Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 477–486

On Q_{p} spaces and pseudoanalytic extension

Konstantin M. Dyakonov and Daniel Girela

Steklov Institute of Mathematics, St. Petersburg Branch (POMI) Fontanka 27, St. Petersburg, 191011, Russia; dyakonov@pdmi.ras.ru

Current address: Departament de Matemática Aplicada i Análisi, Universitat de Barcelona

Gran Via 585, E-08071 Barcelona, Spain; dyakonov@mat.ub.es Universidad de Málaga, Departamento de Análisis Matemático, Facultad de Ciencias E-29071 Málaga, Spain; girela@anamat.cie.uma.es

Abstract. For 0 < p < 1, Q_{p} is the space of those functions f which are analytic in the unit disc Δ = {z ∈ C : |z| < 1} and satisfy sup_{|a|<1} ∫∫_{Δ} |f′(z)|^{2}(g(z, a)^{p} dx dy < ∞, where g( · , ·) is the Green function of Δ. In this paper we obtain a new characterization of Q_{p} -functions in terms of pseudoanalytic extension and, as a corollary, we prove that Q_{p} has the K -property of Havin. The latter means that, for any ψ ∈ H^{∞}, the Toeplitz operator T\overline{ψ} maps Q_{p} into itself. This in turn implies (as usual) that Q_{p} also enjoys the f -property, i.e., division by inner factors preserves membership in Q_{p}.

1. Introduction and statement of results

We denote by Δ the unit disc {z ∈ C : |z| < 1} and by H^{p} (0 < p ≤ ∞) the classical Hardy spaces of analytic functions in Δ (see [8] and [13]).

For a ∈ Δ, let φ_{a} denote the Möbius transformation defined by φ_{a}(z) = (z − a)/(1 − \overline{a}z), z ∈ C, and the Green function g( · , ·) of Δ is given by

g(z, a) = log \frac{1}{|φ_{a}(z)|}, a, z ∈ Δ.

For p ≥ 0, we set

Q_{p} = {f : f is analytic in Δ and sup_{a∈Δ} ∫∫_{Δ} |f′(z)|^{2}g^{p}(z, a) dA(z) < ∞}.

1991 Mathematics Subject Classification: Primary 30D45, 30D50.

This work started while the first author was visiting the University of Málaga. He also acknowledges support from the Russian Foundation for Fundamental Studies (grant # 99-01-00103) and from the Spanish Comisión Interministerial de Ciencia y Tecnología (fellowship # SB97-32363432). The second author has been supported in part by a grant from “El Ministerio de Educación y Cultura, Spain” (PB97-1081) and by a grant from “La Junta de Andalucía” (FQM-

Here and throughout, dA is the Lebesgue measure in C. The Q_{p} spaces arose in [2] in connection with Bloch and normal functions and have been studied by several authors (see e.g. [2], [3], [4], [12] and [17]). Observe that Q_{0} is the Dirichlet space 𝒟, while Q_{1} = BMOA, the space of functions f ∈ H^{1} whose boundary values have bounded mean oscillation on ∂Δ (see [5] and [13]). Further, Aulaskari and Lappan proved in [2] that for all p ∈ (1, ∞), the spaces Q_{p} are the same and equal to the Bloch space

ℬ = {f : f is analytic in Δ and sup_{z∈Δ}(1 − |z|^{2})|f′(z)| < ∞}.

On the other hand Aulaskari, Xiao and Zhao showed in [4] that if 0 ≤ p < q ≤ 1 then Q_{p} ⫋ Q_{q}. In particular, we have

𝒟 ⊂ Q_{p} ⊂ BMOA, 0 ≤ p ≤ 1.

The results of [3] (see also [23] and [26] for the case 1 < p < ∞) show that if 0 < p < ∞ and f is an analytic function in Δ then

(1)  f ∈ Q_{p} ⟺ sup_{|a|<1} ∫∫_{Δ} |f′(z)|^{2}(1 − |φ_{a}(z)|^{2})^{p} dA(z) < ∞.

Our main result, stated as Theorem 1 below, is a new characterization of Q_{p} - spaces (0 < p < 1) in terms of pseudoanalytic continuation. We refer to Dyn’kin’s paper [11] for similar descriptions of classical smoothness spaces, as well as for other important applications of the pseudoanalytic extension method.

In what follows, Δ_{−} denotes the region C \ \overline{Δ}, and we write

z^{∗} =^{def}1/\overline{z}, z ∈ C \ \{0\}.

Finally, we need the Cauchy–Riemann operator

\overline{∂} = \frac{∂}{∂\overline{z}} =^{def} \frac{1}{2}(\frac{∂}{∂x} + i\frac{∂}{∂y} ), z = x + iy.

Theorem 1. If 0 < p < 1 and f ∈ ⋂_{0<q<∞} H^{q}, then the following conditions are equivalent.

(i) f ∈ Q_{p}.

(ii) sup_{|a|<1} ∫∫_{Δ} |f′(z)|^{2}((1/|φ_{a}(z)|^{2}) − 1)^{p} dA(z) < ∞. (iii) There exists a function F ∈ C^{1}(Δ_{−}) satisfying

(2)  F (z) = O(1),  as z → ∞,

On Q_{p} spaces and pseudoanalytic extension 479

(3) lim_{r→1^{+}} F (re^{iθ}) = f(e^{iθ}),  a.e. and in L^{q}([−π, π]) for all q ∈ [1, ∞),

and

(4)  sup_{|a|<1} ∫∫_{Δ_{−}} |\overline{∂}F (z)|^{2}(|φ_{a}(z)|^{2} − 1)^{p} dA(z) < ∞.

We remark that our proof of Theorem 1 will show that the equivalence (i) ⟺ (ii) holds for an arbitrary holomorphic function f (without the a-priori assumption that f ∈ ⋂_{0<q<∞} H^{q}).

To describe some consequences of Theorem 1, we have to introduce further terminology. We recall first that, given a function v ∈ L^{∞}(∂Δ), the associated Toeplitz operator T_{v} is defined by

(T_{v}f)(z) = \frac{1}{2πi} ∫_{∂Δ} \frac{v(ζ)f(ζ)}{ζ − z} dζ (f ∈ H^{1}, z ∈ Δ).

Definition 1. A subspace X of H^{1} is said to have the K -property if T\overline{ψ} (X) ⊂ X for any ψ ∈ H^{∞}.

Definition 2. A subspace X of H^{1} is said to have the f -property if h/I ∈ X whenever h ∈ X and I is an inner function with h/I ∈ H^{1} .

These notions were introduced by Havin in [14]. It was also pointed out in [14] that the K -property implies the f -property: indeed, if h ∈ H^{1} , I is inner and h/I ∈ H^{1} then h/I = T\overline{I}h.

Our next result is

Theorem 2. For 0 < p < 1, the space Q_{p} has the K -property.

In view of the above discussion, this immediately yields

Corollary 1. For 0 < p < 1, the space Q_{p} has the f -property.

Since, as we have mentioned above, the Q_{p} spaces (0 < p < 1) are intermediate spaces between the Dirichlet class 𝒟 and BMOA, we wish to remark that both of these endpoint spaces do have the K -property (and hence also the f -property). The case of 𝒟 is covered by results in [14]; see also [9], [15], [16] and [18] for various extensions dealing with Dirichlet-type spaces. The K -property of BMOA can be established along the lines of [14]: given ψ ∈ H^{∞}, the multiplication map T_{ψ} acts boundedly on H^{1} , whence the adjoint operator T\overline{ψ} must act boundedly on BMOA.

Since, for p > 1, Q_{p} = ℬ and ℬ is not contained in H^{1} , it does not make sense to ask for this range of p’s whether or not Q_{p} has the K - (or f -)property. However, let us mention that H^{∞} ∩ ℬ_{0} fails to possess the f -property (here, ℬ_{0} is the subspace of ℬ defined by the corresponding “little oh” condition). This result, due to Anderson [1], can be also deduced from the fact that ℬ_{0} contains an infinite Blaschke product (see [19]). Some other “Bloch-type” subclasses of H^{∞} without the f -property have been exhibited by Vinogradov in [25] (see also [7]).

For further examples of spaces with or without the K - (or f -)property, the reader is referred to [20], [21] and the bibliography therein.

2. Proofs of the results

We begin by showing how Theorem 2 follows from Theorem 1. The proof of Theorem 1 will be presented afterwards.

Proof of Theorem 2. Let 0 < p < 1, f ∈ Q_{p} and ψ ∈ H^{∞}. We have to

show that g =^{def}T\overline{ψ}f is necessarily in Q_{p}.

Since g is the orthogonal projection of f\overline{ψ} onto H^{2} , one has

f\overline{ψ} = g + \overline{h}

for some h ∈ H_{0}^{2} . (Actually, both g and h lie in ⋂_{0<q<∞} H^{q}. To see why, recall that f ∈ BMO, ψ ∈ L^{∞} and use the boundedness properties of the Riesz projection.) Thus,

(5)  g = f\overline{ψ} − \overline{h}  a.e. on ∂Δ.

Now, since f ∈ Q_{p}, Theorem 1 says that there is a function F ∈ C^{1}(Δ_{−}) satisfying (2), (3) and (4). Further, we set, for z ∈ Δ_{−},

Ψ(z) =^{def}\overline{ψ(z^{∗})}, H(z) =^{def}\overline{h(z^{∗})}

and finally

G(z) =^{def}F (z)Ψ(z) − H(z).

This done, we claim that

(6)  G|_{∂Δ} = g

(the boundary values are again taken in the sense of radial convergence a.e. on ∂Δ and in each L^{q} with q < ∞) and

(7)  |\overline{∂}G(z)| ≤ ∥ψ∥_{∞}|\overline{∂}F (z)|,  z ∈ Δ_{−}.

Indeed, (6) follows from (5) and the facts that

F |_{∂Δ} = f,  Ψ|_{∂Δ} = \overline{ψ},  H|_{∂Δ} = \overline{h},

while (7) holds because Ψ and H are holomorphic in Δ_{−}, and so \overline{∂}G = Ψ · \overline{∂}F on Δ_{−}.

Since G is obviously C^{1} -smooth in Δ_{−} and bounded at ∞, we now conclude from (6) and (7) that the analogues of (2), (3) and (4) hold true with G and g in place of F and f. Another application of Theorem 1 yields g ∈ Q_{p}, as desired.

Now it remains to prove Theorem 1. Before doing so, let us recall that if h is an analytic function in Δ then, as usual, we set

M_{2}(r, h) = (\frac{1}{2π} ∫_{−π}^{π} |h(re^{iθ})|^{2} dθ)^{1/2}, 0 < r < 1.

Proof of Theorem 1. (i) ⟺ (ii) Let us borrow the argument used in the proof of Theorem 2.2 of [3]. By (1), it suffices to prove that there exist positive constants A_{p} and B_{p} such that

A_{p} ∫∫_{Δ} |f′(z)|^{2}(\frac{1}{|φ_{a}(z)|^{2}} − 1)^{p} dA(z) ≤ ∫∫_{Δ} |f′(z)|^{2}(1 − |φ_{a}(z)|^{2})^{p} dA(z)

≤ B_{p} ∫∫_{Δ} |f′(z)|^{2}(\frac{1}{|φ_{a}(z)|^{2}} − 1)^{p} dA(z), (8)

for all a ∈ Δ and any f. By a change of variables argument, it is enough to check (8) for a = 0. This is equivalent to

\begin{aligned}A_{p} ∫_{0}^{1} M_{2}(r, f′)^{2}(\frac{1}{r^{2}} − 1)^{p}r dr & ≤ ∫_{0}^{1} M_{2}(r, f′)^{2}(1 − r^{2})^{p}r dr \\ & ≤ B_{p} ∫_{0}^{1} M_{2}(r, f′)^{2}(\frac{1}{r^{2}} − 1)^{p}r dr, \end{aligned}

that is, to

\begin{aligned}A_{p} ∫_{0}^{1} M_{2}(r, f′)^{2}(1 − r^{2})^{p}r^{1−2p} dr & ≤ ∫_{0}^{1} M_{2}(r, f′)^{2}(1 − r^{2})^{p}r dr \\ & ≤ B_{p} ∫_{0}^{1} M_{2}(r, f′)^{2}(1 − r^{2})^{p}r^{1−2p} dr. \end{aligned}

This follows easily, since 0 < p < 1 and M_{2}(r, f′) is an increasing function of r.

Of course, the second inequality in (8) actually holds with B_{p} = 1. It is the first inequality that we were mainly concerned with.

(i) ⇒ (iii) Let 0 < p < 1 and f ∈ Q_{p}. Set
F (z) = f(z^{⋆}), z ∈ Δ_{−}.

It is clear that F is C^{1} -smooth and satisfies (2) and (3). Now let a ∈ Δ; making the change of variables z = w^{⋆} in the integral which appears in (4) and noting that |\overline{∂}F (z)| = |f′(z^{⋆})| |z^{⋆}|^{2} , we obtain

\begin{aligned}∫∫_{Δ_{−}} |\overline{∂}F (z)|^{2}(|φ_{a}(z)|^{2} − 1)^{p} dA(z) & = ∫∫_{Δ} |f′(w)|^{2}(|φ_{a}(w)^{⋆}|^{2} − 1)^{p} dA(w) \\ & = ∫∫_{Δ} |f′(w)|^{2}(\frac{1}{|φ_{a}(w)|^{2}} − 1)^{p} dA(w). \end{aligned}

Then, since (i) ⟺ (ii), (4) follows.

The proof of the remaining implication (iii) ⇒ (ii) makes use of Calderón– Zygmund operators and Muckenhoupt weights. We refer to [22] and [24] for the notion of a Calderón–Zygmund operator, as well as for the basic terminology and facts listed below.

If q > 1 and ω is a positive measurable function on C, then ω is said to be an A_{q} -weight if

A_{q}(ω) =^{def} sup_{Q}[\frac{1}{|Q|} ∫∫_{Q} ω(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q}(ω(z))^{−q′/q} dA(z)]^{q/q′} < ∞.

Here Q ranges over the discs in C, |Q| denotes the area of Q, and q′ = q/(q −1). The A_{2} -condition has a simpler appearance:

A_{2}(ω) =^{def} sup_{Q}[\frac{1}{|Q|} ∫∫_{Q} ω(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q}(ω(z))^{−1} dA(z)] < ∞.

Now if ω is an A_{2} -weight with A_{2}(ω) ≤ α and if T is a Calderón–Zygmund operator, then we have the weighted inequality

(9) ∫∫_{C} |T g(z)|^{2}ω(z) dA(z) ≤ B_{T,α} ∫∫_{C} |g(z)|^{2}ω(z) dA(z), for all g ∈ L^{2}(ω),

where the constant B_{T,α} depends only on α and ∥T ∥_{L^{2}→L^{2}} , the norm of T in the unweighted L^{2} -space.

We are now in a position to complete the proof of Theorem 1.

(iii) ⇒ (ii) Suppose (iii) holds. We shall argue as in the proof of Lemma 7 on p. 154 of [10]. Fix z ∈ Δ and R > 1. In view of (3), the Cauchy–Green formula applied to the function that equals f in Δ and F in Δ_{−} gives

(10)  f(z) = \frac{1}{2πi} ∫_{|ξ|=R} \frac{F (ξ)}{ξ − z} dξ − \frac{1}{π} ∫∫_{1<|ξ|<R} \frac{\overline{∂}F (ξ)}{ξ − z} dA(ξ).

Differentiating (10) and noticing that the arising contour integral is O(1/R), as R → ∞, we obtain

(11)  f′(z) = −\frac{1}{π} ∫∫_{Δ_{−}} \frac{\overline{∂}F (ξ)}{(ξ − z)^{2}} dA(ξ).

Put

(12)  Φ(z) = \{ \begin{matrix}\overline{∂}F (z), & if z ∈ Δ_{−}, \\ 0, & if z ∈ Δ. \end{matrix}

Let S be the Calderón–Zygmund operator defined by

(13)  Sg(z) = p.v. ∫∫_{C} \frac{g(ξ)}{(ξ − z)^{2}} dA(ξ).

Using (11), (12) and (13), we see that

(14)  f′(z) = −\frac{1}{π} (SΦ)(z),  z ∈ Δ.

Given a ∈ Δ, define

(15)  U_{a}(z) = |1 − \frac{1}{|φ_{a}(z)|^{2}} |^{p} = \frac{(1 − |a|^{2})^{p}||z|^{2} − 1|^{p}}{|z − a|^{2p}} ,  z ∈ C.

We shall prove the following result.

Proposition 1. There exists a positive constant α such that, for every a ∈ Δ, U_{a} is an A_{2} -weight with

(16)  A_{2}(U_{a}) ≤ α,  for all a ∈ Δ.

Once Proposition 1 is established, we can proceed as follows. Taking (12)– (15) into account and using inequality (9), with suitable replacements and in conjunction with (16), we get

∫∫_{Δ} |f′(z)|^{2}(\frac{1}{|φ_{a}(z)|^{2}} − 1)^{p} dA(z) = ∫∫_{Δ} |f′(z)|^{2}U_{a}(z) dA(z)

= \frac{1}{π^{2}} ∫∫_{Δ} |(SΦ)(z)|^{2}U_{a}(z) dA(z)

≤ \frac{1}{π^{2}} ∫∫_{C} |(SΦ)(z)|^{2}U_{a}(z) dA(z)

≤ C ∫∫_{C} |Φ(z)|^{2}U_{a}(z) dA(z) (17)

= C ∫∫_{Δ_{−}} |\overline{∂}F (z)|^{2}U_{a}(z) dA(z)
≤ C ∫∫_{Δ_{−}} |\overline{∂}F (z)|^{2}(|φ_{a}(z)|^{2} − 1)^{p} dA(z),

where C > 0 is a constant independent of a ∈ Δ. To verify the last step, note that |φ_{a}(z)| > 1 for z ∈ Δ_{−}. The resulting inequality from (17) shows that (ii) follows from (4). Consequently, it only remains to prove Proposition 1.

Proof of Proposition 1. Given a ∈ Δ, set

(18)  V_{a}(z) = \frac{||z|^{2} − 1|^{p}}{|z − a|^{2p}} ,  z ∈ C.

It is clear that

(19)  A_{2}(U_{a}) = A_{2}(V_{a}),  for all a ∈ Δ.

We can write

V_{a}(z) = W (z)Y_{a}(z),

where

W (z) = ||z|^{2} − 1|^{p}, Y_{a}(z) = \frac{1}{|z − a|^{2p}} .

It is well known (see e.g. [22, p. 218]) that, since 0 < p < 1, the weight W_{0}(z) = |z|^{−2p} satisfies the A_{s} -condition for all s > 1. Since the Y_{a} are translates of W_{0} , it follows that for every s > 1 there exists a constant α_{s} > 0 such that

(20)  A_{s}(Y_{a}) ≤ α_{s},  for all a ∈ Δ.

Take and fix r ∈ (1, 1/p), and let Q be any disc. Then, for every a ∈ Δ, we have

[\frac{1}{|Q|} ∫∫_{Q} V_{a}(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q} \frac{1}{V_{a}(z)} dA(z)] = [\frac{1}{|Q|} ∫∫_{Q} W (z)Y_{a}(z) dA(z)]

× [\frac{1}{|Q|} ∫∫_{Q} \frac{1}{W (z)Y_{a}(z)} dA(z)]

≤ [sup_{z∈Q} W (z)][∫∫_{Q} Y_{a}(z)\frac{dA(z)}{|Q|}] (21)

× [∫∫_{Q} \frac{1}{W (z)^{r}} \frac{dA(z)}{|Q|}]^{1/r}

× [∫∫_{Q} \frac{1}{Y_{a}(z)^{r′}} \frac{dA(z)}{|Q|}]^{1/r′}.

Now it can be easily proved by direct calculation that there exists a positive constant C such that

[sup_{z∈Q} W (z)][\frac{1}{|Q|} ∫∫_{Q} \frac{1}{W (z)^{r}} dA(z)]^{1/r} ≤ C, for any Q.

Then (21) implies that, for every disc Q and every a ∈ Δ,

(22)  \begin{aligned}[\frac{1}{|Q|} ∫∫_{Q} & V_{a}(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q} \frac{1}{V_{a}(z)} dA(z)] \\ & ≤ C[\frac{1}{|Q|} ∫∫_{Q} Y_{a}(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q} \frac{1}{Y_{a}(z)^{r′}} dA(z)]^{1/r′}. \end{aligned}

Next, we set

s = 1 + \frac{1}{r′} ,

(so that s/s′ = 1/r′) and rewrite (22) as

\begin{aligned}[\frac{1}{|Q|} ∫∫_{Q} & V_{a}(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q} \frac{1}{V_{a}(z)} dA(z)] \\ & ≤ C[\frac{1}{|Q|} ∫∫_{Q} Y_{a}(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q} \frac{1}{Y_{a}(z)^{s′/s}} dA(z)]^{s/s′}. \end{aligned}

Together with (20) this yields

[\frac{1}{|Q|} ∫∫_{Q} V_{a}(z) dA(z)][\frac{1}{|Q|} ∫∫_{Q} \frac{1}{V_{a}(z)} dA(z)] ≤ Cα_{s},

for every disc Q and every a ∈ Δ. Hence

A_{2}(V_{a}) ≤ Cα_{s}, for every a ∈ Δ.

In view of (19), this gives (16) and finishes the proof.

After this work had been completed, Professor Jie Xiao kindly informed us of his (unpublished) proof of Corollary 1, based on ideas different from ours.

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Received 30 April 1999