PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 64 (78), 1998, 36–52

LITTLEWOOD-PALEY TYPE INEQUALITIES FOR M-HARMONIC FUNCTIONS

Miroljub Jevtić and Miroslav Pavlović

Communicated by Stevan Pilipović

Abstract. We prove several Littlewood-Paley type inequalities for Mharmonic and analytic functions on the unit ball B of C^{n}. Further, we give some characterizations of M-harmonic and analytic Hardy spaces on B.

1. Introduction

Let B denote the open unit ball in C^{n}, n > 1, with boundary S. We denote by ν the normalized Lebesgue measure on B and by σ the rotation invariant probability measure on S.

The main purpose of this paper is to prove the following theorem.

Theorem 1.1. If f is a function in L^{p}(S), 1 < p < ∞, and u is a function on B defined via the invariant Poisson integral of f, then

(1.1) (∫_{B} | ˜∇u(z)|^{p}(1 − |z|^{2})^{n}dτ(z) + |u(0)|^{p}) ⪰ ∫_{S} |f(ξ)|^{p}dσ(ξ), for 1 < p ≤ 2,

and

(1.2) (∫_{B} | ˜∇u(z)|^{p}(1 − |z|^{2})^{n}dτ(z) + |u(0)|^{p}) ⪯ ∫_{S} |f(ξ)|^{p}dσ(ξ), for 2 ≤ p < ∞,

where ˜∇ and τ denote the invariant gradient and invariant measure on B.

We also show that | ˜∇u(z)| in (1.1) and (1.2) may be replaced by (1 − |z|^{2}) ×|∇u(z)|, where ∇ denotes the real gradient of u and by (1−|z|^{2})(|Ru(z)|+|R̄u(z)|), where, as usual, R = ∑_{j=1}^{n} z_{j} \frac{∂}{∂z_{j}} is the radial derivative.

AMS Subject Classification (1991): Primary 32A35

Theorem 1.2. If f ∈ L^{p}(S), 1 < p < ∞, and u is the invariant Poisson integral of f, then for 1 < p ≤ 2 we have

(∫_{B} |∇u(z)|^{p}(1 − |z|^{2})^{p−1}dν(z) + |u(0)|^{p}) ⪰ ∫_{S} |f(ξ)|^{p}dσ(ξ) (1.3)

(∫_{B}(|Ru(z)| + |R̄u(z)|)^{p}(1 − |z|^{2})^{p−1}dν(z) + |u(0)|^{p}) ⪰ ∫_{S}|f(ξ)|^{p}dσ(ξ), (1.4)

and for 2 ≤ p < ∞ we have

(∫_{B}(|Ru(z)| + |R̄u(z)|)^{p}(1 − |z|^{2})^{p−1}dν(z) + |u(0)|^{p}) ⪯ ∫_{S} |f(ξ)|^{p}dσ(ξ), (1.5)

(∫_{B} |∇u(z)|^{p}(1 − |z|^{2})^{p−1}dν(z) + |u(0)|^{p}) ⪯ ∫_{S} |f(ξ)|^{p}dσ(ξ). (1.6)

For n = 1 (1.3) and (1.6) are well known inequalities of Littlewood and Paley [11]. Various generalizations of their result are referred to as a Littlawood-Paley type inequalities.

The method of proof of Theorem 1.1 we will present is based on local estimates for M-harmonic functions (which will be defined in Section 2) and the following theorems that allow us to express the L^{p} norm of f in terms of some area integrals, and which are of interest on their own right.

Theorem 1.3. Let u be M-harmonic function on B. If 1 < p < ∞, then

(1.7) \frac{d}{dr} ∫_{S} |u(rξ)|^{p}dσ(ξ) = \frac{c_{p}r^{1−2n}(1 − r^{2})^{n−1}}{2n} ∫_{rB} |u(z)|^{p−2}| ˜∇u(z)|^{2}dτ(z),

where c_{p} = p(p − 1).

For 1 < p < ∞, let H^{p} = H^{p}(B) denote the set of M-harmonic functions u on B, u ∈ M, for which |u|^{p} has an M-harmonic majorant on B. It is well known that u ∈ H^{p}(B) if and only if ∥u∥_{H^{p}} = sup_{0<r<1} M_{p}(r, u) < ∞, where, as usual, M_{p}^{p}(r, u) = ∫_{S} |u(rξ)|^{p}dσ(ξ).

Theorem 1.4. A function u M-harmonic on B belongs to H^{p}, 1 < p < ∞, if and only if

∫_{B} |u(z)|^{p−2}| ˜∇u(z)|^{2}(1 − |z|^{2})^{−1}dν(z) < ∞

Furthermore, if u ∈ H^{p}, 1 < p < ∞, then

(1.8) ∥u∥_{H^{p}}^{p} = ∫_{B} |u(z)|^{p}dν(z) + \frac{p(p − 1)}{4n^{2}} ∫_{B} |u(z)|^{p−2}| ˜∇u(z)|^{2}(1 − |z|^{2})^{−1}dν(z),

Let

(1.9)  G(ρ, r) = \frac{1}{2n} ∫_{ρ}^{r} t^{1−2n}(1 − t^{2})^{n−1}dt, 0 ≤ ρ ≤ r ≤ 1,

As a corollary of Theorem 1.3 we have another characterization of the Hardy space H^{p}.

Theorem 1.5. Let 1 < p < ∞. A function u ∈ M belongs to H^{p} if and only if

∫_{B} |u(z)|^{p−2}| ˜∇u(z)|^{2}G(|z|, 1)dτ(z) < ∞.

Moreover, if u ∈ H^{p}, 1 < p < ∞, then

(1.10)  ∥u∥_{H^{p}}^{p} = |u(0)|^{p} + p(p − 1) ∫_{B} |u(z)|^{p−2}| ˜∇u(z)|^{2}G(|z|, 1)dτ(z).

The method of proof of Theorems concerning M-harmonic functions we will present can also be applied to Hardy spaces H^{p} of holomorphic functions. Recall that a holomorphic function f on B, f ∈ H(B), belongs to the Hardy space H^{p}, 0 < p < ∞, if and only if ∥f∥_{H^{p}} = sup_{0<r<1} M_{p}(r, f) < ∞. An analogue of the identity (1.7) of the Hardy-Stein-Spencer type for analytic functions is as follows.

Theorem 1.6. Let f ∈ H(B). If 0 < p < ∞, then

(1.11) \frac{d}{dr} ∫_{S} |f(rξ)|^{p}dσ(ξ) = \frac{p^{2}}{4} \frac{r^{1−2n}(1 − r^{2})^{n−1}}{2n} ∫_{rB} |f(z)|^{p−2}| ˜∇f(z)|^{2}dτ(z),

An application of the identity (1.11) gives the following characterization of the Hardy spaces H^{p}.

Theorem 1.7. A function f holomorphic on B belongs to H^{p}, 0 < p < ∞, if and only if

∫_{B} |f(z)|^{p−2}| ˜∇f(z)|^{2}(1 − |z|^{2})^{−1}dν(z) < ∞.

Furthermore, if f ∈ H^{p}, 0 < p < ∞, then

(1.12) ∥f∥_{H^{p}}^{p} = ∫_{B} |f(z)|^{p}dν(z) + \frac{p^{2}}{8n} ∫_{B} |f(z)|^{p−2}| ˜∇f(z)|^{2}(1 − |z|^{2})^{−1}dν(z).

Theorem 1.8. Let 0 < p < ∞. A function f ∈ H(B) belongs to H^{p} if and only if

∫_{B} |f(z)|^{p−2}| ˜∇f(z)|^{2}G(|z|, 1) dτ(z) < ∞

Furthermore, if f ∈ H^{p}, 0 < p < ∞, then

(1.13)  ∥f∥_{H^{p}}^{p} = |f(0)|^{p} + \frac{p^{2}}{4} ∫_{B} |f(z)|^{p−2}| ˜∇f(z)|^{2}G(|z|, 1) dτ(z).

The characterizations of Hardy spaces H^{p}, 1 < p < ∞, and H^{p}, 0 < p < ∞, given in Theorems 1.4 and 1.7 are known (see [14]). The proofs we will presente are based on the new identities (1.8) and (1.12) and they are simpler than the proofs given in [14].

For another proof of (1.13) see [4]. See also [14].

The following identity due to Beatrous and Burbea [2] was first proved by Hardy, Stein and Spenser (see [6, p. 42]) for n = 1.

Theorem 1.9, Let f ∈ H(B), 0 < p < ∞ and 0 < r < 1. Then

(1.14)  r \frac{d}{dr} M_{p}^{p}(r, f) = \frac{p^{2}}{2n} ∫_{rB} |z|^{−2n}|f(z)|^{p−2}|Rf(z)|^{2}dν(z).

An application of the identity (1.14) gives the criteria for f holomorphic in B to belong to the Hardy space H^{p}.

Theorem 1.10. [2] Let 0 < p < ∞ and f ∈ H(B). Then the following statements are equivalent:

(i) f ∈ H^{p},

(ii) ∫_{B} |f(z)|^{p−2}|Rf(z)|^{2}(1 − |z|^{2}) dν(z) < ∞.

Since | ˜∇f(z)| ≥ (1 − |z|)^{2}|∇f(z)| ≥ (1 − |z|^{2})|Rf(z)|, (see Section 2) the following theorem is an immediate consequence of Theorems 1.7 and 1.10.

Theorem 1.11. A function f holomorphic on B belongs to H^{p}, 0 < p < ∞, if and only if

∫_{B} |∇f(z)|^{2}|f(z)|^{p−2}(1 − |z|^{2})dν(z) < ∞.

Remark 1. It is authors belief that the results of Theorems 1.10 and 1.11 should hold for the Hardy spaces H^{p}, 1 < p < ∞ of M-harmonic functions

As a final result we have the inequalities of Littlewood-Paley type for analytic functions.

Theorem 1.12. Let f ∈ H(B). If 0 < p ≤ 2, then

∥f∥_{p}^{p} ⪯ (∫_{B} | ˜∇f(z)|^{p}(1 − |z|^{2})^{−1}dν(z) + |f(0)|^{p}), (1.15)

∥f∥_{p}^{p} ⪯ (∫_{B} |∇f(z)|^{p}(1 − |z|^{2})^{p−1}dν(z) + |f(0)|^{p}), (1.16)

∥f∥_{p}^{p} ⪯ (∫_{B} |Rf(z)|^{p}(1 − |z|^{2})^{p−1}dν(z) + |f(0)|^{p}), (1.17) and if 2 ≤ p < ∞, then

(|f(0)|^{p} + ∫_{B} | ˜∇f(z)|^{p}(1 − |z|^{2})^{−1}dν(z)) ⪯ ∥f∥_{p}^{p}, (1.18)

(|f(0)|^{p} + ∫_{B} |∇f(z)|^{p}(1 − |z|^{2})^{p−1}dν(z)) ⪯ ∥f∥_{p}^{p}, (1.19)

(|f(0)|^{p} + ∫_{B} |Rf(z)|^{p}(1 − |z|^{2})^{p−1}dν(z)) ⪯ ∥f∥_{p}^{p}. (1.20)

The usual method of proof of the inequality (1.20) ((1.19), resp. ) is to apply the Riesz Convexity Theorem to the operator f → |Rf(z)|(1 − |z|^{2}) (f → |∇f(z)|(1 − |z|^{2})) acting on functions f on the measure space (S, σ) and taking them to functions on (B, (1 − |z|^{2})^{−1}dν(z)). It is relatively easy to show that this operator is of type (2.2) as well as (∞, ∞) and the Riesz theorem produces (1.20) ((1.19), resp.). By duality we have (1.17) and (1.16) for 1 < p ≤ 2. Additional considerations show that (1.17) holds also for 0 < p ≤ 1. See [3]. Obviously (1.15) and (1.16) are consequences of (1.17). For the inequality (1.18) see [5].

Our argument is different. We show that

∫_{B} | ˜∇f(z)|^{p}(1 − |z|^{2})^{−1}dν(z) ⪯ ∫_{B} |f(z)|^{p−2}| ˜∇f(z)|^{2}(1 − |z|^{2})^{−1}dν(z),

for 2 ≤ p < ∞, and that the reverse inequality holds for 0 < p ≤ 2, and then we apply the identity (1.12) established in Theorem 1.7.

2. Notations and preliminary results

Let Δ̃ be the invariant Laplacian on B. That is, Δ̃u(z) = Δ(u∘φ_{z})(0), where Δ is the ordinary Laplacian and φ_{z} the standard authomorphism of B taking 0 to z [13]. (It can be shown that Δ̃ is equal (n + 1) times the Laplacian with respect to the Bergman metric). As in [13],

Δ̃ = 4(1 − |z|^{2}) ∑_{i,j}(δ_{i,j} − z_{i}z̄_{j})\frac{∂^{2}}{∂z_{i}∂z̄_{j}} .

The real valued functions annihilated by Δ̃ are called invariantly harmonic or M-harmonic.

The operator Δ̃ is M-invariant in the sense that Δ̃(f ∘ φ) = (Δ̃f) ∘ φ, for φ ∈ Aut(B) [13, Theorem 4.1.2]. This implies that the class M of M- harmonic functions on B is M-invariant.

For a function u ∈ C^{1}(B) let D̃u(a) = D(u ∘ φ_{a})(0) and ˜∇u(a) = ∇(u ∘ φ_{a})(0) be the invariant complex and invariant real gradient respectively, where Du = (\frac{∂u}{∂z_{1}} ,…, \frac{∂u}{∂z_{n}} ) denotes the complex gradiant and ∇u = (\frac{∂u}{∂x_{1}} ,…, \frac{∂u}{∂x_{2n}} ), z_{k} = x_{2k−1} + ix_{2k}, k = 1,…, n, is the real gradient of u.

For u ∈ C^{1}(B) and ϕ ∈ Aut(B) we have |D̃(u∘ϕ)| = |(D̃u)∘ϕ| and | ˜∇(u∘ϕ)| = |( ˜∇u) ∘ ϕ|. In other words, |D̃| and | ˜∇| are M-invariant. See [12].

The length of the invariant gradient of a function u ∈ C^{1}(B) is given in coordinates by

(2.1)  | ˜∇u(z)|^{2} = 2(1 − |z|^{2})(|Du(z)|^{2} − |Ru(z)|^{2} + |Dū(z)|^{2} − |Rū(z)|^{2}).

It is easy to check that

|z|^{2}|Du(z)|^{2} = |Ru(z)|^{2} + ∑_{i<j} |T_{i,j}u(z)|^{2}, u ∈ C^{2}(B),

where T_{i,j}u = z̄_{i} \frac{∂u}{∂z_{j}} − z̄_{j} \frac{∂u}{∂z_{i}} are tangential derivatives of u. Using this and (2.1) we find that

(2.2)  \begin{aligned}|z|^{2}| ˜∇u(z)|^{2} & = 2(1 − |z|)^{2}[(1 − |z|^{2})(|Ru(z)|^{2} + |Rū(z)|^{2}) \\ & + ∑_{i<j} |T_{i,j}u(z)|^{2} + ∑_{i<j} |T_{i,j}ū(z)|^{2}], \end{aligned}

For z ∈ B and r between 0 and 1, E_{r}(z) = {w ∈ B : |φ_{z}(w)| < r}.

In this note we follow the custom of using the letter C to stand for a positive constant which changes its value from one appearence to another while remaining independent of the important variables.

We write A ⪯ B, or equivalently B ⪰ A, when there is a constant C such that A ≤ CB, and A ≅ B when A ⪯ B and B ⪯ A.

Lemma 2.1. [12], [1] Let 0 < p < ∞ and 0 < r < 1. There exist constants C_{1} > 0 and C_{2} > 0 so that if u ∈ M, then

(a) | ˜∇u(z)|^{p} ≤ \frac{C_{1}}{(1−|z|^{2})^{n+1}} ∫_{E_{r}(z)} | ˜∇u(w)|^{p}dν(w), z ∈ B, and

(b) |∇u(z)|^{p} ≤ \frac{C_{2}}{(1−|z|^{2})^{n+1}} ∫_{E_{r}(z)} |∇u(w)|^{p}dν(w), z ∈ B.

Lemma 2.2. [9], [8] Let 0 < r < 1. There is a constant C sach that if u ∈ M then

(a) |T_{i,j}Ru(z)| ≤ C(1 − |z|^{2})^{−1/2} ∫_{E_{r}(z)} |Ru(w)|dτ(w), z ∈ B,

(b) |T_{i,j}R̄u(z)| ≤ C(1 − |z|^{2})^{−1/2} ∫_{E_{r}(z)} |R̄u(w)|dτ(w), z ∈ B,

(c) |T_{i,j}u(z)| ≤ C(1 − |z|^{2})^{−1/2} ∫_{E_{r}(z)} |u(w)|dτ(w), z ∈ B.

Here, dτ(z) = (1 − |z|^{2})^{−n−1}dν(z) is the Mobius invariant measure on B. The following lemma gives some basic properties of G which will be needed

later.

Lemma 2.3. Let 0 < δ < \frac{1}{2} be fixed. Then G(|z|, 1) satisfies the following:

(a) G(|z|, 1) ≥ c_{n}(1 − |z|^{2})^{n}, for all z ∈ B.

(b) G(|z|, 1) ≤ c_{δ}(1 − |z|^{2})^{n} for all z ∈ B, |z| ≥ δ, where c_{δ} is a positive constant depending only on δ.

Furthermore, for all z, |z| ≤ δ, G(|z|, 1) ≅ |z|^{2−2n}, n > 1.

The proof is a routine estimation of the integral in (1.9), and thus is omitted.

Lemma 2.4. Let 0 < δ < 1. There exists a constant K = K(δ) > 0 such that |φ_{a}(z)| ≥ K|z − a|, for all a, |a| ≤ δ, and z ∈ B.

Proof. By [13, p. 26]

|φ_{a}(z)|^{2} = 1 − \frac{(1 − |a|^{2})(1 − |z|^{2})}{|1 − zā|^{2}} = \frac{|z − a|^{2} + |zā|^{2} − |z|^{2}|a|^{2}}{|1 − zā|^{2}}

Thus

|φ_{a}(z)|^{2} ≥ \frac{|z − a|^{2} + |zā|^{2} − |z|^{2}|a|^{2}}{(1 + δ)^{2}}

= \frac{|z − a|^{2} + |(z − a)ā|^{2} − |a|^{2}|z − a|^{2}}{(1 + δ)^{2}}

≥ \frac{(1 − δ^{2})|z − a|^{2}}{(1 + δ)^{2}}

3. Characterizations of the Hardy space H^{p}

Proof of Theorem 1.3. Let u_{ϵ} = (u^{2} + ϵ^{2})^{p/2}, ϵ > 0. Then u_{ϵ} ∈ C^{2}(B) and

Δ̃u_{ϵ} = p(p − 2)(u^{2} + ϵ^{2})^{p/2−2}u^{2}| ˜∇u|^{2} + p(u^{2} + ϵ^{2})^{p/2−1}(| ˜∇u|^{2} + uΔ̃u).

By the Green formula we have

(3.1) \frac{d}{dr} ∫_{S} u_{ϵ}(rξ) dσ(ξ) = \frac{r^{1−2n}(1 − r^{2})^{n−1}}{2n} ∫_{rB} Δ̃u_{ϵ}(z) dτ(z), 0 < r < 1,

(see [12])

Since u is M-harmonic, we have

(3.2)  Δ̃u_{ϵ} = p(p − 2)(u^{2} + ϵ^{2})^{p/2−2}u^{2}| ˜∇u|^{2} + p(u^{2} + ϵ^{2})^{p/2−1}| ˜∇u|^{2},

and

(3.3)  lim_{ϵ→0} \frac{d}{dr} ∫_{S} u_{ϵ}(rξ) dσ(ξ) = \frac{d}{dr} ∫_{S} |u(rξ)|^{p}dσ(ξ),

Littlewood-Paley type inequalities... 43

The latter holds because the function

(ϵ, z) → (u^{2}(z) + ϵ^{2})^{p/2} = |u(z) + iϵ|^{p}

is of class C^{1}, (ϵ, z) ∈ (−∞, ∞) × B.

If 1 < p ≤ 2, then it follows from (3.2) that

Δ̃u_{ϵ} ≥ p(p − 2)(u^{2} + ϵ^{2})^{p/2−2}(u^{2} + ϵ^{2})| ˜∇u|^{2}
+ p(u^{2} + ϵ^{2})^{p/2−1}| ˜∇u|^{2}

= p(p − 1)(u^{2} + ϵ^{2})^{p/2−1}| ˜∇u|^{2} (3.4)

Hence, by (3.1),

c_{p} \frac{r^{1−2n}(1 − r^{2})^{n−1}}{2n} ∫_{rB}(u(z)^{2} + ϵ^{2})^{p/2−1}| ˜∇u(z)|^{2}dτ(z) ≤ \frac{d}{dr} ∫_{S} u_{ϵ}(rξ) dσ(ξ)

where c_{p} = p(p − 1).

Using Fatou’s lemma we find that

(3.5) \frac{p(p − 1)}{2n}r^{1−2n}(1−r^{2})^{n−1} ∫_{rB} |u(z)|^{p−2}| ˜∇u(z)|^{2}dτ(z) ≤ \frac{d}{dr} ∫_{S} |u(rξ)|^{p}dσ(ξ).

If 2 ≤ p < ∞, we find that

(3.6) \frac{p}{2n}(1 − r^{2})^{n−1}r^{1−2n} ∫_{rB} |u(z)|^{p−2}| ˜∇u(z)|^{2}dτ(z) ≤ \frac{d}{dr} ∫_{S} |u(rξ)|^{p}dσ(ξ).

Hence by (3.5) and (3.6) we conclude that the function |u(z)|^{p−2}| ˜∇u(z)|^{2} is locally integrable.

By (3.4) and (3.2) Δ̃u_{ϵ} ≥ 0, for every z ∈ B. (Note that p > 1). It is easy to see that

Δ̃u_{ϵ}(z) ≤ c_{p}p(u^{2}(z) + ϵ^{2})^{p/2−1}| ˜∇u(z)|^{2}, z ∈ B,

where c_{p} = 1 if 1 < p ≤ 2 and c_{p} = p − 1 if p ≥ 2.

Hence by (3.1), (3.2), (3.3) and the Lebesgue theorem, we obtain

\frac{d}{dr} ∫_{S} |u(rξ)|^{p}dσ(ξ) = lim_{ϵ→0} ∫_{rB}(Δ̃u_{ϵ}(z)) dτ(z)

= \frac{p(p − 1)}{2n}(1 − r^{2})^{n−1}r^{1−2n} ∫_{rB} |u(z)|^{p−2}| ˜∇u(z)|^{2}dτ(z)

To prove Theorem 1.4 we need the following proposition

44 Jevtić and Pavlović

Proposition 3.1. Let u be M-harmonic and p > 1. Then u ∈ H^{p} if and only if

\overline{lim_{r→1}}[M_{p}^{p}(r, u) − r^{−2n} ∫_{rB} |u(z)|^{p}dν(z)]< ∞

Proof. “Only if” is trivial. To prove the “if” let

φ(r) = ∫_{rB} |u(z)|^{p}dν(z) = 2n ∫_{0}^{r} ρ^{2n−1}M_{p}^{p}(ρ, u) dρ.

Hence,

M_{p}^{p}(r, u) − r^{−2n} ∫_{rB} |u(z)|dν(z) = \frac{r^{1−2n}}{2n} φ′(r) − r^{−2n}φ(r) = \frac{r}{2n}(φ(r)r^{−2n})′.

From this the result follows easily.

Proof of Theorem 1.4. By (1.7) we have

r^{2n} \frac{d}{dr} M_{p}^{p}(r, u) = \frac{p(p − 1)}{2n}(1 − r^{2})^{n−1}r ∫_{rB} |u(z)|^{p−2}| ˜∇u(z)|^{2}dτ(z)

Integrating this from 0 to ϵ we get, by integration by parts and Fubini’s theorem,

ϵ^{2n}M_{p}^{p}(ϵ, u) − ∫_{ϵB} |u(z)|^{p}dν(z)

= \frac{c_{p}}{2n} ∫_{0}^{ϵ}[r(1 − r^{2})^{n−1} ∫_{rB} |u(z)|^{p−2}| ˜∇u(z)|^{2}dτ(z)]dr

= \frac{c_{p}}{2n} ∫_{ϵB} |u(z)|^{p−2}| ˜∇u(z)|^{2}dτ(z) ∫_{|z|}^{ϵ}(1 − r^{2})^{n−1}rdr

= \frac{c_{p}}{2n} ∫_{ϵB} |u(z)|^{p−2}| ˜∇u(z)|^{2} \frac{(1 − |z|^{2})^{n} − (1 − ϵ^{2})^{n}}{2n} dτ(z), (3.7)

where c_{p} = p(p − 1).

If u ∈ H^{p}, by taking limit in (3.7), ϵ → 1, we obtain (1.8).

Conversely, if

∫_{B} |u(z)|^{p−2}| ˜∇u(z)|^{2}(1 − |z|)^{−1}dν(z) < ∞

then

\overline{lim_{ϵ→1}}[M_{p}^{p}(ϵ, u) − ϵ^{−2n} ∫_{ϵB} |u(z)|^{p}dν(z)]< ∞

and u ∈ H^{p}, by Proposition 3.1.

Proof of Theorem 1.5. We have, by (1.7) and Fubini’s theorem,

(3.8)  M_{p}^{p}(r, u) = |u(0)|^{p} + p(p − 1) ∫_{rB} |u(z)|^{p−2}| ˜∇u(z)|^{2}G(|z|, r) dτ(z).

If M_{p}^{p}(r, u) ≤ C < ∞, then we apply Fatou‘s lemma to get that

|u(0)|^{p} + p(p − 1) ∫_{B} |u(z)|^{p−2}| ˜∇u(z)|^{2}G(|z|, 1)dτ(z) ≤ C

In the opposite direction we use the Lebesgue theorem and the inequality

G(|z|, r) ≤ G(|z|, 1).

Remark 2. An upper-semicontinuous function u : B → [−∞, ∞) is M-sub-harmonic or invariant subharmonic if for each a ∈ B

u(a) ≤ ∫_{S} u(φ_{a}(rξ)) dσ(ξ), 0 < r < 1,

provided that none of the integrals is −∞.

Recall that the Riesz measure of M-subharmonic function u on B is the non-negative regular Borel measure μ_{u} satisfying ∫_{B} ϕdμ_{u} = ∫_{B} uΔ̃ϕdτ, for all ϕ ∈ C_{c}^{2}(B), the class of twice continuosly differentiable functions on B with compact support.

If u is M-harmonic on B and 1 < p < ∞, then the Riesz measure of |u(z)|^{p} is given by u_{p}^{⋆}dτ, where u_{p}^{⋆}(z) = p(p − 1)|u(z)|^{p−2}| ˜∇u(z)|^{2}, (see [14]).

If u ∈ H^{p}(B), 1 < p < ∞, then the least M-harmonic majorant of |u|^{p} is given by

P [u^{⋆}](z) = ∫_{S} P (z, ξ)u^{⋆}(ξ) dσ(ξ),

where P (z, ξ) = |1 − zξ̄|^{−2n}(1 − |z|^{2})^{n} is the invariant Poisson kernel on B, and u^{⋆}(ξ) = lim_{r→1} u(rξ) a.e. on S, [13]. Thus by the Riesz Decomposition Theorem [14], [15] with z = 0, we again have the equality (1.10):

∥u∥_{H^{p}}^{p} = |u(0)|^{p} + p(p − 1) ∫_{B} |u(z)|^{p−2}| ˜∇u(z)|^{2}G(|z|, 1) dτ(z)

For a similar argument see [4], [14]. Our argument in proving (1.10) is much simpler. We only use Green‘s theorem, as well as in the proof of Theorem 1.4.We note that the characterization of the Hardy space H^{p} given in Theorem 1.4 is a part of Proposition 5, [14]. The proof given in [14] is also based on the computation of the Riesz measure of the function |u|^{p} and the Riesz decomposition theorem.

4. A Littlewood-Paley type inequalities

Proof of Theorem 1.1. Without loss of generality we may suppose that f is real. It is well known that a M-harmonic function v belongs to H^{p}, 1 < p < ∞, if and only if v = P [g], for some g ∈ L^{p}(S), 1 < p < ∞. Furthemore, ∥v∥_{H^{p}} = ∥g∥_{L^{p}(S)}. Thus to prove (1.1), by (1.8), it is sufficient to show that

I_{1} = ∫_{B} |u(z)|^{p}dν(z) ⪯ (|u(0)|^{p} + ∫_{B} | ˜∇u(z)|^{p}(1 − |z|^{2})^{−1}dν(z))

and

I_{2} = ∫_{B} | ˜∇u(z)|^{2}|u(z)|^{p−2}(1 − |z|^{2})^{−1}dν(z) ⪯ ∫_{B} | ˜∇u(z)|^{p}(1 − |z|^{2})^{−1}dν(z).

Let I denote the last integral. Using Lemma 2.1, (b), we find that

|u(z)|^{p} = |∫_{0}^{1} \frac{d}{dt}u(tz) dt + u(0)|^{p}
≤ C(∫_{0}^{1} |∇u(tz)|^{p}dt + |u(0)|^{p})
≤ C(∫_{0}^{1}(∫_{E_{r}(tz)} |∇u(w)|^{p}dτ(w)) dt + |u(0)|^{p})

Since 1 − |w|^{2} ≅ 1 − t^{2}|z|^{2} ≅ |1 − tzw̄| for w ∈ E_{r}(tz), we obtain

|u(z)|^{p} ≤ C(∫_{0}^{1}(∫_{E_{r}(tz)} \frac{|∇u(w)|^{p}(1 − |w|^{2})^{α−n−1}}{|1 − tzw̄|^{α}} dν(w))dt + |u(0)|^{p})

≤ C(∫_{0}^{1}(∫_{B} \frac{|∇u(w)|^{p}(1 − |w|^{2})^{α−n−1}}{|1 − tzw̄|^{α}} dν(w)) dt + |u(0)|^{p})

≤ C(∫_{B} \frac{|∇u(w)|^{p}(1 − |w|^{2})^{α−n−1}}{|1 − zw̄|^{α−1}} dν(w) + |u(0)|^{p})

We may suppose that α > n + 2.

By (2.1) |∇u(w)| ≤ \frac{| ˜∇u(w)|}{1−|w|^{2}}. Thus

I_{1} ≤ C(|u(0)|^{p} + ∫_{B} dν(z) ∫_{B} \frac{| ˜∇u(w)|^{p}(1 − |w|^{2})^{α−n−1−p}}{|1 − zw̄|^{α−1}} dν(w))

= C(|u(0)|^{p} + ∫_{B} | ˜∇u(w)|^{p}(1 − |w|^{2})^{α−n−1−p}dν(w) ∫_{B} \frac{dν(z)}{|1 − zw̄|^{α−1}} )

By standard estimates (see [13, p. 17])

∫_{B} \frac{dν(z)}{|1 − zw̄|^{α−1}} ≤ \frac{C}{(1 − |w|^{2})^{α−n−2}} .

Hence,

I_{1} ≤ C(|u(0)|^{p} + ∫_{B} | ˜∇u(w)|^{p}(1 − |w|^{2})^{1−p}dν(w))

Since 1 < p ≤ 2, we have (1−|w|^{2})^{1−p} ≤ (1−|w|^{2})^{−1}. Therefore, I_{1} ⪯ (|u(0)|^{p} +I).

Let r ∈ (0, 1/2) be fixed. Set ϵ = r/4 and δ = r/2. By Fubini’s theorem we have

(4.1)  I_{2} ≤ C ∫_{B}(1 − |a|^{2})^{n}dτ(a) ∫_{E_{ϵ}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}dτ(z).

It is easy to see that if |z| ≤ r/4 then G(|z|, r/2) ≥ C > 0 and therefore

∫_{ϵB} | ˜∇u(z)|^{2}|u(z)|^{p−2}dτ(z) ≤ C ∫_{δB} | ˜∇u(z)|^{2}|u(z)|^{p−2}G(|z|, r/2) dτ(z).

Applying this to u ∘ φ_{a} and using the M-invariance of | ˜∇|, we see that (4.2)

∫_{E_{ϵ}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}dτ(z) ≤ C ∫_{E_{δ}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}G(|φ_{a}(z)|, δ)dτ(z).

Let c_{p} = p(p − 1). Using (3.8) and Jensen‘s inequality we get

c_{p} ∫_{δB} | ˜∇u(z)|^{2}|u(z)|^{p−2}G(|z|, r/2) dτ(z) = ∫_{S} |u(δξ)|^{p}dσ(ξ) − |u(0)|^{p}
= ∫_{S}(|u(δξ)|^{2})^{p/2} − (|u(0)|^{2})^{p/2}

≤ (∫_{S} |u(δξ)|^{2}dσ(ξ))^{p/2}−(|u(0)|^{2})^{p/2}. (4.3)

If 0 < α ≤ 1 and 0 ≤ b ≤ a then a^{α} − b^{α} ≤ (a − b)^{α}. Thus

(∫_{S} |u(δξ)|^{2}dσ(ξ))^{p/2} − (|u(0)|^{2})^{p/2} ≤ (∫_{S} |u(δξ)|^{2}dσ(ξ) − |u(0)|^{2})^{p/2}

= (2 ∫_{δB} | ˜∇u(z)|^{2}G(|z|, r/2)dτ(z))^{p/2}

≤ C sup_{z∈δB} | ˜∇u(z)|^{p}(∫_{δB} G(|z|, r/2)dτ(z))^{p/2}

≤ C sup_{z∈δB} | ˜∇u(z)|^{p}. (4.4)

Using M-harmonic behaviour of | ˜∇u(z)|^{p}, (Lemma 2.1), we find that

(4.5) sup_{z∈δB} | ˜∇u(z)|^{p} ≤ C sup_{z∈δB} ∫_{E_{δ}(z)} | ˜∇u(w)|^{p}dτ(w) ≤ C ∫_{rB} | ˜∇u(w)|^{p}dτ(w).

Combining (4.3), (4.4) and (4.5) we obtain

∫_{δB} | ˜∇u(z)|^{2}|u(z)|^{p−2}G(|z|, r/2) dτ(z) ≤ C ∫_{rB} | ˜∇u(z)|^{p}dτ(z)

Applying this again to u ∘ φ_{a} we get

(4.6) ∫_{E_{δ}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}G(|φ_{a}(z)|, r/2)dτ(z) ≤ C ∫_{E_{r}(a)} | ˜∇u(z)|^{p}dτ(z)

Finally, from (4.1), (4.2) and (4.6) we see that

I_{2} ≤ C ∫_{B}(1 − |a|^{2})^{n}dτ(a) ∫_{E_{r}(a)} | ˜∇u(z)|^{p}dτ(z)
= C ∫_{B} | ˜∇u(z)|^{p}(1 − |z|^{2})^{−1}dν(z) = CI,

by Fubini’s theorem.

This finishes the proof of the inequality (1.1). Since the function |u|^{p} is M-subharmonic we have

|u(0)|^{p} ≤ ∫_{B} |u(z)|^{p}dν(z).

Hence by (1.8) to prove (1.2) it sufficies to show that I_{2} ≥ CI.

By Fubini’s theorem

(4.7)  I_{2} ≅ ∫_{B}(1 − |a|^{2})^{n}dτ(a) ∫_{E_{r}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}dτ(z)

Using (3.8) and Jensen’s inequality we find that

c_{p} ∫_{δB} | ˜∇u(z)|^{2}|u(z)|^{p−2}G(|z|, δ) dτ(z) = ∫_{S} |u(δξ)|^{p}dσ(ξ) − |u(0)|^{p}

= ∫_{S}(|u(δξ)|^{2})^{p/2}dσ(ξ) − |u(0)|^{p}

≥ (∫_{S} |u(δξ)|^{2}dσ(ξ))^{p/2} − (|u(0)|^{2})^{p/2}.

Littlewood-Paley type inequalities... 49

For 1 ≤ α < ∞ and 0 ≤ b ≤ a, we have (a − b)^{α} ≤ a^{α} − b^{α}. Hence, by using (3.8) and Lemma 2.1,

(∫_{S} |u(δξ)|^{2}dσ(ξ))^{p/2} − (|u(0)|^{2})^{p/2} ≥ (∫_{S} |u(δξ)|^{2}dσ(ξ) − |u(0)|^{2})^{p/2}
= (2 ∫_{δB} | ˜∇u(z)|^{2}G(|z|, δ)dτ(z))^{p/2}
≥ C(∫_{δB} | ˜∇u(z)|^{2}dτ(z))^{p/2} ≥ C| ˜∇u(0)|^{p}

Therefore,

| ˜∇u(0)|^{p} ≤ C ∫_{δB} | ˜∇u(z)|^{2}|u(z)|^{p−2}G(|z|, δ) dτ(z)

Since G(|z|, δ) ≤ C|z|^{2−2n}, for |z| ≤ δ, (note that n > 1), we get | ˜∇u(0)|^{p} ≤ C ∫_{δB} | ˜∇u(z)|^{2}|u(z)|^{p−2}|z|^{2−2n}dτ(z).

Applying this to u ∘ φ_{a}, where |a| < δ, we find that | ˜∇u(a)|^{p} ≤ C ∫_{E_{δ}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}|φ_{a}(z)|^{2−2n}dτ(z)

By Lemma 2.4 |φ_{a}(z)| ≥ C|z − a|. Thus

| ˜∇u(a)|^{p} ≤ C ∫_{E_{δ}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}|z − a|^{2−2n}dτ(z).

Integrating the last inequality over |a| < δ with respect to ν we obtain ∫_{δB} | ˜∇u(a)|^{p}dν(a) ≤ C ∫_{rB} | ˜∇u(z)|^{2}|u(z)|^{p−2}dτ(z) ∫_{δB} |z − a|^{2−2n}dν(a). Since, for z ∈ rB,

∫_{δB} |z − a|^{2−2n}dν(a) ≤ ∫_{{a:|z−a|<2r}} |z − a|^{2−2n}dν ≤ C

where C is independent of z, we have | ˜∇u(0)|^{p} ≤ C ∫_{δB} | ˜∇u(a)|^{p}dτ(a) ≤ C ∫_{rB} | ˜∇u(z)|^{2}|u(z)|^{p−2}dτ(z)

Applying this again to u ∘ φ_{a} we get

(4.8)  | ˜∇u(a)|^{p} ≤ C ∫_{E_{r}(a)} | ˜∇u(z)|^{2}|u(z)|^{p−2}dτ(z)

From (4.7) and (4.8) it follows that I_{2} ≥ CI. This finishes the proof of Theorem 1.1

Now, using Theorem 1.1 we get another version of Littlewood-Paley theorem.

Theorem 4.1. Let f ∈ L^{p}(S) and u = P [f]. Then

(4.9)  ∥f∥_{p}^{p} ⪯ (|u(0)|^{p} + ∫_{B} | ˜∇u(z)|^{p}G(|z|, 1) dτ(z)), 1 < p ≤ 2

and

(4.10)  ∥f∥_{p}^{p} ⪰ (|u(0)|^{p} + ∫_{B} | ˜∇u(z)|^{p}G(|z|, 1)dτ(z)), 2 ≤ p < ∞.

Proof. By Lemma 2.3 G(|z|, 1) ≥ c_{n}(1 − |z|)^{n} for all z ∈ B. Hence, (4.9) is a consequence of (1.1).

Let 0 < δ < 1/2 be fixed. Then using again Lemma 2.3 we find that

∫_{B} | ˜∇u(z)|^{p}G(|z|, 1) dτ(z)

≤ C(∫_{\frac{1}{2} δB} | ˜∇u(z)|^{p}|z|^{2−2n}dτ(z) + ∫_{( \frac{1}{2} δB)^{c}} | ˜∇u(z)|^{p}(1 − |z|^{2})^{n}dτ(z)).

Thus, (4.10) follows from (1.2) since

∫_{\frac{1}{2} δB} | ˜∇u(z)|^{p}|z|^{2−2n}dτ(z) ≤ C ∫_{δB} | ˜∇u(z)|^{p}(1 − |z|^{2})^{−1}dν(z).

Theorem 1.2 is a corollary of Theorem 1.1 and the following theorem

Theorem 4.2. Let 1 < p < ∞ and u ∈ M. Then the following statements are equivalent:

(i) ∫_{B} | ˜∇u(z)|^{p}(1 − |z|^{2})^{−1}dν(z) < ∞.

(ii) ∫_{B} |∇u(z)|^{p}(1 − |z|^{2})^{p−1}dν(z) < ∞.

(iii) ∫_{B} |Ru(z)|^{p}(1 − |z|^{2})^{p−1}dν(z) < ∞.

Proof. Since | ˜∇u(z)| ≥ (1 − |z|^{2})|∇u(z)| ≥ (1 − |z|^{2})|Ru(z)|, we have (i) ⇒ (ii) ⇒ (iii). So it remains to show that (iii) ⇒ (i).

By (2.2) to prove that (iii) ⇒ (i) it is sufficient to prove that

∫_{B}(1 − |z|^{2})^{p/2−1}|T_{i,j}u(z)|^{p}dν(z) ≤ C∫_{B}(1 − |z|^{2})^{p−1}|Ru(z)|^{p}dν(z), 1 ≤ i < j ≤ n.

An integration by parts show that

u(z) = ∫_{0}^{1}[Ru(tz) + R̄u(tz) + u(tz)]dt

From this we conclude that it is sufficient to prove that

I = ∫_{B}(1 − |z|^{2})^{p/2−1}(∫_{0}^{1} |T_{i,j}v(tz)|dt)^{p}dν(z) ≤ C ∫_{B}(1 − |z|^{2})^{p−1}|Ru(z)|^{p}dν(z)

where v is either Ru or R̄u or u. We prove this for v = Ru. The remaining cases may be treated analogously.

Let J = ∫_{0}^{1} |T_{i,j}Ru(tz)|dt. Using Lemma 2.2 and Fubini’s theorem we find that for any s > 0

J ≤ C ∫_{0}^{1}(∫_{E_{r}(tz)} \frac{|Ru(w)|(1 − |w|^{2})^{s}dν(w)}{|1 − tzw̄|^{n+s+3/2}})dt

≤ C ∫_{0}^{1}(∫_{B} \frac{|Ru(w)|(1 − |w|^{2})^{s}dν(w)}{|1 − tzw̄|^{n+s+3/2}})dt

= ∫_{B} |Ru(w)|(1 − |w|^{2})^{s}(∫_{0}^{1} \frac{dt}{|1 − tzw̄|^{n+s+3/2}} )dν(w)

≤ C ∫_{B} \frac{|Ru(w)|(1 − |w|^{2})^{s}dν(w)}{|1 − zw̄|^{n+s+1/2}}.

Now we apply Lemma 4.1 [3] to conclude that

I ≤ C(∫_{B}(1 − |z|^{2})^{p/2−1}(∫_{B} \frac{|Ru(w)|(1 − |w|^{2})^{s}dν(w)}{|1 − zw̄|^{n+s+1/2}})^{p}dν(z))
≤ C ∫_{B}(1 − |w|^{2})^{p−1}|Ru(w)|^{p}dν(w)

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Matematički fakultet (Received 02 07 1998) Studentski trg 16 11001 Beograd, p. p. 550 Yugoslavia