1} l(∞, q),

and consequently S(H^{p,q}) = D^{(1/p)−1} l(∞, q), by Theorem L. □

2. The solid hull of mixed norm space H_{0}^{p,∞,α} , 0 < p ⩽ ⩽ ⩽ 1, 0 < α < ∞

If f(z) = ∑ _{k=0}^{∞} f̂(k)z^{k} and g(z) = ∑ _{k=0}^{∞} ĝ(k)z^{k} are holomorphic functions in U, then the function f ⋆ g is defined by (f ⋆ g)(z) = ∑ _{k=0}^{∞} f̂(k)ĝ(k)z^{k}.

The main tool for proving Theorem 3 are polynomials W_{n}, n ⩾ 0, constructed in [JP1 ]and [JP3 ]. Recall the construction and some of their properties.

Let ω : R → R be a nonincreasing function of class C^{∞} such that ω(t) = 1, for t ⩽ 1, and ω(t) = 0, for t ⩾ 2. We define polynomials W_{n} = W_{n}^{ω}, n ⩾ 0, in the following way:

W_{0}(z) = ∑_{k=0}^{∞} ω(k)z^{k} and W_{n}(z) = ∑_{k=2^{n−1}}^{2^{n+1}} φ( \frac{k}{2^{n−1}} ) z^{k}, for n ⩾ 1,

where φ(t) = ω(t/2) − ω(t), t ∈ R.

The coefficients Ŵ_{n}(k) of these polynomials have the following properties:

(4)  supp{Ŵ_{n}} ⊂ [2^{n−1} , 2^{n+1} ];
(5)  0 ⩽ Ŵ_{n}(k) ⩽ 1, for all k,
(6)  ∑_{n=0}^{∞} Ŵ_{n}(k) = 1, for all k,
(7)  Ŵ_{n}(k) + Ŵ_{n+1} (k) = 1, for 2^{n} ⩽ k ⩽ 2^{n+1} , n ⩾ 0.

Property (5) implies that

f(z) = ∑_{n=0}^{∞} (W_{n} ⋆ f)(z), f ∈ H(U),

the series being uniformly convergent on compact subsets of U.

If 0 < p < 1, then there exists a constant C > 0 depending only on p such that

(8)  ∥W_{n}∥_{p}^{p} ⩽ C_{p}2^{−n(1−p)} , n ⩾ 0.

Proof of Theorem 3. Let f ∈ H_{0}^{p,∞,α} , 0 < p < 1, 0 < α < ∞. By using the familiar inequality

M_{p}(r, f) ⩾ C(1 − r)^{(1/p)−1} M_{1}(r^{2}, f), 0 < p ⩽ 1,

(see [Du , Theorem 5.9]), we obtain

sup_{k∈I_{n}} |f̂(k)|r^{2k} ⩽ M_{1}(r^{2}, f) ⩽ CM_{p}(r, f)(1 − r)^{1−(1/p)} , 0 < r < 1.

Now we take r_{n} = 1 − 2^{−n} and let n → ∞, to get {f̂(k)} ∈ D^{α+(1/p)−1} c_{0}. Thus H_{0}^{p,∞,α} ⊂ D^{α+(1/p)−1} c_{0}.

To show that D^{α+(1/p)−1} c_{0} is the solid hull of H_{0}^{p,∞,α} , it is enough to prove that if {a_{n}} ∈ D^{α+(1/p)−1} c_{0}, then there exists {b_{n}} ∈ H_{0}^{p,∞,α} such that |b_{n}| ⩾ |a_{n}|, for all n.

Let {a_{n}} ∈ D^{α+(1/p)−1} c_{0}. Define

g(z) = ∑_{j=0}^{∞} B_{j} (W_{j} (z) + W_{j+1} (z)) = ∑_{k=0}^{∞} c_{k}z^{k},

where B_{j} = sup_{2^{j} ⩽k<2^{j+1}} |a_{k}|. Using (4) and (8) we find that

M_{p}^{p}(r, g) ⩽ ∑_{j=0}^{∞} B_{j}^{p}(M_{p}^{p}(r, W_{j}) + M_{p}^{p}(r, W_{j+1} )) ⩽ C( B_{0}^{p} + ∑_{j=1}^{∞} B_{j}^{p}r^{p2^{j−1}} 2^{−j(1−p)} )

Set B_{j}^{p}2^{−j(αp+1−p)} = λ_{j}. Then

M_{p}^{p}(r, g) ⩽ C( λ_{0} + ∑_{j=1}^{∞} λ_{j} r^{p2^{j−1}} 2^{jαp} ) ,

where λ_{j} → 0, as j → ∞. From this it easily follows that (1 − r)^{αp} M_{p}^{p}(r, g) → 0, as r → 1. Thus g ∈ H_{0}^{p,∞,α} .

To prove that |c_{k}| ⩾ |a_{k}|, k = 1, 2, …, choose n so that 2^{n} ⩽ k < 2^{n+1} . It follows from (7)

c_{k} = ∑_{j=0}^{∞} B_{j} (Ŵ_{j}(k) + Ŵ_{j+1} (k)) ⩾ B_{n}(Ŵ_{n}(k) + Ŵ_{n+1} (k))

= B_{n} = sup_{2^{n} ⩽j<2^{n+1}} |a_{j} | ⩾ |a_{k}|.

Now the function h(z) = ∑ _{n=0}^{∞} b_{n}z^{n}, where b_{0} = a_{0} and b_{n} = c_{n}, for n ⩾ 1, belongs to H_{0}^{p,∞,α} and |b_{n}| ⩾ |a_{n}| for all n ⩾ 0. This finishes the proof of Theorem 3. □

3. The solid hull of the space H_{0}^{p,∞} , 0 < p < 1

Proof of Theorem 2. Let 0 < p < 1. Choose p_{0} and s so that p_{0} < p < s ⩽ 1 and β ∈ R so that β + (1/p) − (1/p_{0}) > 0. Then

S(D^{−β} H_{0}^{p_{0} ,∞,β+(1/p)−(1/p_{0} )} ) = D^{(1/p)−1} c_{0},
S(H_{0}^{s,∞,(1/p)−(1/s)} ) = D^{(1/p)−1} c_{0},

by Theorem 3. By Theorem L we have S(H_{0}^{p,∞} ) = D^{(1/p)−1} c_{0}. □

4. Applications to multipliers

As it was noticed in the introduction, another objective of this paper is to extend some of the results given in [Le, Section 5].

The next lemma due to Kellog (see [K]) (who states it for exponents no smaller than 1, but it then follows for all exponents, since {λ_{n}} ∈ (l(a, b), l(c, d))) if and only if {λ_{n}^{(1/t)} )} ∈ (l(at, bt), l(ct, dt)).

Lemma 1. If 0 < a, b, c, d ⩽ ∞, then (l(a, b), l(c, d)) = l(a ⊝ c, b ⊝ d), where a ⊝ c = ∞ if a ⩽ c, b ⊝ d = ∞, if b ⩽ d, and

\frac{1}{a ⊝ c} = \frac{1}{c} − \frac{1}{a}, for 0 < c < a,

\frac{1}{b ⊝ d} = \frac{1}{d} − \frac{1}{b}, for 0 < d < b.

In particular, (l^{∞} , l(u, v)) = l(u, v). Also, it is known that (c_{0}, l(u, v)) = l(u, v). In [AS ] it is proved that if X is any solid space and A any vector space of

sequences, then (A, X) = (S(A), X).

Since l(u, v) are solid spaces, we have (H^{p,q}, l(u, v)) = (S(H^{p,q}), l(u, v)) and (H_{0}^{p,∞} , l(u, v)) = (S(H_{0}^{p,∞} ), l(u, v)). Using this, Lemma 1, Theorem 1 and Theorem 2 we get

Theorem 4. Let 0 < p < 1 and 0 < q ⩽ ∞. Then

(H^{p,q}, l(u, v)) = D^{1−(1/p)} l(u, q ⊝ v).

Theorem 5. Let 0 < p < 1. Then

(H_{0}^{p,∞} , l(u, v)) = D^{1−(1/p)} l(u, v).

In particular, (H^{p,∞} , l(u, v)) = D^{1−(1/p)} l(u, v). In fact more is true.

Theorem 6. Let 0 < p < 1 and let X be a solid space. Then

(H^{p,∞} , X) = D^{1−(1/p)} X.

Proof. Since X is a solid space, we have (l^{∞} , X) = X. Hence, using Theorem 1 we get

(H^{p,∞} , X) = (S(H^{p,∞} ), X) = (D^{(1/p)−1} l^{∞} , X)
= D^{1−(1/p)} (l^{∞} , X) = D^{1−(1/p)} X.  □

References

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[AS] J. M. Anderson and A. Shields, Coefficient multipliers of Bloch functions, Trans. Am. Math. Soc. 224 (1976), 255–265.

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[Du] P. L. Duren, Theory of H^{p} Spaces, Academic Press, New York 1970; reprinted by Dover, Mineola, NY, 2000.

[Le] M. Lengfield, A nested embedding theorem for Hardy–Lorentz spaces with applications to coefficient multiplier problem, Rocky Mount. J. Math. 38(4) (2008), 1215–1251.

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[JP2] M. Jevtić and M. Pavlović, Coefficient multipliers on spaces of analytic functions, Acta Sci. Math. (Szeged) 64 (1998), 531–545.

[JP3] M. Jevtić and M. Pavlović, On multipliers from H^{p} to l^{q}, 0 < q < p < 1, Arch. Math. 56 (1991), 174–180.

[K] C. N. Kellog, An extension of the Hausdorff–Young theorem, Michigan. Math. J. 18 (1971), 121–127.

[P] M. Pavlović, Introduction to Function Spaces on the Disk, Matematički Institut, Beograd, 2004.

Matematički fakultet (Received 12 05 2008) Studentski trg 16 11000 Beograd, p.p. 550 Serbia

 jevtic@matf.bg.ac.yu
pavlovic@matf.bg.ac.yu