����������� ������ 58 (2006), 97–110

UDK 517.55

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SOME GENERALIZATIONS OF LITTLEWOOD-PALEY INEQUALITY IN THE POLYDISC

K. L. Avetisyan and R. F. Shamoyan

Abstract. The paper generalizes the well-known inequality of Littlewood-Paley in the polydisc. We establish a family of inequalities which are analogues and extensions of Littlewood-Paley type inequalities proved by Sh. Yamashita and D. Luecking in the unit disk. Some other generalizations of the Littlewood-Paley inequality are stated in terms of anisotropic Triebel-Lizorkin spaces. With the help of an extension of Hardy-Stein identity, we also obtain area inequalities and representations for quasi-norms in weighted spaces of holomorphic functions in the polydisc.

1. Introduction

Let D^{n} = {z = (z_{1}, …, z_{n}) ∈ C^{n} : |z_{j}| < 1, 1 ≤ j ≤ n} be the unit polydisc in C^{n}, and T^{n} = {ξ = (ξ_{1}, …, ξ_{n}) ∈ C^{n} : |ξ_{j}| = 1, 1 ≤ j ≤ n} be the ndimensional torus, the distinguished boundary of D^{n}. Denote by H(D^{n}) the set of all holomorphic functions in D^{n}. If f(z) = f(rξ) is a measurable function in D^{n}, then

M_{p}(f, r) = [\frac{1}{(2π)^{n}} ∫_{T^{n}} |f(rξ)|^{p}dm_{n}(ξ)]^{1/p} , r = (r_{1}, …, r_{n}) ∈ I^{n},

where 0 < p < ∞, I^{n} = (0, 1)^{n}, m_{n} is the n-dimensional Lebesgue measure on T^{n}. The collection of holomorphic functions f(z), for which ∥f∥_{H^{p}} = sup_{r∈I^{n}} M_{p}(f, r) <

+∞, is the usual Hardy space H^{p}. For a radial weight function ω(r) = ∏_{j=1}^{n} ω_{j}(r_{j}) the quasi-normed space L_{ω}^{p} (0 < p < ∞) is the set of those functions f(z) measurable in the polydisc D^{n}, for which the quasi-norm

∥f∥_{L_{ω}^{p}} = (C_{ω} ∫_{D^{n}} |f(z)|^{p} ∏_{j=1}^{n} ω_{j}(|z_{j}|) dm_{2n}(z))^{1/p}

is finite. Here dm_{2n}(z) = r dr dm_{n}(ξ) is the Lebesgue measure on D^{n}, and the constant C_{ω} is chosen so that ∥1∥_{L_{ω}^{p}} = 1. For the subspace of L_{ω}^{p} consisting of holomorphic functions let A_{ω}^{p} = H(D^{n}) ∩ L_{ω}^{p}. We will write L_{α}^{p}, A_{α}^{p} instead of L_{ω}^{p}, A_{ω}^{p} if ω_{j}(r_{j}) = (1 − r_{j})^{α_{j}} (α_{j} > −1, 1 ≤ j ≤ n).

AMS Subject Classification: 32A37, 32A36.

Keywords and phrases: Littlewood-Paley inequalities, polydisc, Triebel-Lizorkin spaces, weighted spaces.

The classical inequality of Littlewood and Paley for functions holomorphic in the unit disk D = D^{1} (see, e.g., [23]) is well known.

Theorem A. (Littlewood-Paley) If 2 ≤ p < ∞, then for any f ∈ H^{p}(D)

∫_{D} |f′(z)|^{p}(1 − |z|)^{p−1} dm_{2}(z) ≤ C∥f∥_{H^{p}}^{p}.  (1.1)

Many generalizations and extensions of Theorem A are known, see, for example, [1–2, 8–13, 17–22]. The next theorem is Luecking’s [9] generalization of (1.1).

Theorem B. (Luecking) Let 0 < p, s < ∞. Then

∫_{D} |f(z)|^{p−s}|f′(z)|^{s}(1 − |z|)^{s−1} dm_{2}(z) ≤ C∥f∥_{H^{p}}^{p}  (1.2)

for any f ∈ H^{p}(D) if and only if 2 ≤ s < p + 2.

We see that the case 0 < s < 2 is omitted. So, it would be of interest to obtain analogues of (1.2) for 0 < s < 2.

The present paper is organized as follows. Theorem 1 deals with Luecking’s integral (1.2) in the polydisc for 0 < s < 2. We obtain a family of inequalities reducing to the Littlewood-Paley inequality in the limiting case s, p → 2. Note that the proof of D. Luecking [9] essentially uses some one variable methods which are not extendible to the polydisc case by a direct iteration. We exploit function spaces introduced by R. Coifman, Y. Meyer and E. Stein [3] and apply methods for estimating of Luecking’s integral, which are quite different from those of [9]. In Theorem 2 we prove another extension of the Littlewood-Paley inequality in terms of anisotropic Triebel-Lizorkin spaces. Then we consider in Theorem 3 fractional derivatives of arbitrary order and estimate more general integrals for all 0 < s ≤ p < ∞. We establish in Theorem 4 other analogues of (1.2) by means of general weight functions ω(r). To this end, we extend to the polydisc the well-known Hardy-Stein identity. Finally, in Theorem 5 we give a characterization of weighted spaces A_{ω}^{p} on the polydisc with the use of (1.2) type integrals.

2. Notation and main theorems

We will use the conventional multi-index notations: rζ = (r_{1}ζ_{1}, …, r_{n}ζ_{n}), dr = dr_{1} ⋯ dr_{n}, (1 − |ζ|)^{α} = ∏_{j=1}^{n}(1 − |ζ_{j}|)^{α_{j}}, ζ^{α} = ∏_{j=1}^{n} ζ_{j}^{α_{j}}, αq + 1 = (α_{1}q + 1, …, α_{n}q + 1) for ζ ∈ C^{n}, r ∈ I^{n}, q ∈ R and a multi-index α = (α_{1}, …, α_{n}). Let Z_{+}^{n} denote the set of all multi-indices k = (k_{1}, …, k_{n}) with nonnegative integers k_{j} ∈ Z_{+}. Any inequality (or equality) A ≤ B quoted or proved is to be interpreted as meaning ’if B is finite, then A is finite, and A ≤ B’. Throughout the paper, the letters C(α, β, …), C_{α} etc. stand for positive constants possibly different at different places and depending only on the parameters indicated. For A, B > 0 we will write A ≲ B, if there exists an inessential constant c > 0 independent of variables involved such that A ≤ cB. The symbol A ≍ B means A ≲ B and B ≲ A. For any p, 1 ≤ p ≤ ∞, we define the conjugate index p′ as p′ = p/(p − 1) (we interpret 1/∞ = 0 and 1/0 = +∞).

For every function f ∈ H(D^{n}) having a series expansion f(z) = ∑_{k∈Z_{+}^{n}} a_{k}r^{k}ξ^{k}, where z = rξ, r ∈ I^{n}, ξ ∈ T^{n}, we define the radial fractional integro-differentiation of arbitrary order α = (α_{1}, …, α_{n}), α_{j} ∈ R by

D^{α}f(z) ≡ D_{r}^{α}f(z) = ∑_{k∈Z_{+}^{n}} ∏_{j=1}^{n}(1 + k_{j})^{α_{j}}a_{k}r^{k}ξ^{k}.

It is easily seen that D_{r}^{α}f(z) = D_{r_{1}}^{α_{1}}D_{r_{2}}^{α_{2}} … D_{r_{n}}^{α_{n}}f, where D_{r_{j}}^{α_{j}} means the same operator acting in the variable r_{j} only.

We now formulate the main theorems of the paper. First we establish a family of inequalities which are analogues of Littlewood-Paley type inequalities (1.2) proved by Sh. Yamashita [22] and D. Luecking [9] in the unit disk.

Theorem 1. Let 0 < α < s < 2, s < p. Then for any λ > (p − s)/α

∫_{D^{n}} |f(z)|^{p−s}|D^{1}f(z)|^{s}(1 − |z|)^{s−1}dm_{2n}(z) ≲ ∥f∥_{H^{λ}}^{p−s}∥D^{α/s}f∥_{H^{s}}^{s}.  (2.1)

Remark 1. Taking p = 2 in (2.1) and formally passing to the limit as s → 2− and α → +0, we get the classical Littlewood-Paley inequality (1.1) for p = 2 in the polydisc.

Recall now anisotropic Triebel-Lizorkin spaces on the polydisc, see [5], [11], [12], [15], [16]]. The function f(z) holomorphic in D^{n}, is said to belong to the space F_{α}^{pq} (0 < p < ∞, 0 < q ≤ ∞, α = (α_{1}, …, α_{n}), α_{j} ≥ 0), if for some multiindex β = (β_{1}, …, β_{n}), β_{j} > α_{j} the (quasi-)norm

∥f∥_{F_{α}^{pq}} = \{ \begin{matrix}[∫_{T^{n}}(∫_{I^{n}}(1 − r)^{(β−α)q−1}|D^{β}f(rξ)|^{q}dr)^{p/q}dm_{n}(ξ)]^{1/p}, & 0 < q < ∞, \\ [∫_{T^{n}}( sup_{r∈I^{n}}(1 − r)^{β−α}|D^{β}f(rξ)|)^{p}dm_{n}(ξ)]^{1/p}, & q = ∞, \end{matrix}

is finite. For different β (β_{j} > α_{j}) equivalent norms appear. Many well-studied function spaces are included in the Triebel–Lizorkin spaces. For p = q the space F_{α}^{pp} coincides with the holomorphic Besov space; for q = 2 Hardy-Sobolev spaces arise, and for q = 2, α_{j} = 0 the space F_{0}^{p2} coincides with H^{p}.

Theorem 2. For any 0 < p < ∞, 0 < q ≤ q_{1} ≤ ∞, α = (α_{1}, …, α_{n}), α_{j} ≥ 0 the following inclusion is continuous

F_{α}^{pq} ⊂ F_{α}^{pq_{1}}.  (2.2)

Remark 2. The inclusion (2.2) is proved in [11] in the setting of the unit ball of C^{n}. For the polydisc, (2.2) is a generalization of the inclusion F_{0}^{p2} ⊂ F_{0}^{p∞} proved in [6] as well as of that in [1], where α_{j} = 0 and n-harmonic functions are considered. In particular, for α_{j} = 0, q = 2, p = q_{1} the inclusion (2.2) reduces to (1.1).

In the next theorem the fractional derivative of the first order is replaced by the same operator D^{α} of arbitrary order α = (α_{1}, …, α_{n}), α_{j} > 0 and more general integrals are studied.

Theorem 3. Let 0 < s ≤ p < ∞, α_{j} > 0 (1 ≤ j ≤ n), and f(z) is a function of Hardy space H^{p}(D^{n}), and a function g(z) belongs to the mixed norm space H(p, s, α), that is

∥g∥_{H(p,s,α)}^{s} = ∫_{I^{n}} M_{p}^{s}(g, r)(1 − r)^{αs−1} dr < +∞.

Then

\frac{1}{(2π)^{n}} ∫_{D^{n}} |f(z)|^{p−s}|g(z)|^{s}(1 − |z|)^{αs−1} dm_{2n}(z) ≤ ∥f∥_{H^{p}}^{p−s}∥g∥_{H(p,s,α)}^{s}.

In particular, if D^{α}f ∈ H(p, s, α), then

\frac{1}{(2π)^{n}} ∫_{D^{n}} |f(z)|^{p−s}|D^{α}f(z)|^{s}(1 − |z|)^{αs−1} dm_{2n}(z) ≤ ∥f∥_{H^{p}}^{p−s}∥D^{α}f∥_{H(p,s,α)}^{s}. (2.3)

Theorem 4. (i) Let f(z) be a holomorphic function in D^{n}, 0 < p < ∞, ω_{j}(r_{j}), j = 1, …, n be weight functions positive and continuously differentiable in [0, 1) such that

ω_{j}(r_{j})\frac{∂}{∂r_{j}} M_{p}^{p}(f, r) = o(1)  as  r_{j} → 1 − .  (2.4)

Then the following identity holds:

∫_{D^{n}} ∏_{j=1}^{n} ω_{j}(r_{j}) · f^{#}(z) dm_{2n}(z) = (−1)^{n} ∫_{D^{n}} ∏_{j=1}^{n} ω′_{j}(r_{j})\frac{∂^{n}}{∂r_{1} ⋯ ∂r_{n}} |f(z)|^{p} dm_{2n}(z),

(2.5)

where f^{#}(z) = Δ_{z_{1}}Δ_{z_{2}} … Δ_{z_{n}}|f(z)|^{p}, and Δ_{z_{j}} is the usual Laplacian in the variable z_{j}. For the standard weight functions ω_{j}(r_{j}) = (1 − r_{j})^{α_{j}} (α_{j} > 0) the assumptions (2.4) can be dropped.

(ii) For n = 1 the following improvements of (2.5) are valid: The identity

∫_{D}(1 − |z|)^{α}f^{#}(z) dm_{2}(z) = α ∫_{D}(1 − |z|)^{α−1} \frac{∂}{∂r} |f(z)|^{p} dm_{2}(z), p > 0, α > 0,

(2.6)

holds if one of the integrals in (2.6) exists. Here

f^{#}(z) = Δ|f(z)|^{p} = p^{2}|f(z)|^{p−2}|f′(z)|^{2}.  (2.7)

(iii) The integrals

A(f; p, α) = ∫_{D} |f(z)|^{p−2}|f′(z)|^{2}(1 − |z|)^{α} dm_{2}(z),
B(f; p, α) = ∫_{D} |f(z)|^{p−1}|f′(z)|(1 − |z|)^{α−1} dm_{2}(z)

are comparable. More precisely,

– If p > 0, α > 0, then
A(f; p, α) ≤ \frac{α}{p} B(f; p, α),  (2.8)

where the constant α/p is sharp.

– If p > 0, α > 1, then there exists a constant C_{α,p} > 0 such that

B(f; p, α) ≤ C_{α,p} A(f; p, α).  (2.9)

Remark 3. The inequalities (2.8) and (2.9) for p = 2 are proved in [21]. Their analogues for integers p (p ≥ 2) in the unit disk and in the unit ball of C^{n} are proved in [17], [18] in another way.

The next theorem gives a characterization of weighted Bergman spaces A_{ω}^{p} on the bidisc and a representation for (quasi-)norms in A_{ω}^{p} with the use of (1.2) type integrals.

Theorem 5. Let 0 < p < ∞, f(z) ∈ H(D^{2}), ω_{j}(r_{j}) ∈ L^{1}(0, 1), ω_{j}(r_{j}) > 0, j = 1, 2. Then the following representations are valid:

∥f∥_{A_{ω}^{p}(D^{2})}^{p} ≍ |f(0, 0)|^{p} + ∫_{D^{2}}(Δ_{z_{1}}Δ_{z_{2}}|f(z_{1}, z_{2})|^{p} + Δ_{z_{1}}|f(z_{1}, 0)|^{p}+

+ Δ_{z_{2}}|f(0, z_{2})|^{p}) ∏_{j=1}^{2} h_{ω_{j}}(|z_{j}|) dm_{4}(z),  (2.10)
∥f∥_{A_{ω}^{p}(D^{2})}^{p} + |f(0, 0)|^{p} = ∥f(·, 0)∥_{A_{ω_{1}}^{p}}^{p} + ∥f(0, ·)∥_{A_{ω_{2}}^{p}}^{p} +
+ C_{ω} ∫_{D^{2}} f^{#}(z_{1}, z_{2}) ∏_{j=1}^{2} h_{ω_{j}}(|z_{j}|) dm_{4}(z), (2.11)

where A_{ω_{j}}^{p} is the weighted Bergman space in the variable z_{j}, and h_{ω_{j}} is the weight function

h_{ω_{j}}(|z_{j}|) = ∫_{|z_{j}|}^{1}(∫_{ρ_{j}}^{1} ω_{j}(x)x dx)\frac{dρ_{j}}{ρ_{j}} .

In particular, f ∈ A_{α}^{p}(D^{2}) if and only if f^{#} ∈ L_{α+2}^{1}(D^{2}) (α_{j} > −1).

Remark 4. For n = 1 and ω(r) = (1 − r)^{α} (α > −1) and by virtue of the formula (2.7), the relation (2.10) in the limiting case α → −1 coincides with Yamashita’s [22] characterization of Hardy spaces H^{p}(D), while some analogues of (2.10) and (2.11) for the unit ball of C^{n} are established in [2], [10], [20].

Without loss of generality and to simplify notation, we may assume that n = 2 everywhere below in the proofs.

3. Preliminaries and proof of Theorem 1

Let us introduce some more notation i order to formulate several auxiliary lemmas. In what follows, for a fixed δ > 1 let Γ_{δ}(ξ) = \{z ∈ D : |1−\overline{ξ}z| ≤ δ(1−|z|)\} be the admissible approach region whose vertex is at ξ ∈ T. For any arc I ⊂ T of the length |I| define the Carleson square over I to be □I = \{z ∈ D; \frac{z}{|z|} ∈ I, 1−|z| ≤ \frac{1}{2π} |I|\}. Following [3], consider the functions

A_{p}(f)(ξ) = (∫_{Γ_{δ}(ξ)} \frac{|f(z)|^{p}}{(1 − |z|)^{2}} dm_{2}(z))^{1/p},  p < ∞,
A_{∞}(f)(ξ) = sup{|f(z)|; z ∈ Γ_{δ}(ξ)},
C_{p}(f)(ξ) = sup_{I⊃ ξ} (\frac{1}{|I|} ∫_{□I} \frac{|f(z)|^{p}}{1 − |z|} dm_{2}(z))^{1/p} ,  p < ∞, ξ ∈ T.

Lemma C. ([3], [12]) For any functions f(z) and g(z) measurable in the unit disk

∫_{D} \frac{|f(z)|}{1 − |z|} dm_{2}(z) ≲ ∫_{T} (∫_{Γ_{δ}(ξ)} \frac{|f(z)|}{(1 − |z|)^{2}} dm_{2}(z)) dm(ξ),  (3.1)

∫_{D} \frac{|f(z)||g(z)|}{1 − |z|} dm_{2}(z) ≲ ∫_{T} A_{p}(f)(ξ) C_{p′}(g)(ξ) dm(ξ), 1 < p ≤ ∞, (3.2)

where dm(ξ) = dm_{1}(ξ) is the Lebesgue measure on the circle T.

For a proof of Lemma C see [3, pp. 313, 316, 326], [12, Th. 2.1].

Lemma D. ([3], [12]) For 0 < q < ∞, α > 0, β > 0 and a function f(z) measurable in the unit disk

∥C_{q}(|f(z)|(1 − |z|)^{α})∥_{L^{∞}}^{q} ≍ sup_{w∈D}(1 − |w|)^{β} ∫_{D} \frac{|f(z)|^{q}(1 − |z|)^{αq−1}}{|1 − \overline{w}z|^{β+1}} dm_{2}(z). (3.3)

For a proof of Lemma D including estimates of Carleson measures see [12, pp. 736–737], and also [4, Ch. VI, Sec. 3].

Define a version of Lusin’s area integral (see, e.g., [23])

S(f)(ξ) = (∫_{Γ_{δ}(ξ)} |D^{1}f(z)|^{2} dm_{2}(z))^{1/2} , ξ ∈ T, δ > 1.

Lemma E. (Lusin [23]) If f ∈ H(D), 0 < p < ∞, then ∥S(f)∥_{L^{p}(T)} ≍ ∥f∥_{H^{p}}.

We now turn to the proof of Theorem 1. Denote by L the integral on the left-hand side of (2.1) and write

L = ∫_{D}(1 − |z_{2}|)^{s−1} [∫_{D} |f(z)|^{p−s}|D^{1}f(z)|^{s}(1 − |z_{1}|)^{s−1} dm_{2}(z_{1})] dm_{2}(z_{2}). (3.4)

Denote also the inner integral in (3.4) by J. Choosing any α, 0 < α < s, we estimate J by Lemma C:

J = ∫_{D} |D^{1}f(z)|^{s}(1 − |z_{1}|)^{s−α} · |f(z)|^{p−s}(1 − |z_{1}|)^{α} \frac{dm_{2}(z_{1})}{1 − |z_{1}|}

≲ ∫_{T} A_{2/s}(|D^{1}f(z)|^{s}(1 − |z_{1}|)^{s−α})(ξ_{1}) · C_{(2/s)′}(|f(z)|^{p−s}(1 − |z_{1}|)^{α})(ξ_{1}) dm(ξ_{1})

≤ ∥C_{(2/s)′}(|f(z)|^{p−s}(1 − |z_{1}|)^{α})∥_{L^{∞}} ∫_{T} A_{2/s}(|D^{1}f(z)|^{s}(1 − |z_{1}|)^{s−α})(ξ_{1}) dm(ξ_{1}). (3.5)

Estimate the last integral separately:

J_{1} ≡ ∫_{T} A_{2/s}(|D^{1}f(z)|^{s}(1 − |z_{1}|)^{s−α})(ξ_{1}) dm(ξ_{1})

= ∫_{T} [∫_{Γ_{δ}(ξ_{1})} |D^{1}f(z)|^{2}(1 − |z_{1}|)^{−2α/s}dm_{2}(z_{1})]^{s/2} dm(ξ_{1}).

According to a result of [11, pp. 179, 186] on fractional differentiation and then by Lemma E

J_{1} ≲ ∫_{T} [∫_{Γ_{δ}(ξ_{1})} |D_{r_{1}}^{α/s}D^{1}f(z)|^{2} dm_{2}(z_{1})]^{s/2} dm(ξ_{1}) ≲ ∥D_{r_{2}}^{1}D_{r_{1}}^{α/s}f∥_{H_{z_{1}}^{s}}^{s} , (3.6)

where H_{z_{1}}^{s} means the Hardy space in the variable z_{1}. Combining the inequalities (3.4)–(3.6), we conclude that

L ≲ ∫_{D}(1 − |z_{2}|)^{s−1}∥C_{(2/s)′}(|f(z)|^{p−s}(1 − |z_{1}|)^{α})(ξ_{1})∥_{L^{∞}} ∥D_{r_{2}}^{1}D_{r_{1}}^{α/s}f∥_{H_{z_{1}}^{s}}^{s} dm_{2}(z_{2}).

By Fatou’s lemma and Lemma C

L ≲ lim inf_{r_{1}→1} ∫_{T} ∫_{D} (1 − |z_{2}|)^{s−1}∥C_{( \frac{2}{s})′}(|f(z)|^{p−s}(1 − |z_{1}|)^{α})∥_{L^{∞}}×
× |D_{r_{2}}^{1}D_{r_{1}}^{α/s}f|^{s}dm(ξ_{1}) dm_{2}(z_{2})

≲ ∥C_{(2/s)′}(∥C_{(2/s)′}(|f(z)|^{p−s}(1 − |z_{1}|)^{α})∥_{L^{∞}}(1 − |z_{2}|)^{α})(ξ_{2})∥_{L^{∞}}×

× lim inf_{r_{1}→1} ∫_{T} ∫_{T} A_{2/s}(|D_{r_{2}}^{1}D_{r_{1}}^{α/s}f|^{s}(1 − |z_{2}|)^{s−α})(ξ_{2}) dm(ξ_{2}) dm(ξ_{1}) ≡ J_{2} · J_{3}.

Let us now evaluate each factor J_{2} and J_{3} separately. Applying again the rule of fractional differentiation [11, pp. 179, 186], Lemma E, Fatou’s lemma and using the equality D_{r}^{γ_{1}}D_{r}^{γ_{2}} = D_{r}^{γ_{2}}D_{r}^{γ_{1}}, we get

J_{3} = lim inf_{r_{1}→1} ∫_{T} ∫_{T}[∫_{Γ_{δ}(ξ_{2})} |D_{r_{2}}^{1}D_{r_{1}}^{α/s}f|^{2}(1 − |z_{2}|)^{−2α/s}dm_{2}(z_{2})]^{s/2}dm(ξ_{2}) dm(ξ_{1})

≲ lim inf_{r_{1}→1} ∫_{T} ∫_{T}[∫_{Γ_{δ}(ξ_{2})} |D_{r_{2}}^{α/s}D_{r_{2}}^{1}D_{r_{1}}^{α/s}f|^{2}dm_{2}(z_{2})]^{s/2}dm(ξ_{2}) dm(ξ_{1})

≲ lim inf_{r_{1}→1} ∫_{T} ∥D^{α/s}f∥_{H_{z_{2}}^{s}}^{s} dm(ξ_{1}) = ∥D^{α/s}f∥_{H^{s}}^{s}.

Estimate now J_{2} choosing β > 0 large enough:

J_{2} = ∥C_{(2/s)′}(∥C_{(2/s)′}(|f(z)|^{p−s}(1 − |z_{1}|)^{α})∥_{L^{∞}}(1 − |z_{2}|)^{α})(ξ_{2})∥_{L^{∞}}.

By Lemma D, the inner norm can be estimated as follows

∥C_{2/(2−s)}(|f(z)|^{p−s}(1 − |z_{1}|)^{α})∥_{L^{∞}}^{2/(2−s)}

≲ sup_{w∈D}(1 − |w|)^{β} ∫_{D} |f(z_{1}, z_{2})|^{2(p−s)/(2−s)} \frac{(1 − |z_{1}|)^{2α/(2−s)−1}}{|1 − \overline{w}z_{1}|^{β+1}} dm_{2}(z_{1})

≤ ∥f∥_{H_{z_{1}}^{λ}}^{2(p−s)/(2−s)} sup_{w∈D}(1 − |w|)^{β} ∫_{D} \frac{(1 − |z_{1}|)^{2α/(2−s)−(2/λ)(p−s)/(2−s)−1}}{|1 − \overline{w}z_{1}|^{β+1}} dm_{2}(z_{1})

≲ ∥f∥_{H_{z_{1}}^{λ}}^{2(p−s)/(2−s)},

where the inequality |f(ζ)| ≲ ∥f∥_{H^{q}}(1 − |ζ|)^{−1/q}, ζ ∈ D, and another well-known inequality ([14, Sec. 1.4.10]) are used. Hence

J_{2} ≲ ∥C_{2/(2−s)}(∥f∥_{H_{z_{1}}^{λ}}^{p−s}(1 − |z_{2}|)^{α})(ξ_{2})∥_{L^{∞}}

≲ [ sup_{w∈D}(1 − |w|)^{β} ∫_{D} ∥f(z_{1}, z_{2})∥_{H_{z_{1}}^{λ}}^{2(p−s)/(2−s)} \frac{(1 − |z_{2}|)^{2α/(2−s)−1}}{|1 − \overline{w}z_{2}|^{β+1}} dm_{2}(z_{2})]^{\frac{2−s}{2}}

≲ ∥f∥_{H^{λ}(D^{2})}^{p−s}[ sup_{w∈D}(1 − |w|)^{β} ∫_{D} \frac{(1 − |z_{2}|)^{2α/(2−s)−(2/λ)(p−s)/(2−s)−1}}{|1 − \overline{w}z_{2}|^{β+1}} dm_{2}(z_{2})]^{\frac{2−s}{2}}

≲ ∥f∥_{H^{λ}}^{p−s}.

Thus, for any λ > (p − s)/α

L ≲ ∥f∥_{H^{λ}}^{p−s}∥D^{α/s}f∥_{H^{s}}^{s}.

This completes the proof of Theorem 1.

4. Proof of Theorems 2 and 3

We begin by proving the inclusion (2.2) for q_{1} = ∞, i.e.

∥f∥_{F_{α}^{p∞}} ≲ ∥f∥_{F_{α}^{pq}}.  (4.1)

Throughout the proof, J_{ξ} denotes the arc on T centered at ξ ∈ T

J_{ξ}(t) = {η ∈ T; |1 − ξ̄η| < t}.

On the torus T^{n} the symbol J_{ξ}(t) means J_{ξ}(t) = J_{ξ_{1}}(t_{1}) × ⋯ × J_{ξ_{n}}(t_{n}), ξ = (ξ_{1}, …, ξ_{n}) ∈ T^{n}, t = (t_{1}, …, t_{n}). Consider a version of Hardy-Littlewood maximal function on the circle:

M(ψ)(ξ) = sup_{t>0} \frac{1}{|J_{ξ}(t)|} ∫_{J_{ξ}(t)} |ψ(η)| dm(η), ξ ∈ T.

It is well known (see, e.g., [23]) that the operator M is bounded in L^{p} for p > 1.

Let f(rξ) be a function of the space F_{α}^{pq} on the bidisc. For ε, 0 < ε < min{p, q}, in view of 2-subharmonicity, we can find small numbers c, c′ ∈ (0, 1) such that

|D^{β}f(rξ)|^{ε} ≲ \frac{1}{(1 − r)^{2}} ∫_{J_{ξ}(c(1−r))} ∫_{r−c(1−r)}^{r+c′(1−r)}|D^{β}f(tη)|^{ε} dt dm_{2}(η), r ∈ I^{2}, ξ ∈ T^{2}.

A similar argument in the setting of the unit ball of C^{n} can be found in [11, p. 189].

Then an application of Hölder’s inequality with indices q/ε and q/(q − ε) leads to

(1 − r)^{ε(β−α)}|D^{β}f(rξ)|^{ε}

≲ \frac{1}{(1 − r)^{2}} ∫_{J_{ξ}(c(1−r))} ∫_{r−c(1−r)}^{r+c′(1−r)}(1 − t)^{ε(β−α)}|D^{β}f(tη)|^{ε}dt dm_{2}(η)

≲ \frac{1}{1 − r} ∫_{J_{ξ}(c(1−r))}( ∫_{r−c(1−r)}^{r+c′(1−r)}(1 − t)^{q(β−α)−1}|D^{β}f(tη)|^{q}dt)^{ε/q}dm_{2}(η).

Denoting

ψ(η_{1}, η_{2}) = (∫_{I^{2}}(1 − t)^{q(β−α)−1}|D^{β}f(tη)|^{q}dt)^{ε/q} ,

we get

(1 − r)^{p(β−α)}|D^{β}f(rξ)|^{p} ≲ [\frac{1}{1 − r} ∫_{J_{ξ}(c(1−r))} ψ(η_{1}, η_{2}) dm_{2}(η)]^{p/ε}

≲ [\frac{1}{|J_{ξ_{1}}(c(1 − r_{1}))|}  ∫_{J_{ξ_{1}} (c(1−r_{1}))}(\frac{1}{|J_{ξ_{2}}(c(1 − r_{2}))|}  ∫_{J_{ξ_{2}} (c(1−r_{2}))} ψ(η_{1}, η_{2})dm(η_{2}))dm(η_{1})]^{p/ε}.

Taking supremum over all r ∈ I^{2}, and then integrating the inequality in ξ_{1}, ξ_{2}, and twice applying the boundedness of the Hardy-Littlewood operator M in L^{p/ε}, we obtain

∫_{T^{2}} sup_{r∈I^{n}}(1 − r)^{p(β−α)}|D^{β}f(rξ)|^{p} dm_{2}(ξ) ≲ ∫_{T^{2}} ψ^{p/ε}(η_{1}, η_{2}) dm(η_{1}) dm(η_{2}) = ∥f∥_{F_{α}^{pq}}^{p}.

The inclusion (4.1) is proved. The general case 0 < q ≤ q_{1} < ∞ follows easily from (4.1). Indeed, an application of Hölder’s inequality with indices q_{1}/q and q_{1}/(q_{1}−q) gives

∥f∥_{F_{α}^{pq_{1}}}^{p}

= ∫_{T^{2}} (∫_{I^{2}} (1 − r)^{(β−α)(q_{1}−q)}(1 − r)^{(β−α)q−1}|D^{β}f(rξ)|^{q_{1}−q}|D^{β}f(rξ)|^{q}dr)^{p/q_{1}}dm_{2}(ξ)

≲ ∥f∥_{F_{α}^{pq}}^{pq/q_{1}}(∫_{T^{2}} sup_{r∈I^{2}}(1 − r)^{p(β−α)}|D^{β}f(rξ)|^{p}dm_{2}(ξ))^{(q_{1}−q)/q_{1}}.

Thus,

∥f∥_{F_{α}^{pq_{1}}} ≲ ∥f∥_{F_{α}^{pq}}^{q/q_{1}}∥f∥_{F_{α}^{p∞}}^{(q_{1}−q)/q_{1}} ≲ ∥f∥_{F_{α}^{pq}}^{q/q_{1}}∥f∥_{F_{α}^{pq}}^{(q_{1}−q)/q_{1}} = ∥f∥_{F_{α}^{pq}},

and this completes the proof of Theorem 2.

Proof of Theorem 3. Assuming that ∥f∥_{H^{p}} ≠ 0, we can apply Jensen’s inequality to the integral

\frac{1}{(2π)^{n}} ∫_{T^{n}} |f(rξ)|^{p−s}|g(rξ)|^{s}dm_{n}(ξ)

= M_{p}^{p}(f, r) [\frac{1}{M_{p}^{p}(f, r)} ∫_{T^{n}} |\frac{g(rξ)}{f(rξ)}|^{s} |f(rξ)|^{p} \frac{dm_{n}(ξ)}{(2π)^{n}}]^{\frac{p}{s} \frac{s}{p}}

≤ M_{p}^{p}(f, r) [\frac{1}{M_{p}^{p}(f, r)} ∫_{T^{n}} |\frac{g(rξ)}{f(rξ)}|^{p} |f(rξ)|^{p} \frac{dm_{n}(ξ)}{(2π)^{n}}]^{s/p}

= M_{p}^{p−s}(f, r) [∫_{T^{n}} |g(rξ)|^{p} \frac{dm_{n}(ξ)}{(2π)^{n}}]^{s/p} = M_{p}^{p−s}(f, r)M_{p}^{s}(g, r).

A similar method is applied in the proof of Theorem 4 of [19]. Further, a weighted integration leads now to

\frac{1}{(2π)^{n}} ∫_{D^{n}} |f(z)|^{p−s}|g(z)|^{s}(1 − |z|)^{αs−1}dm_{2n}(z)
≤ ∫_{I^{n}} M_{p}^{p−s}(f, r)M_{p}^{s}(g, r)(1 − r)^{αs−1}dr
≤ ∥f∥_{H^{p}}^{p−s} ∫_{I^{n}} M_{p}^{s}(g, r)(1 − r)^{αs−1}dr,

and the proof is complete.

Some generalizations of Littlewood-Paley inequality in the polydisc 107

5. Proof of Theorems 4 and 5 We need the next lemma which extends the well-known Hardy-Stein identity

(see, e.g., [7]) to the polydisc.

Lemma 1. Suppose that f(z) ∈ H(D^{n}), 0 < p < ∞. Then for any r = (r_{1}, …, r_{n}) ∈ I^{n}

∏_{j=1}^{n} r_{j} · \frac{∂^{n}}{∂r_{1} … ∂r_{n}} M_{p}^{p}(f, r) = \frac{1}{(2π)^{n}} ∫_{|z_{1}|<r_{1}} ⋯ ∫_{|z_{n}|<r_{n}} f^{#}(z)dm_{2n}(z),  (5.1)

where f^{#}(z) = Δ_{z_{1}}Δ_{z_{2}} … Δ_{z_{n}}|f(z)|^{p}, and Δ_{z_{j}} is the usual Laplacian in the variable z_{j}.

Proof. Fix z_{2} for a moment and apply Green’s formula (see, e.g., [4], [23]) to the function |f(z_{1}, z_{2})|^{p} in |z_{1}| < r_{1}:

∫_{|z_{1}|=r_{1}} \frac{∂}{∂r_{1}} |f(z_{1}, z_{2})|^{p}dℓ = ∫_{|z_{1}|<r_{1}} Δ_{z_{1}}|f(z_{1}, z_{2})|^{p}dm_{2}(z_{1}),

where dℓ means arc length integration. With respect to the function

ψ(z_{2}) = r_{1} \frac{∂}{∂r_{1}} ∫_{T} |f(r_{1}ξ_{1}, z_{2})|^{p}dm(ξ_{1}) = ∫_{|z_{1}|<r_{1}} Δ_{z_{1}}|f(z_{1}, z_{2})|^{p}dm_{2}(z_{1}),

we can again apply Green’s formula in |z_{2}| < r_{2}:

∫_{|z_{2}|=r_{2}} \frac{∂}{∂r_{2}} ψ(z_{2})dℓ = ∫_{|z_{2}|<r_{2}} Δ_{z_{2}}ψ(z_{2}) dm_{2}(z_{2}).

Combining these equalities, we obtain

r_{1}r_{2} \frac{∂^{2}}{∂r_{1}∂r_{2}} ∫_{T} ∫_{T} |f(r_{1}ξ_{1}, r_{2}ξ_{2})|^{p} dm(ξ_{1}) dm(ξ_{2})

= ∫_{|z_{1}|<r_{1}} ∫_{|z_{2}|<r_{2}} Δ_{z_{1}}Δ_{z_{2}}|f(z_{1}, z_{2})|^{p} dm_{4}(z),

which finishes the proof.

Remark 5. For n = 1 (5.1) coincides with the well-known Hardy-Stein identity [7] in view of formula (2.7).

Proof of Theorem 4. Lemma 1 enables us to establish another identity r_{1}r_{2} ∫_{T^{2}} Δ_{z_{1}}Δ_{z_{2}}|f(z_{1}, z_{2})|^{p}dm(ξ_{1}) dm(ξ_{2})

= \frac{∂^{2}}{∂r_{1}∂r_{2}} ∫_{0}^{r_{1}} ∫_{0}^{r_{2}} ∫_{T^{2}} Δ_{z_{1}}Δ_{z_{2}}|f(z_{1}, z_{2})|^{p}ρ_{1}ρ_{2}dm(ξ_{1}) dm(ξ_{2}) dρ_{1} dρ_{2}

= (2π)^{2} \frac{∂^{2}}{∂r_{1}∂r_{2}} [r_{1}r_{2} \frac{∂^{2}}{∂r_{1}∂r_{2}} M_{p}^{p}(f, r_{1}, r_{2})] .  (5.2)

First we prove the identity (2.6), i.e. the one variable version. We transform the left integral of (2.6), integrating by parts and using the identity (5.2):

\frac{1}{2π} ∫_{D}(1 − |z|)^{α}f^{#}(z)dm_{2}(z)

= \frac{1}{2π} ∫_{0}^{1}(1 − r)^{α} [∫_{−π}^{π} Δ|f(re^{iθ})|^{p}dθ] r dr

= ∫_{0}^{1}(1 − r)^{α} [\frac{∂}{∂r} (r \frac{∂}{∂r} M_{p}^{p}(f, r))] dr

= lim_{r→1−}(1 − r)^{α} r \frac{∂}{∂r} M_{p}^{p}(f, r) + α ∫_{0}^{1}(1 − r)^{α−1} r \frac{∂}{∂r} M_{p}^{p}(f, r) dr. (5.3)

If the right-hand side integral in (2.6) or (5.3) exists, then the limit in (5.3) vanishes. Indeed, by the Hardy-Stein identity, the function r \frac{∂}{∂r} M_{p}^{p}(f, r) is increasing in r ∈ (0, 1). Hence for any ρ ∈ (0, 1)

∫_{ρ}^{(1+ρ)/2}(1 − r)^{α−1} r \frac{∂}{∂r} M_{p}^{p}(f, r)dr ≥ C_{α} ρ (1 − ρ)^{α} \frac{∂}{∂ρ}M_{p}^{p}(f, ρ).

By the Cauchy criterion for convergence

lim_{ρ→1−}(1 − ρ)^{α} \frac{∂}{∂ρ}M_{p}^{p}(f, ρ) = 0.

It follows from (5.3) that

\frac{1}{2π} ∫_{D}(1 − |z|)^{α}f^{#}(z)dm_{2}(z) = α ∫_{0}^{1}(1 − r)^{α−1} r \frac{∂}{∂r} M_{p}^{p}(f, r) dr.

Part (ii) of the theorem is proved.

Proceeding to the proof of the inequality (2.8), note that the example f(z) = z shows the sharpness of the constant α/p. Then the identity (2.6) can be written as follows

A(f; p, α) = \frac{α}{p} ∫_{0}^{1} ∫_{−π}^{π} |f(re^{iθ})|^{p−1} (\frac{∂}{∂r} |f(re^{iθ})|) (1 − r)^{α−1}r dr dθ.  (5.4)

Since ||f(re^{iθ})|−|f(ρe^{iθ})|| ≤ |f(re^{iθ})−f(ρe^{iθ})|, we have |\frac{∂}{∂r} |f(re^{iθ})|| ≤ |f′(re^{iθ})|. Hence, (2.8) follows.

We now turn to the proof of the inequality (2.9). By the Cauchy-Schwarz inequality,

B(f; p, α) ≤ \sqrt{A(f; p, α)} (∫_{D} |f(z)|^{p}(1 − |z|)^{α−2}dm_{2}(z))^{1/2} .

Therefore, it only remains to verify the inequality

∫_{D} |f(z)|^{p}(1 − |z|)^{α−2}dm_{2}(z) ≲ A(f; p, α),  p > 0, α > 1.  (5.5)

To this end, we integrate by parts to get

\frac{p^{2}}{2πα}A(f; p, α) = ∫_{0}^{1}(1 − r)^{α−1} r \frac{∂}{∂r} M_{p}^{p}(f, r) dr

= lim_{r→1−}(1 − r)^{α−1} r M_{p}^{p}(f, r) − ∫_{0}^{1} M_{p}^{p}(f, r) d(r(1 − r)^{α−1})

= lim_{r→1−}(1 − r)^{α−1} r M_{p}^{p}(f, r) + \frac{1}{2π} ∫_{0}^{1} ∫_{−π}^{π} |f(re^{iθ})|^{p}(1 − r)^{α−2}(αr − 1) dr dθ.

This equality shows that if A(f; p, α) exists, then the function f(z) is in the Bergman space A_{α−2}^{p}(D). Consequently lim_{r→1−}(1 − r)^{α−1} M_{p}^{p}(f, r) = 0. So, we get

A(f; p, α) = \frac{α}{p^{2}} ∫_{0}^{1} ∫_{−π}^{π} |f(re^{iθ})|^{p}(1 − r)^{α−2}(αr − 1) dr dθ

≥ \frac{α(α − 1)}{2p^{2}} ∫_{(α+1)/(2α)}^{1} ∫_{−π}^{π} |f(re^{iθ})|^{p}(1 − r)^{α−2}dr dθ

≥ C(α, p) ∫_{D} |f(z)|^{p}(1 − |z|)^{α−2}dm_{2}(z),

and this gives the desired result. Part (iii) of the theorem is proved. Part (i) of the theorem can be proved from (5.2) similarly, so we omit the details.

Proof of Theorem 5. The integrated Hardy-Stein identity (see Lemma 1)

M_{p}^{p}(f, r_{1}, r_{2}) + |f(0, 0)|^{p} = M_{p}^{p}(f, 0, r_{2}) + M_{p}^{p}(f, r_{1}, 0)+

+ \frac{1}{(2π)^{2}} ∫_{0}^{r_{1}} ∫_{0}^{r_{2}}(∫_{|z_{1}|<ρ_{1}} ∫_{|z_{2}|<ρ_{2}} f^{#}(z_{1}, z_{2})dm_{4}(z))\frac{dρ_{1}dρ_{2}}{ρ_{1}ρ_{2}}

can be integrated again with respect to the measure (2π)^{2}C_{ω_{1}}C_{ω_{2}}ω_{1}(r_{1})ω_{2}(r_{2}) × r_{1}r_{2} dr_{1} dr_{2}. We thus have

∥f∥_{A_{ω}^{p}}^{p} + |f(0, 0)|^{p} = J_{1} + J_{2} + J_{3},

where

J_{1} = ∥f(z_{1}, 0)∥_{A_{ω_{1}}^{p} (D)}^{p} = |f(0, 0)|^{p} + 2πC_{ω_{1}} ∫_{0}^{1}M_{1}(Δ_{z_{1}}|f(z_{1}, 0)|^{p}, r_{1})h_{ω_{1}}(r_{1})r_{1}dr_{1},
J_{2} = ∥f(0, z_{2})∥_{A_{ω_{2}}^{p} (D)}^{p} = |f(0, 0)|^{p} + 2πC_{ω_{2}} ∫_{0}^{1}M_{1}(Δ_{z_{2}}|f(0, z_{2})|^{p}, r_{2})h_{ω_{2}}(r_{2})r_{2}dr_{2}.

Besides, a further application of Fubini’s theorem shows that

J_{3} = C_{ω_{1}}C_{ω_{2}} ∫_{I^{2}} [∫_{0}^{r_{1}} ∫_{0}^{r_{2}}( ∫_{|z_{1}|<ρ_{1}} ∫_{|z_{2}|<ρ_{2}} f^{#}(z_{1}, z_{2})dm_{4}(z))\frac{dρ_{1}dρ_{2}}{ρ_{1}ρ_{2}} ]ω(r)rdr

= (2π)^{2}C_{ω_{1}}C_{ω_{2}} ∫_{0}^{1} ∫_{0}^{1} M_{1}(f^{#}(z_{1}, z_{2}), r_{1}, r_{2})h_{ω_{1}}(r_{1})h_{ω_{2}}(r_{2})r_{1}r_{2}dr_{1}dr_{2}.

This completes the proof of Theorem 5.

Acknowledgment. The authors thank the referees for their useful suggestions and comments.

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(received 14.01.2005, in revised form 20.02.2006)

K. L. Avetisyan, Yerevan State University, Yerevan, Armenia, E-mail: avetkaren@ysu.am R. F. Shamoyan, Bryansk Pedagogical University, Bryansk, Russia, E-mail: rshamoyan@mail.ru