We give new characterizations of the analytic Besov spaces Bp on the unit ball B of Cnin terms of oscillations and integral means over some Euclidian balls contained in B

A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXV, NO. 2, 2011 SECTIO A 87–97

MAŁGORZATA MICHALSKA, MARIA NOWAK and PAWEŁ SOBOLEWSKI

M¨obius invariant Besov spaces on the unit ball of Cⁿ

Dedicated to the memory of Professor Jan G. Krzyż

Abstract. We give new characterizations of the analytic Besov spaces B^p on the unit ball B of Cⁿin terms of oscillations and integral means over some Euclidian balls contained in B.

1. Introduction. Let B = {z ∈ Cⁿ : |z| < 1} denote the open unit ball in Cⁿ and H(B) be the set of all holomorphic functions on B. By Aut(B) we mean the group of all automorphisms of B. It is known that Aut(B) is generated by the unitary operators and involutions of the form

ϕw(z) = w − Pw(z) − swQw(z) 1 − hz, wi ,

where w ∈ B, sw = (1 − |w|²)^1/2, P_w is the orthogonal projection of Cⁿ to the subspace spanned by w, i.e.

P_w(z) = hz, wi

|w|² w for w 6= 0, and P₀(z) = 0,

2000 Mathematics Subject Classification. 30H25.

Key words and phrases. Besov spaces, conformal M¨obius transformation.

and Q_w = I − P_w (see, e.g. [13, 16] for definition and properties of the automorphism group of B). The mapping ϕa is called the M¨obius transfor- mation. It is known that ρ(z, w) = |ϕ_z(w)| is a metric on B, the so-called pseudo-hyperbolic metric (see, e.g. [9, 15, 16]).

Let dυ be the Lebesgue measure on B normalized so that υ(B) = 1 and let dτ (z) = _(1−|z|^dv(z)2)ⁿ⁺¹ be the invariant measure on B.

For f ∈ H(B), set

Q_f(z) = sup

06=x∈Cⁿ

|h∇f (z), xi|

H_z(x, x)^1/2, z ∈ B,

where ∇f (z) = (∂f /∂z₁, ∂f /∂z₂, . . . , ∂f /∂z_n) is the complex gradient of f and H_z(x, x) is the Bergman metric on B, that is

H_z(x, x) = n + 1 2

(1 − |z|²)|x|²+ |hx, zi|² (1 − |z|²)² .

The M¨obius invariant Besov space B_p, 1 < p ≤ ∞, consists of all holomor- phic functions on B for which Qf ∈ L^p(B, dτ ). In the case p = ∞ the space B∞ is the Bloch space B; so

B = B∞= {f ∈ H(B) : kf kB < ∞}, where

kf k_B = sup

z∈B

Qf(z).

If 1 < p ≤ ∞ the space B_p is the Banach space with the norm kf k_Bp = |f (0)| + (p − 1)kQ_fk_Lp(dτ ).

Hahn and Youssfi [3] proved that for n > 1 the Besov space B_p is nontrivial and contains all polynomials if and only if p > 2n. Moreover, it is known that for f ∈ H(B), the following conditions are equivalent

(i) f ∈ B_p,

(ii) |∇f (z)|(1 − |z|²) ∈ L^p(B, dτ ),

(iii) | e∇f (z)| ∈ L^p(B, dτ ) where | e∇f (z)| = |∇(f ◦ ϕ_z)(0)|.

The proofs can be found in [3, 8, 16].

The following results for the space B_p are reminiscences of Holland and Walsh characterization of the Bloch space [6].

In the case n = 1 Stroethoff [14] proved that for 2 < p < ∞, f ∈ B_p ⇔

Z

B

Z

B

f (z) − f (w) z − w

p

(1 − |z|²)^p2(1 − |w|²)^p2dτ (w)dτ (z) < ∞.

This equivalence has been generalized to the unit ball case in [8], where the following result has been obtained. If 2n < p < ∞, then

(1)

f ∈ B_p ⇔ Z

B

Z

B

 |f (z)−f (w)|

|w−P_w(z)−s_wQ_w(z)|

p

(1−|z|²)^p2(1−|w|²)^p2dτ (w)dτ (z) < ∞.

Let B(a, r) denote a Euclidian ball of radius r and centered at a ∈ Cⁿ. For a ∈ B and 0 < r < 1 let

E(a, r) = {z ∈ B : |ϕa(z)| < r} = ϕ_a(B(0, r))

be the pseudo-hyperbolic (or Bergman) metric ball centered at z. Then E(a, r) is an elipsoid in Cⁿ. We will often use the following property of E(a, r).

There exists a positive constant C (dependent on r, but not on z and a) such that

(2) C⁻¹(1 − |z|²) ≤ |1 − hz, ai| ≤ C(1 − |a|²) for all z ∈ E(a, r).

Using equivalence (1), we easily obtain the following

Theorem 1. Assume that f ∈ H(B) and 2n < p < ∞. Then f ∈ Bp if and only if

(3) Z

B

Z

B

 |f (z) − f (w)|

|1 − hz, wi|

p

(1 − |z|²)^p2(1 − |w|²)^p2dτ (w)dτ (z) < ∞.

Proof. Assume that for f ∈ H(B) condition (3) is satisfied. Since

| e∇f (z)|^p≤ C Z

E(z,r)

|f (w) − f (z)|^p

|1 − hw, zi|ⁿ⁺¹dv(w), (see, e.g. [8]) and for w ∈ E(z, r),

(4) 1 − r² < 1 − |ϕ_z(w)|² = (1 − |z|²)(1 − |w|²)

|1 − hw, zi|² , we get, using (2),

Z

B

| e∇f (z)|^pdτ (z)

≤ C Z

B

Z

E(z,r)

|f (w) − f (z)|^p

|1 − hw, zi|ⁿ⁺¹

(1 − |z|²)^p/2(1 − |w|²)^p/2

|1 − hw, zi|^p dv(w)dτ (z)

≤ C Z

B

Z

E(z,r)

|f (w) − f (z)|^p (1 − |w|²)ⁿ⁺¹

(1 − |z|²)^p/2(1 − |w|²)^p/2

|1 − hw, zi|^p dv(w)dτ (z)

≤ C Z

B

Z

B

|f (w) − f (z)|^p(1 − |z|²)^p/2(1 − |w|²)^p/2

|1 − hw, zi|^p dτ (w)dτ (z).

Hence (3) implies f ∈ B_p. The other implication follows from (1) and from the inequality

|w − P_w(z) − swQw(z)| ≤ |1 − hz, wi|, z, w ∈ B.  For α > −1 we define the weighted volume measure dv_α(z) = c_α(1 −

|z|²)^αdv(z), where cα is a positive constant such that v_α(B) = 1.

We remark that condition (3) can be written in the form Z

B

Z

B

 |f (z) − f (w)|

|1 − hz, wi|

p

dvα(z)dvα(w) < ∞, where α = −n − 1 + p/2.

Moreover, the inequality

|w − P_w(z) − swQw(z)| ≤ |z − w|, z, w ∈ B, and equivalence (3) imply that if f ∈ B_p, then

(5) Z

B

Z

B

 |f (z) − f (w)|

|z − w|

p

dv_α(z)dv_α(w) < ∞, α = −n − 1 + p/2.

We do not know if condition (5) is sufficient for f to belong to B_p. The sufficiency of (5) has been claimed in [4]. Unfortunately, the proof given there is not correct.

For p = ∞, condition (5) is understood as kf k_B˜= sup

z,w∈B,z6=w

|f (z) − f (w)|

|z − w| (1 − |z|²)¹²(1 − |w|²)¹² < ∞

and is necessary and sufficient for containment in the Bloch space B as shown in [12]. For the proof of the last result the authors [12] used the so-called conformal M¨obius transformation. We also will discuss this transformation in the next section.

Recently, M. Pavlović [10, 11] considered a more general space of C¹ functions in the unit ball for which two Bloch norms can be defined as follows

kf k_B1 = sup

x∈B

(1 − |x|²)kdf (x)k, (6)

kf k_B2 = sup

x∈B

k ˜df (x)k, (7)

where kdf (x)k is the norm of the differential of f at x and k ˜df (x)k = kd(f ◦ ϕ_x)(0)k. It is proved in [10, 11] that

kf k_B1 = kf k_B˜ and

kf k_B2 = sup

z,w∈B,z6=w

|f (z) − f (w)|

|w − P_w(z) − s_wQ_w(z)|(1 − |z|²)¹²(1 − |w|²)¹².

Here we get one more criterion for containment in the Bloch space.

Namely, if f ∈ H(B), then f ∈ B ⇔ sup

z,w∈B,z6=w

|f (z) − f (w)|

|1 − hz, wi| (1 − |z|²)¹²(1 − |w|²)¹² < ∞.

Finally, it is worth noting that characterizations of weighted Bergman spaces on the unit ball in terms of double integrals of the functions |f (z) − f (w)|/|1 − hz, wi| and |f (z) − f (w)|/|z − w| have been recently obtained in [5] and [7].

2. Characterizations in terms of oscillation and integral means.

For f ∈ H(B), z ∈ B and 0 < r < 1 we put

ωr(f )(z) = sup{|f (z) − f (w)| : w ∈ E(z, r)}

and

M O_r(f )(z) = 1 v(E(z, r))

Z

E(z,r)

|f (w) − f_z,r|dv(w), where

fz,r= 1 v(E(z, r))

Z

E(z,r)

f (u)dv(u).

ω_r(f ) and M O_r(f ) are, respectively, the oscillation and the mean oscillation of f in the Bergman metric at the point z.

The following characterizations of the space B_p in terms of ω_r(f ) and M Or(f ) can be found in [16].

Theorem A. Let f ∈ H(B) and 2n < p, and 0 < r < 1. Then the following conditions are equivalent

(i) f ∈ B_p,

(ii) ω_r(f ) ∈ L^p(B, dτ ), (iii) M O_r(f ) ∈ L^p(B, dτ ).

We will prove similar characterizations of B_p in terms of oscillations, but in a different metric. The metric will be connected with the conformal M¨obius transformation on B given by

ϕ^c_a(z) = |z − a|²a − (1 − |a|²)(z − a)

||a|z − a⁰|² ,

where a ∈ B, a⁰ = _|a|^a for a 6= 0 and a⁰ = (1, 0, . . . , 0), when a = 0. The mapping ϕ^c_a is an involution automorphism of B such that ϕ^ca(0) = a and ϕ^c_a(a) = 0. Moreover,

|ϕ_a(z)| ≤ |ϕ^c_a(z)|, a, z ∈ B.

Also, it is easy to check that (8) 1 − |ϕ^c_a(z)|²

|ϕ^c_a(z)|² = (1 − |z|²)(1 − |a|²)

|z − a|² .

We refer the reader to [1] and [12] for further properties of ϕ^c_a.

Analogously to the M¨obius transformations case, the formula ρ^c(a, z) =

|ϕ^c_a(z)| defines a metric on B. We give the proof of this fact, probably known, because we do not know a reference. By the definition of ϕ^c_a, we get

|ϕ^c_a(z)| = |z − a|

||a|z − a⁰| = |a − z|

||z|a − z⁰|= |ϕ^c_z(a)|.

It is also obvious that

|ϕ^c_a(z)| = 0 ⇔ z = a.

The invariance of ρ^c(a, z) under the conformal M¨obius transformations fol- lows immediately from formula (38) in [1]. So, we have

ρ^c(a, z) = |ϕ^c_a(z)| = |ϕ^c_ϕc

w(a)(ϕ^c_w(z))| = ρ^c(ϕ^c_w(a), ϕ^c_w(z)).

In view of this, it is enough to show that

(9) ρ^c(a, z) ≤ |a| + |z|.

Using the inequality

1 − (x + y)² ≤ 1 − x²

1 − y²
(1 + xy)² , for x, y ∈ [0, 1], (see, e.g. [15]), we obtain

1 − (|a| + |z|)² ≤ 1 − |a|²

1 − |z|²
(1 + |a||z|)²

≤ 1 − |a|²

1 − |z|²

||a|z − a⁰|² = 1 − |ϕ^c_a(z)|², which proves (9).

For a ∈ B and 0 < r < 1 let

E^c(a, r) = {z ∈ B : |ϕ^c_a(z)| < r} = ϕ^c_a(B(0, r)).

The set E^c(a, r) is a Euclidian ball in R²ⁿ centered at _1−r^(1−r2|a|^2)a2 and of the radius ^(1−|a|_1−r2|a|^2)r₂. Note that if z ∈ B(a,^r₂(1 − |a|²)), then

|ϕ^c_a(z)| = |z − a|

||a|z − a⁰| ≤ |z − a|

|a⁰| − |a||z| ≤ |z − a|

1 − |a| ≤ 2|z − a|

1 − |a|² < r.

It follows immediately that

(10) B

 a,r

2(1 − |a|²)



⊂ E^c(a, r) ⊂ E(a, r).

Now, for f ∈ H(B) and z ∈ B, we define

ω^c_r(f )(z) = sup{|f (z) − f (w)| : w ∈ E^c(z, r)}

and

M O^c_r(f )(z) = 1 v(E^c(z, r))

Z

E^c(z,r)

|f (w) − f_z,r^c |dv(w), where

f_z,r^c = 1 v(E^c(z, r))

Z

E^c(z,r)

f (u)dv(u).

We get the following analogue of Theorem A.

Theorem 2. Let 2n < p < ∞ and 0 < r < 1. Then the following state- ments are equivalent

(i) f ∈ B_p,

(ii) ω^c_r(f ) ∈ L^p(B, dτ ), (iii) M O^c_r(f ) ∈ L^p(B, dτ ).

Proof. (i)⇒(ii) If f ∈ Bp, then inclusion (10) and Theorem A imply that ω_r^c(f ) ∈ L^p(B, dτ ).

(ii)⇒(iii) Since

f (w) − f_z,r^c = f (w) − f (z) − (f_z,r^c − f (z)) and

f_z,r^c − f (z) = 1 v(E^c(z, r))

Z

E^c(z,r)

(f (w) − f (z))dv(w), we get

M O_r^c(f )(z) ≤ 2 v(E^c(z, r))

Z

E^c(z,r)

|f (w) − f (z)|dv(w) ≤ 2ω_r^c(f )(z).

(iii)⇒(i) It follows from the subharmonicity of |F |^p, F ∈ H(B), that for any 0 < s < 1, 0 < p < ∞ and B(z, s) ⊂ B,

(11) |∇F (z)|^ps^p ≤ Cs⁻²ⁿ Z

B(z,s)

|F (w)|^pdv(w), z ∈ B.

Applying inequality (11) with s = ^r₂(1 − |z|²) to the function F (w) = f (z + w) − f_z,r^c and using inclusion (10), we see that

|∇f (z)|(1 − |z|²) ≤ C Z

E^c(z,r)

|f (w) − f_z,r^c | dv(w)

(1 − |w|²)²ⁿ ≤ CM O_r^c(f )(z)

and the proof is complete. 

Moreover, we have

Theorem 3. Assume that f ∈ H(B), 2n < p < ∞, r ∈ (0, 1). Then f ∈ B_p ⇔

Z

E^c(a,r)

|∇f (z)| dv(z)

(1 − |z|²)²ⁿ⁻¹ = (Mf )(a) ∈ L^p(B, dτ ).

Proof. By subharmonicity of   

∂f

∂zi

 we have 

∂f

∂zi

(z)    

≤ Z

B

∂f

∂zi

(z + δw)    

dv(w) = 1 δ²ⁿ

Z

B(z,δ)

∂f

∂zi

(w)    

dv(w)

for z ∈ B and 0 ≤ δ < 1 − |z|. Thus for r ∈ (0, 1), 

∂f

∂z_i(z)    

≤ 2²ⁿ

r²ⁿ(1 − |z|²)²ⁿ Z

B(z,^r₂(1−|z|²))

∂f

∂z_i(w)    

dv(w)

≤ C Z

B(z,^r₂(1−|z|²))

∂f

∂zi

(w)    

dv(w) (1 − |w|²)²ⁿ. Consequently,

|∇f (z)|(1 − |z|²) ≤ C Z

B(z,^r₂(1−|z|²))

|∇f (w)| dv(w) (1 − |w|²)²ⁿ⁻¹, which proves the implication “⇒”.

Now, let f ∈ B_p. Then ω_r(f ) ∈ L^p(B, dτ ) by Theorem A. It follows from the proof of Theorem 2 that

|∇f (z)|(1 − |z|²) ≤ Cω^c_r(f )(z) ≤ Cωr(f )(z).

Hence Z

B

(Mf )^p(a)dτ (a)

= Z

B

Z

E^c(a,r)

|∇f (z)|(1 − |z|²) dv(z) (1 − |z|²)²ⁿ

!p

dτ (a)

≤ C Z

B

Z

E^c(a,r)

ωr(f )(z) dv(z) (1 − |z|²)²ⁿ

!p

dτ (a)

= C Z

B

Z

E^c(a,r)

sup

w∈E(z,r)

|f (z) − f (w)|

! dv(z) (1 − |z|²)²ⁿ

!p

dτ (a).

To complete the proof, we apply the following triangle inequalities for the pseudo-hyperbolic metric ρ(z, a) = |ϕ_a(z)| (see, e.g. [2])

(12) |ρ(z, a) − ρ(a, w)|

1 − ρ(z, a)ρ(a, w) ≤ ρ(z, w) ≤ ρ(z, a) + ρ(a, w)

1 + ρ(z, a)ρ(a, w), z, w, a ∈ B.

This inequality implies that if w ∈ E(a, r) and a ∈ E(z, r), then w ∈ E(z, 2r/(1 + r²)). Consequently, using inclusion (10),

Z

E^c(a,r)

sup

w∈E(z,r)

|f (z) − f (w)|

! dv(z) (1 − |z|²)²ⁿ

≤ C sup

z∈E^c(a,r)

sup

w∈E(z,r)

(|f (z) − f (a)| + |f (w) − f (a)|)

!

≤ C sup

z∈E(a, ^2r

1+r2)

|f (z) − f (a)| + sup

w∈E(a, ^2r

1+r2)

|f (a) − f (w)|

≤ 2C sup

w∈E(a, ^2r

1+r2)

|f (a) − f (w)|

= 2Cω ^2r

1+r2

(f )(a) ∈ L^p(B, dτ ). 

We remark that the last theorem is equivalent to the statement that a function f holomorphic on B is in Bp if and only if the integral mean of f at a given by

(M f )(a) = 1 v(E^c(a, r))

Z

E^c(a,r)

|∇f (z)|(1 − |z|²)dv(z)

is in L^p(B, dτ ).

Let us define (Hf )(a) = Z

E^c(a,r)

|f (z) − f (a)|

|z − a| (1 − |z|²)¹²(1 − |a|²)¹² dv(z) (1 − |z|²)²ⁿ. Our last theorem refers to Holland–Walsh characterization of the Bloch space.

Theorem 4. Let f be a holomorphic function in B and r ∈ (0, 1). Then the following statements are equivalent

(i) f ∈ B_p,

(ii) Hf ∈ L^p(B, dτ ), (iii)

Z

B

Z

E^c(a,r)

|f (z) − f (a)|^p

|z − a|^p (1−|z|²)^p2(1−|a|²)^p2 dv(z)

(1 − |z|²)²ⁿdτ (a) < ∞.

Proof. (i)⇒(ii) Suppose f ∈ Bp. Using the invariance of the measure

dv(z)

(1−|z|²)²ⁿ under the map ϕ^c_a(z), we obtain

Z

E^c(a,r)

|f (z) − f (a)|

|z − a| (1 − |z|²)¹²(1 − |a|²)¹² dv(z) (1 − |z|²)²ⁿ

≤ Z

E^c(a,r)

sup

z∈E^c(a,r)

|f (z) − f (a)|

!(1 − |z|²)¹²(1 − |a|²)¹²

|z − a|

dv(z) (1 − |z|²)²ⁿ

= ω_r^c(f )(a) Z

E^c(a,r)

p1 − |ϕ^c_a(z)|²

|ϕ^c_a(z)|

dv(z) (1 − |z|²)²ⁿ

= ω_r^c(f )(a) Z

E^c(0,r)

p1 − |z|²

|z|

dv(z)

(1 − |z|²)²ⁿ = Cω_r^c(f )(a) ∈ L^p(B, dτ ).

(ii)⇒(iii) It is enough to apply the Jensen inequality.

(iii)⇒(i) From (8) we see that if |ϕ^c_a(z)| < r, then (1 − |z|²)¹²(1 − |a|²)¹²

|z − a| = p1 − |ϕ^c_a(z)|²

|ϕ^c_a(z)| ≥

√ 1 − r²

r .

This and (11) imply

|∇f (a)|^p(1 − |a|²)^p

≤ C Z

E^c(a,r)

|f (z) − f (a)|^p

|z − a|^p (1 − |z|²)^p2(1 − |a|²)^p2 dv(z) (1 − |z|²)²ⁿ,

which proves the implication. 

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Małgorzata Michalska Department of Mathematics Maria Curie-Skłodowska University pl. Marii Curie-Skłodowskiej 1 20-031 Lublin

Poland

e-mail: malgorzata.michalska@poczta.umcs.lublin.pl Maria Nowak

Department of Mathematics Maria Curie-Skłodowska University pl. Marii Curie-Skłodowskiej 1 20-031 Lublin

Poland

e-mail: mt.nowak@poczta.umcs.lublin.pl Paweł Sobolewski

Department of Mathematics Maria Curie-Skłodowska University pl. Marii Curie-Skłodowskiej 1 20-031 Lublin

Poland

e-mail: pawel.sobolewski@umcs.eu Received August 11, 2011