Potential AnalDOI 10.1007/s11118-013-9361-x

Addendum to “Besov–Lipschitzand Mean Besov–Lipschitz Spaces”:The Ahern–Schneider Inequality

Miroslav Pavlovic

Received: 23 December 2012 / Accepted: 25 June 2013© Springer Science+Business Media Dordrecht 2013

Abstract Let f be a function holomorphic in the unit ball of CN , and R f theradial derivative of f . It is proved that the Ahern–Schneider inequality ‖∇ f‖H p ≤Cp‖R f‖H p holds for 0 < p < 1. This fills a gap in the proof of the main result in thepaper “Besov–Lipschitz and mean Besov–Lipschitz spaces of holomorphic functionson the unit ball” [Potential Analysis] by Jevtic and Pavlovic.

Keyword Ahern–Schneider inequality

Mathematics Subject Classifications (2010) 32A35 · 32A36 · 32A37

Let BN denote the unit ball in CN and let SN = ∂BN , where N is a positiveinteger. For a point z = (z1, . . . , zN) ∈ CN we write |z| = (|z1|2 + . . . + |zN|2)1/2. Thenormalized Lebesgue measures on BN and SN will be denoted by dv = dvN anddσ = dσN , respectively. The Lp-mean over the sphere |z| = r (0 < r ≤ 1) of a Borelfunction f on BN is defined by

Mp(r, f ) =(∫

SN

| f (rζ )|p dσ(ζ )

)1/p

(0 < p < ∞).

Let H(BN) denote the space of all holomorphic functions in BN . In proving someof the results in [2] the following fact is used:

Theorem 1 Let 0 < p < ∞. If f ∈ H(BN), then Mp(r, |∇ f |) ≤ CMp(r,R f ), f or1/4 < r < 1.

Supported by Ministry of Sciences Serbia, Project ON174017.

M. Pavlovic (B)Faculty of Mathematics, University of Belgrade,Studentski trg 16, p.p. 550, 11000 Belgrade, Serbiae-mail: pavlovic@matf.bg.ac.rs

M. Pavlovic

Here R f denotes radial derivative of a function f ∈ H(BN),

R f (z) =N∑

j=1

z j∂ f∂z j

,

and ∇ f = (∂1 f, . . . , ∂N f ) is the euclidean gradient of f . All unexplained notationand facts can be found in [2] and [5].

The validity of Theorem 1 is known only for p ≥ 1, and it is in fact a reformulationof a result of Ahern and Schneider [1]. Here we prove the theorem for p ≤ 1 and sofill the gap in the proofs of several results of [2]: Theorems 1.4 and 1.6, Corollary 1.7,and Lemmas 4.1, 5.5, and 5.6. In proving the other results Theorem 1 could beavoided.

Our proof works for 0 < p ≤ 2 and is based on the following Hardy–Stein typeinequality.

Lemma 2 If f = ( f1, . . . , fN) : D �→ CN, is a function holomorphic in D = B1 andcontinuous on D, then, for 0 < p ≤ 2,

p2

2

∫D

| f (z)|p−2| f ′(z)|2 log1

|z| dA(z)

≤ Mpp(1, f ) − | f (0)|p ≤ p

∫D

| f (z)|p−2| f ′(z)|2 log1

|z| dA(z), (1)

where dA(z) = dv1(z).

Proof We apply the Green formula

−∫ 2π

0u

(eiθ ) dθ − u(0) = 1

2

∫D

�u(z) log1

|z| dA(z)

to the function

u = | f |p = (| f1|2 + . . . + | fN|2)p/2.

After a direct, although tedious computation, we get

�(| f |p) = p2

( p2

− 1)

U p/2−2|∇U |2 + p2

U p/2−1�U,

where U = | f1|2 + . . . + | fN|2, and ∇U denotes the ordinary euclidean gradient ofthe real-valued function U. Since �U = 4| f ′(z)|2 and p/2 − 1 ≤ 0, we see that theright-hand side inequality of Eq. 1 holds. On the other hand, we have |∇U |2 = 4|∂U |2and so

|∇U |2 = 4∣∣ f1 f ′

1 + . . . + fN f ′N

∣∣2 ≤ 4| f |2| f ′|2,whence

�(| f |p) ≥ p(p − 2)| f |p−2| f ′|2 + 2p| f |p−2| f ′|2 = p2| f |p−2| f ′|2.The result follows. (The problems with possible non-differentiability of | f |p are fixedin the standard way; see [4].) ��

Addendum to “Besov–Lipschitz and Mean Besov–Lipschitz Spaces”

Arguing in the same way as in the proof of Theorem 1.2 (and Theorem 1.1) of [4],we obtain:

Lemma 3 (with the hypotheses of Lemma 2) Let γ > 0 and

Hp,γ ( f ) =∫

D

|z|γ | f (z)|p−2| f ′(z)|2 (1 − |z|2) dA(z).

Then Mpp(1, f ) | f (0)|p + Hp,γ ( f ).

As usual, we write A( f ) B( f ) to denote that A( f )/B( f ) lies between twopositive constants that depend only on p, N, and γ. We note that one of the mainingredient in the proof is the inequality, due to Hardy and Littlewood,

∫D

| f |p dA(z) | f (0)|p +∫

D

| f ′(z)|p(1 − |z|)p dA(z), (2)

which holds for vector-valued functions, which can be seen from the proofs ofTheorem 9.1.2 and Proposition 7.1.6 in [3]; in our case, Eq. 2 can be obtained byapplication of the classical inequality to the coordinates of f . Applying the lastlemma to the function zf (z) we get

Lemma 4 (with the hypotheses of Lemma 2) We have

Mpp(1, f )

∫D

|z|p−2+γ | f (z)|p−2|zf ′(z) + f (z)|2 (1 − |z|2) dA(z). (3)

Now we consider a mapping F : BN �→ CN holomorphic in BN and continuous onthe closure of BN . We use the formula

∫SN

|F|p dσN =∫

SN

dσN(ζ )−∫ 2π

0|F (

eitζ) |p dt (4)

(integration by slices) together with integration in polar coordinates,

∫BN

g dvn = 2N∫ 1

0r2N−1 M1(r, g) dr (g ≥ 0 measurable) (5)

to get:

Lemma 5 (F is as above, 0 < p ≤ 2) We have∫

SN

|F|p dσN ∫

BN

|z|p−2N+γ |F(z)|p−2|F(z) + RF(z)|2 (1 − |z|2) dvN(z),

where

RF(z) =N∑

j=1

z j∂ jF(z).

M. Pavlovic

Proof For a fixed ζ ∈ SN , let f (z) = F(zζ ). By Eq. 3, we have

−∫ 2π

0|F(eitζ )|p dt

∫D

|z|p−2+γ |F(zζ )|p−2|F(zζ ) + zd

dzF(zζ )|2 (

1 − |z|2) dA(z)

=∫

D

|z|p−2+γ |F(zζ )|p−2|F(zζ ) + RF(zζ )|2 (1 − |z|2) dA(z).

where we have used the chain rule: z(d/dz)F(zζ ) = zF ′(zζ )ζ = (RF)(zζ ). Writingthe last integral as 2N

∫ 10 r2N−1 dr−

∫ 2π

0 r p−2N+γ . . . dt and then using the formulas 4and 5, we obtain the desired result. ��

As an application of this lemma and the identity ∇ f + R(∇ f ) = ∇(R f ), wheref ∈ H(BN), we have:

Lemma 6 If f ∈ H(BN) and ∇ f is continuous up to the boundary of BN, then

Mpp(1, |∇ f |)

∫BN

|∇ f (z)|p−2|∇(R f )|2 (1 − |z|2) dvN(z). (6)

Now we are ready to prove Theorem 1 for 0 < p ≤ 2. Assume, as we may, thatr = 1. By [4, Theorem 5.2], we have

Mpp(1,R f )

∫BN

|R f |p−2|∇(R f )|2 (1 − |z|2) dvN(z). (7)

Since |∇ f | ≥ |R f | and p − 2 ≤ 0, we have |∇ f |p−2 ≤ |R f |p−2 so the desired inequal-ity follows from Eqs. 6 and 7.

References

1. Ahern, P., Schneider, R.: Holomorphic Lipschitz functions in pseudoconvex domains. Amer. J.Math. 101, 543–565 (1979)

2. Jevtic, M., Pavlovic, M.: Besov–Lipschitz and mean Besov–Lipschitz spaces of holomorphic func-tions on the unit ball. Potential Anal. 38(4), 1187–1206 (2013)

3. Pavlovic, M.: Introduction to Function Spaces on the Disk. Posebna Izdanja [Special Editions],vol. 20, Matematicki Institut SANU, Belgrade (2004)

4. Pavlovic, M.: Green’s formula and the Hardy–Stein identities. Filomat (Niš) 23(3), 135–153 (2009)5. Rudin, W.: Function Theory in the Unit Ball in C

n. Springer, New York (1980)