GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 3, 1996, 217-232

THE WEIGHTED BMO CONDITION AND A CONSTRUCTIVE DESCRIPTION OF CLASSES OF

ANALYTIC FUNCTIONS SATISFYING THIS CONDITION

L. CHIKVINIDZE

Abstract. The problem of local polynomial approximation of analytic functions prescribed in finite domains with a quasiconformal boundary is investigated in weighted plane integral metrics; a constructive description of the class of analytic functions satisfying a weak version of the known BMO condition is obtained.

The First results of the investigation of the problem (formulated by V. I. Belyi) dealing with a local polynomial approximation of analytic functions prescribed in finite domains with quasiconformal boundary have been described in [1, 2] for weighted plane integral metrics. This problem is investigated in [3] for the nonweighted case, where a constructive description of Hölder classes as well as of some other classes of analytic functions has been obtained. In the present paper we continue the investigation of the above-mentioned problem for the weighted case; moreover, a constructive description of one more class of analytic functions in weighted plane integral metrics is obtained.

1. Notation and Definitions. The Basic Results

Let G be the domain with a quasiconformal boundary ∂G = Γ, and let y = y(ζ)− be a quasiconformal reflection across the curve Γ [4]. We will be concerned only with the special, so-called canonical, quasiconformal reflection (see relations (2.1) and (2.2)). Let w be some weight function (i.e., nonnegative and measurable) defined in the domain Γ. Let us introduce the

217 notation

1991 Mathematics Subject Classification. 30E10, 30C62, 41A10.

Key words and phrases. Domain with quasiconformal boundary, weighted local polynomial approximation, weak versions of the Muckenhoupt and the BMO conditions.

1072-947X/96/0500-0217$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

H′(G) = {f : f holomorphic in G},

L_{p}(G, w) = {f : |f|^{p}w ∈ L_{1}(G)}, H′_{p}(G, w) = L_{p}(G, w) ∩ H′(G) (p ≥ 1).

Furthermore, let σ be the plane Lebesgue measure, and let μ be the Borel measure defined by the equality

μ(E) = ∫∫_{E} w(z)dσ_{z} (E ⊂ G).  (1.1)

The integral with respect to the measure μ of the function f will be denoted by the symbols

∫∫_{E} f(z)dμ_{z} = ∫∫_{E} fdμ.

If μ = σ, then (in cases where this does not cause misunderstanding) we shall use the brief notation

∫∫_{E} f(z)dσ_{z} = ∫∫_{E} f.

Let Q = Q(z, r) be an open square with center at the point z, whose sides are parallel to the coordinate axes and have the length r, and let

F (G) = {Q = Q(z, r) : z ∈ Γ, r > 0}.

For a square Q denote |Q| = σ(Q).

Let the function f and the weight function w be defined in the domain G, let the measure μ be defined by equality (1.1), and

f_{μ,Q∩G} = \frac{1}{μ(Q ∩ G)} ∫∫_{Q∩G} fdμ.  (1.2)

We say that f satisfies the weighted BMO condition BMO_{p}(G, w) (briefly, f ∈ BMO_{p}(G, w)), if

sup_{Q∈F (G)}(\frac{1}{μ(Q ∩ G)}∫∫_{Q∩G} |f − f_{μ,Q∩G}|^{p}dμ)^{\frac{1}{p}} =^{df} ∥f∥_{BMO_{p}(G,w)}< ∞. (1.3)

When p = 1 and w(z) = 1 everywhere in G, we shall use the usual notation f ∈ BMO(G) and ∥f∥_{BMO(G)} respectively.

The BMO(G) condition is a weaker analogue of the well-known BMO (bounded mean oscillation) condition (see, e.g., [5, Ch. VI]).

Next, we say that the weight function w given in the domain G satisfies the condition A_{p}(F (G)) (1 < p < ∞) (briefly w ∈ A_{p}(F (G))), if (assuming 0 · ∞ = 0)

sup_{Q∈F (G)} (\frac{1}{|Q ∩ G|} ∫∫_{Q∩G} w)(\frac{1}{|Q ∩ G|} ∫∫_{Q∩G} w^{− \frac{1}{p−1}} )^{p−1} < ∞.

The condition A_{p}(F (G)), introduced for the first time in [6] (with the unit circle as G), is a weaker analogue of the well-known Muckenhoupt condition (A_{p}) [7].

Let z_{0} ∈ Γ, ρ_{n}(z_{0}) (n ∈ N) be the distance from the point z_{0} to the external level line Γ_{1+1/n} of G, u(z_{0}, r) = {z : |z − z_{0}| < r}, c_{0} > 0. Set

G(z_{0}, c_{0}) = {z ∈ G : |ζ − z| ≥ c_{0}|ζ − z_{0}| ∀ζ ∈ CG},
G_{n}(z_{0}, c_{0}) = G(z_{0}, c_{0}) ∪ {u(z_{0}, ρ_{n}(z_{0})) ∩ G}.

The set G(z_{0}, c_{0}) is a kind of a “nontangential” subset of G with the vertex at the point z_{0}.

In the sequel, for brevity we shall write

( ∫∫_{G_{n}(z_{0},c_{0})} |f(z) − P_{n}(z)|^{p}w(z)dσ_{z})^{\frac{1}{p}} =^{df} ∥f − P_{n}∥_{z_{0},p,w}.

Let us now formulate the basic results in which G denotes a finite domain with a quasiconformal boundary Γ, the weight function w ∈ A_{p}(F (G)) (1 < p < ∞), and μ is the measure defined by equality (1.1).

Theorem 1. For the function f to belong to the class BMO_{p}(G, w) ∩ H′(G) (neglecting its values on the set of measure zero), it is necessary and sufficient that a sequence of algebraic polynomials P_{n} of order not higher than n exist such that for all z_{0} ∈ Γ and n ∈ N the relation

∥f − P_{n}∥_{z_{0},p,w} ≤ c(c_{0})(μ\{u(z_{0}, ρ_{n}(z_{0})) ∩ G\})^{\frac{1}{p}}  (1.4)

holda, where the constant c(c_{0}) does not depend on z_{0} and n.

Theorem 2. Let f ∈ H′(G). The following conditions are equivalent:

(a) f ∈ BMO(G);

(b) f ∈ BMO_{p}(G, w).

This theorem is an analogue of the well-known John and Nirenberg theorem (see, e.g., [5]).

Obviously, Theorem 2 allows us to formulate Theorem 1 as follows:

Theorem 1^{∗}. Theorem 1 remains valid when the class BMO_{p}(G) is replaced by the class BMO(G).

Thus we have given the constructive description of the class of functions BMO(G) ∩ H′(G) in the weighted plane integral metrics.

2. Auxiliary Results

Let G be the domain with a quasiconformal boundary Γ, and let 0 ∈ G, y = y(ζ) be a quasiconformal reflection across the curve Γ [4]. As follows from the Ahlfors theorem [4] (see also [8]), the reflection y = y(ζ) can always be chosen to be canonical in the sense that it is differentiable for ζ ∉ Γ, and for any fixed sufficiently small δ > 0 it will satisfy the relations

|yζ̄(ζ)| ≍ M, |y_{ζ}(ζ)| ≼ M, δ < |ζ| < 1/δ, ζ ≠ Γ,  (2.1)
|yζ̄(ζ)| ≼ M|ζ|^{−2}, |y(ζ) ≍ | M|ζ|^{−1}, |ζ| ≤ δ, |ζ| ≥ \frac{1}{δ},  (2.2)

where M = M(δ, Γ) is a constant depending only on δ and Γ.

The symbol A ≼ B for the numbers A and B depending on some parameters denotes that A ≤ cB, where c = const > 0 does not depend on those parameters; the symbol A ≽ B means that B ≼ A; A ≍ B if simultaneously A ≼ B and A ≽ B.

Let w ∈ A_{p}(F (G)) (1 < p < ∞), and let y = y(ζ) be a canonical quasiconformal reflection across the curve Γ = ∂G. Let us introduce the notation

w^{∗}(z) = \begin{cases}w(z) & for z ∈ G, \\ w(y(z)) & for z ∉ G, \end{cases} μ^{∗}(E) = ∫∫_{E} w^{∗}dσ.  (2.3)

It is clear that if E ⊂ G, then μ(E) = μ^{∗}(E). Suppose further that

ρ(E_{1}, E_{2}) = inf {|z_{1} − z_{2}| : z_{1} ∈ E_{1}, z_{2} ∈ E_{2}},
diam E = sup {|z_{1} − z_{2}| : z_{1}, z_{2} ∈ E},
F (G, k) = {Q : diam Q ≥ kρ(Q, Γ), Q ∩ G ≠ ∅} (k > 0). (2.4)

Let w ∈ A_{p}(F (G)) (1 < p < ∞), and let w^{∗}(z) be the function defined by equality (2.3). Then owing to relations (2.1) and (2.2), we can conclude that for all Q ∈ F (G, k), diam Q < k_{0} (k_{0}, k > 0 are arbitrary fixed numbers) the inequality

(\frac{1}{|Q|} ∫∫_{Q} w^{∗})(\frac{1}{|Q|} ∫∫_{Q} w^{∗− \frac{1}{p−1}} )^{p−1} ≤ c(k, k_{0}) < ∞  (2.5)

holds, where c(k, k_{0}) is a constant independent of Q.

Lemma 1 ([9], [10]). Let w ∈ A_{p}(F (G)), and let w^{∗}(z) be the function defined by equality (2.3). There exist numbers 0 < δ = δ(k, k_{0}) < 1, 0 < ε = ε(k, k_{0}) < 1 such that for every e ⊂ Q the inequality |e| < δ|Q| implies

∫∫_{e} w^{∗} < ε ∫∫_{Q} w^{∗}

for all Q ∈ F (G, k), diam Q < k_{0}.

Next, by virtue of the Hölder inequality and relation (2.5), we find that for all Q(z, r), Q(z, R) ∈ F (G, k) (0 < r ≤ R ≤ k_{0} < ∞) the inequality

(\frac{r}{R})^{2} ≤ c(k, k_{0})\frac{μ^{∗}(Q(z, r))}{μ^{∗}(Q(z, R))}  (2.6)

holds, where μ^{∗} is defined by equality (2.3).

In particular, it follows from (2.6) that μ^{∗} satisfies the known “doubling” condition

μ^{∗}(2Q) ≼ μ^{∗}(Q), (Q ∈ F (G, k), diam Q < k_{0}).  (2.7)

Let us now prove that for all squares Q ∈ F (G) (diam Q < k_{0}) the relation

μ(Q ∩ G) ≽ μ^{∗}(Q)  (2.8)

holds, where μ and μ^{∗} are defined respectively by equalities (1.1) and (2.3).

Indeed, let Q = Q(z_{0}, r) (z_{0} ∈ Γ, r > 0), diam Q < k_{0}. Owing to relations (2.1) and (2.2), we get

μ^{∗}(Q(z_{0}, (1/M)r) ∩ CG) ≼ c(M)μ^{∗}(Q(z_{0}, r) ∩ G) = c(M)μ(Q ∩ G),

where M > 1 is the constant from (2.1) and (2.2), and CG is the complement to the domain G. But then, using the “doubling” condition (2.7), we get

μ^{∗}(Q) ≼ μ^{∗}(Q(z_{0}, (1/M)r) ∩ G) + μ^{∗}(Q(z_{0}, (1/M)r) ∩ CG) ≼

≼ c(M)μ^{∗}(Q(z_{0}, r) ∩ G) ≼ μ(Q ∩ G).

Let w be a weight function, and let μ^{∗} be the measure defined by equality (2.3). Let f be a function given in the domain G, and let Q be a square. Introduce the notation

f^{∗}(z) = \begin{cases}f(z) & for z ∈ G, \\ f(y(z)) & for z ∉ G, \end{cases} f_{μ^{∗},Q}^{∗} = \frac{1}{μ^{∗}(Q)} ∫∫_{Q} f^{∗}dμ^{∗}. (2.9)

In the case of the Lebesgue measure σ, we shall use f_{Q}^{∗} instead of f_{σ^{∗},Q}^{∗}.

Lemma 2. Let w be some weight function, p > 1, f ∈ BMO_{p}(G, w). Then for all squares Q ∈ F (G, k), diam Q < k_{0} (k, k_{0} > 0 are fixed numbers) the relation

(\frac{1}{μ^{∗}(Q)} ∫∫_{Q} |f^{∗} − f_{μ^{∗},Q}^{∗}|^{p}dμ^{∗})^{\frac{1}{p}} ≤ c^{∗}∥f∥_{BMO_{p}(G,w)}  (2.10)

holds, where c^{∗} is a constant independent of Q, p, f, and w.

Proof. Assume first that Q ∈ F (G), diam Q < k_{0}. Let M > 1 be the number from relations (2.1) and (2.2), and let MQ be the square obtained by an M-fold increase of the square Q. It follows from the “doubling” condition (2.7) that

μ^{∗}(Q) ≽ c(M)μ^{∗}(MQ) ≥ c(M)μ(MQ ∩ G).  (2.11)

On account of relations (2.1) and (2.2) we obtain

( ∫∫_{Q∩CG} |f^{∗} − f_{μ,MQ}|^{p}dμ^{∗})^{\frac{1}{p}} ≼ M^{2}( ∫∫_{MQ∩G} |f − f_{μ,MQ}|^{p}dμ)^{\frac{1}{p}} . (2.12)

Hence, using the Minkowsky inequality and relations (2.11), (2.12), and (1.3), we obtain

(\frac{1}{μ^{∗}(Q)} ∫∫_{Q} |f^{∗} − f_{μ^{∗},Q}^{∗}|^{p}dμ^{∗})^{\frac{1}{p}} ≼ (\frac{2}{μ^{∗}(Q)} ∫∫_{Q} |f^{∗} − f_{μ,MQ∩G}|^{p}dμ^{∗})^{\frac{1}{p}} ≼

≼ (\frac{M^{2} + 1}{c(M)} \frac{1}{μ(MQ ∩ G)} ∫∫_{MQ∩G} |f − f_{μ,MQ∩G}|^{p}dμ)^{\frac{1}{p}} ≼ c^{∗}∥f∥_{BMO_{p}(G,w)}.

Thus we have proved that inequality (2.10) is true for all Q ∈ F (G), diam Q < k_{0}. Using the “doubling” condition (2.7), it is not difficult to show that (2.10) holds for all Q ∈ F (G, k) as well.

Lemma 3. Let f ∈ BMO(G) be an analytic function in the domain G, ζ ∈ G, Q = Q(ζ, a) ∉ F (G, k) (k ≤ 1), Q ⊂ G. Then

|f(ζ) − f(z)| ≤ 4∥f∥_{BMO(G)} ∀(z ∈ Q(ζ, a)).

Proof. Since Q = Q(ζ, a) ∉ F (G, k) (k ≤ 1), it follows from the definition of the set F (G, k) (see (2.4)) that diam Q < kρ(Q, Γ) (k ≤ 1). Hence

|ζ − z| ≤ \frac{1}{2} diam Q < \frac{1}{2}kρ(Q, Γ) < \frac{1}{2}kρ(ζ, Γ) < \frac{1}{2}ρ(ζ, Γ)

for all z ∈ Q. Then assuming for brevity that ρ(ζ, Γ) = ρ, owing to the mean value theorem and the condition f ∈ BMO(G), we obtain

|f(z) − f(ζ)| ≤ \frac{1}{|u(z, ρ/2)|} ∫∫_{u(z,ρ/2)} |f(ξ) − f(ζ)|dσ_{ξ} ≤

≤ \frac{4}{|u(ζ, ρ)}| ∫∫_{u(ζ,ρ)} |f(ξ) − f_{u(ζ,ρ)}|dσ_{ξ} ≤ 4∥f∥_{BMO(G)}.

for all z ∈ Q(ζ, a).

Lemma 4. Let f ∈ BMO(G) be an analytic function in the domain G, and let f^{∗} and f_{Q}^{∗} be defined by equalities (2.9) (the case μ = σ), Q ∈ F (G, k) (k ≤ 1), α > c^{∗}∥f∥_{BMO(G)} (c^{∗} is a constant from (2.10)). Then there exists at most a countable set of nonintersecting squares A = {Q_{j}} such that Q_{j} ∈ F (G, k), and

(1) |f^{∗}(z) − f_{Q}^{∗}| ≤ 12α ∀(z ∈ Q\Q_{j}, Q_{j} ∈ A); (2.13)

(2) α ≤ \frac{1}{|Q_{j}|} ∫∫_{Q_{j}} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} < 4α ∀(Q_{j} ∈ A); (2.14)

(3) ∑_{Q_{j}∈A} |Q_{j}| ≤ \frac{1}{α}∥f∥_{BMO(G)}|Q|. (2.15)

Proof. It is obvious that the conditions (2.10) and α > c^{∗}∥f∥_{BMO(G)} yield

\frac{1}{|Q|} ∫∫_{Q} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} < α.

Let Q^{∗} be the square obtained by partitioning Q into four equal squares. In the case Q^{∗} ∈ F (G, k) we insert Q^{∗} in A if ∫∫_{Q^{∗}} |f(z)−f_{Q}|dσ_{z} ≥ α|Q^{∗}|,

while when the opposite inequality holds we again partition Q^{∗} in four equal squares and argue as above.

Let us show that the squares Q_{j} ∈ A obtained in such a way satisfy all the requirements of Lemma 2.

Let ζ ∈ {Q ∩ G}\ ∪ {Q_{j} : Q_{j} ∈ A}. Then, obviously, there exist the squares Q_{1} and Q_{2} from the above-mentioned partitioning such that ζ ∈ Q_{1} ⊂ Q_{2}, Q_{1} ∉ F (G, k), Q_{2} ∈ F (G, k), and

\frac{1}{|Q_{2}|} ∫∫_{Q_{2}} |f^{∗}(z) − f_{Q}^{∗}| < α,

whence it follows that

\frac{1}{|Q_{1}|} ∫∫_{Q_{1}} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} < \frac{4}{|Q_{2}|} ∫∫_{Q_{2}} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} < 4α.

Then, denoting by z_{1} the center of the square Q_{1} and using Lemma 3 and the mean value theorem, we obtain

|f(ζ) − f_{Q_{1}}| ≤ |f(ζ) − f(z_{1})| + |f(z_{1}) − f_{Q_{1}}| ≤

≤ 4∥f∥_{BMO(G)} + \frac{2}{|Q_{1}|} ∫∫_{Q_{1}} |f(z) − f_{Q_{1}}|dσ_{z} ≤ 12α.

Thus the validity of the first requirement of Lemma 2 is proved.

Further, it is evident that the left-hand side of the “double” inequality (2.14) holds for all Q_{j} ∈ A. Let us show that the right-side of that inequality is also valid.

Let Q_{j}^{∗} be a square whose partitioning into four equal squares gives the square Q_{j} ∈ A. Clearly, Q_{j}^{∗} ⊃ Q_{j}, and

\frac{1}{|Q_{j}^{∗}|} ∫∫_{Q_{j}^{∗}} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} < α.

Taking into account the above inequality, we obtain

\frac{1}{|Q_{j}|} ∫∫_{Q_{j}} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} < \frac{4}{|Q_{j}^{∗}|} ∫∫_{Q_{j}^{∗}} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} < 4α.

Thus relation (2.14) is proved.

Finally, using the already proven relation (2.14) and inequality (2.10) (the case where μ = σ is the Lebesgue measure), we get

| ∪_{A} Q_{j}| = ∑_{Q_{j}∈A} |Q_{j}| ≤ \frac{1}{α} ∑_{Q_{j}∈A} ∫∫_{Q_{j}} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} ≤

≤ \frac{1}{α} ∫∫_{Q} |f^{∗}(z) − f_{Q}^{∗}|dσ_{z} ≤ \frac{1}{α}|Q| ∥f∥_{BMO(G)}.

The proof of the following lemma can be found in [11]. Let us formulate it in a way convenient for us.

Lemma 5. Let G be a finite domain with a quasiconformal boundary Γ, z_{0} ∈ Γ, n, m ∈ N, n > m. Then

(\frac{m}{n})^{2} ≼ \frac{ρ_{n}(z_{0})}{ρ_{m}(z_{0})} ≼ (\frac{m}{n})^{β},  (2.16)

where β = β(G) > 0 is a constant depending only on G.

In particular, from relation (2.16) we obtain the known inequality

ρ_{n}(z_{0}) ≽ (\frac{1}{n})^{2}.  (2.17)

Lemma 6. Let G be a finite domain with a quasi-conformal boundary Γ, p > 1, w ∈ A_{p}(F (G)), z_{0} ∈ Γ, u(z_{0}, r) = {z : |z − z_{0}| < r}, let μ be a measure defined by equality (1.1), and let {Π_{n}(z)}_{n=1}^{∞} be a sequence of algebraic polynomials of order not higher than n such that

∥Π_{n}∥_{z_{0},p,w} ≤ c_{1}(μ\{u(z_{0}, ρ_{n}(z_{0}))\})^{\frac{1}{p}} ,

where c_{1} is a constant not depending on z_{0} and n.

Then for all z ∈ u(z_{0}, ρ_{n}(z_{0})) we have the inequality

|Π′_{n}(z)| ≤ c_{2}|ρ_{n}(z_{0})|^{−1},

where c_{2} is a constant not depending on z_{0} and n.

This lemma is the analogue of the well-known theorem on the derivatives of algebraic polynomials [12, p.420], [13] which can be proved analogously to the result of [3, p.14].

3. Proofs of the Basic Results

Proof of Theorem 1. For brevity we shall use the notation μ{u(z_{0}, t) ∩ G} = μ(z_{0}, t) (t ≥ 0).

Let us prove first the necessity. Assume that f ∈ BMO_{p}(G, w) ∩ H′(G) and let us show that relation (1.4) holds.

Let n ∈ N, z_{0} ∈ Γ, Q = Q(z_{0}, ρ_{n}(z_{0})), and μ and μ^{∗} be the measures defined by the equalities (1.1) and (2.3). Relations (2.8), (2.6), and (2.17) yield

μ(Q ∩ G) ≽ μ(Q^{∗}) ≽ [ρ_{n}(z_{0})]^{2} ≽ (\frac{1}{n})^{4},

but then, obviously, we shall have

f_{μ,Q∩G} ≼ \frac{1}{μ(Q ∩ G)} ∫∫_{G} |f|dμ ≼ c(f, μ)(\frac{1}{n})^{−4}.  (3.1)

Clearly, f ∈ H′_{p}(G, w). But then, repeating the arguments (and taking into account (3.1)) cited in [2, pp. 174, 182], we can see that there exists a sequence of algebraic polynomials P_{n} of order not higher than n, such that

∥f − P_{n}∥_{z_{0},p,w} ≤ cρ_{n}(z_{0})(μ(z_{0}, ρ_{n}(z_{0})))^{\frac{1}{p}} ×

× ∫_{ρ_{n}(z_{0})}^{∞} \frac{σ_{p}(f − f_{μ,Q∩G}, w, z_{0}, t)}{t^{2}μ^{1/p}(z_{0}, t)}dt.  (3.2)

Now let us estimate the value σ_{p}(f − f_{μ,Q∩G}, w, z_{0}, t) for t ≥ ρ_{n}(z_{0}). Let Q_{m} = Q(z_{0}, 2^{m}ρ_{n}(z_{0})) (m ∈ N), Q_{0} = Q. Using relations (2.8) and

(2.3) for all m ≥ 1, we get

μ(Q_{m−1} ∩ G) ≽ μ^{∗}(Q_{m−1}) ≽ μ^{∗}(2Q_{m−1}) = μ^{∗}(Q_{m}) ≥ μ(Q_{m} ∩ G).

Then, on account of (1.3), we have

|f_{μ,Q_{m}∩G}−f_{μ,Q_{m−1}∩G}|≼ \frac{1}{μ(Q_{m} ∩ G)} ∫∫_{Q_{m}∩G}|f −f_{μ,Q−m∩G}|dμ≤∥f∥_{BMO_{p}(G,w)}.

Thus, using the Minkowsky inequality and relation (1.3) for all k ≥ 1, we obtain

σ_{p}(f − f_{μ,Q∩G}, w, z_{0}, 2^{k}ρ_{n}(z_{0})) ≤ ( ∫∫_{Q_{k}∩G} |f − f_{μ,Q_{k}∩G}|^{p}dμ)^{\frac{1}{p}} +

+ ∑_{m=1}^{k} |f_{μ,Q_{m}∩G} − f_{μ,Q_{m−1}∩G}|( ∫∫_{Q_{k}∩G} dμ)^{\frac{1}{p}} ≤

≤ ∥f∥_{BMO_{p}(G,w)}(1 + k)( ∫∫_{Q_{k}∩G} dμ)^{\frac{1}{p}} =

= ∥f∥_{BMO_{p}(G,w)}(1 + log_{2} \frac{2^{k}ρ_{n}(z_{0})}{ρ_{n}(z_{0})}) · μ^{\frac{1}{p}} (z_{0}, 2^{k}ρ_{n}(z_{0})),

whence, obviously,

σ_{p}(f − f_{μ,Q∩G}, w, z_{0}, t) ≤ ∥f∥_{BMO_{p}(G,w)}(1 + log_{2} \frac{t}{ρ_{n}(z_{0})}) · μ^{\frac{1}{p}} (z_{0}, t)

for all t ≥ ρ_{n}(z_{0}).

Consequently, owing to (3.2), we have

∥f − P_{n}∥_{z_{0},p,ω} ≼

≼ ∥f∥_{BMO_{p}(G,w)}ρ_{n}(z_{0})(μ(z_{0}, ρ_{n}(z_{0})))^{\frac{1}{p}} ∫_{ρ_{n}(z_{0})}^{∞} \frac{(1 + log_{2} \frac{t}{ρ_{n}(z_{0})})}{t^{2}}dt =

= ∥f∥_{BMO_{p}(G,w)}(μ(z_{0}, ρ_{n}(z_{0})))^{\frac{1}{p}} ∫_{1}^{∞} \frac{(1 + log_{2} τ)}{τ^{2}}dτ ≼

≼ ∥f∥_{BMO_{p}(G,w)}(μ\{u(z_{0}, ρ_{n}(z_{0})) ∩ G\})^{\frac{1}{p}} .

Assume now that relation (1.4) is fulfilled for some function f given in G. Then, obviously, f ∈ L_{p}(G, w). Moreover, f ∈ H′(G) if we neglect the values of the function f on the set of measure zero. Indeed, if z ∈ G is an arbitrary point and z^{∗} ∈ Γ is a point such that |z − z^{∗}| = ρ(z, Γ), then it is easy to check that u(z, ρ(z, Γ)) ⊂ G_{n}(z^{∗}, c_{0}) (c_{0} ≤ \frac{1}{2}).Taking into account relation (1.4) it is not difficult to prove that the polynomials P_{n} converge uniformly on u(z, \frac{1}{2}ρ(z, Γ)). Cleary, the limiting analytic function coincides with the functions f a.e.

Further, let z_{0} ∈ Γ, r > 0, Q = Q(z_{0}, r), and let n ∈ N be a number such that

ρ_{2^{n+1}}(z_{0}) < r ≤ ρ_{2^{n}}(z_{0}).  (3.3)

Using the Minkowsky inequality, we can see that

( ∫∫_{Q∩G} |f(z) − f_{μ,Q∩G}|^{p}dμ_{z})^{\frac{1}{p}} ≤ ( ∫∫_{Q∩G} |f(z) − P_{2^{n}}(z)|^{p}dμ_{z})^{\frac{1}{p}} +

( ∫∫_{Q∩G} |P_{2^{n}}(z) − f_{μ,Q∩G}|^{p}dμ_{z})^{\frac{1}{p}} =^{df} I_{1} + I_{2}.

By virtue of (1.4),

I_{1} ≤ const μ^{1/p}(z_{0}, ρ_{n}(z_{0})).  (3.4)

It remains to estimate I_{2}. Evidently,

I_{2} ≼ μ^{1/p}(Q ∩ G) · max_{z∈Q∩G} |P_{2^{n}}(z) − f_{μ,Q∩G}|.  (3.5)

Then it is obvious that for all z ∈ Q ∩ G

|P_{2^{n}}(z) − f_{μ,Q∩G}| ≤ \frac{1}{μ(Q ∩ G)} ∫∫_{Q∩G} |f(ζ) − P_{2^{n}}(z)|dμ_{ζ} ≤

≤ \frac{1}{μ(Q ∩ G)} ∫∫_{Q∩G} |f(ζ) − P_{2^{n}}(ζ)|dμ_{ζ} +

+ max_{ζ∈Q∩G} |P_{2^{n}}(ζ) − P_{2^{n}}(z)| =^{df} I′_{2} + I′′_{2}.

Using the Hölder inequality and relation (1.4), we obtain

I′_{2} ≤ \frac{1}{μ(Q ∩ G)}( ∫∫_{Q∩G} |f − P_{2^{n}}|^{p}dμ)^{\frac{1}{p}} ( ∫∫_{Q∩G} dμ)^{1− \frac{1}{p}} ≼ const . (3.6)

To estimate I′′_{1}, let us consider the polynomial

Π_{2^{k}}(z) = P_{2^{k}}(z) − P_{2^{k−1}}(z) (k ≥ 1).

By (2.15), the Minkowsky inequality and relation (1.4) imply

∥Π_{2^{k}}∥_{z_{0},p,w} ≤ ∥f − P_{2^{k}}∥_{z_{0},p,w} + ∥f − P_{2^{k−1}}∥_{z_{0},p,w} ≼ μ^{1/p}(z_{0}, ρ_{n}(z_{0})).

But then, according to Lemma 6, we have

|Π′_{2^{k}}(z)| ≼ |ρ_{2^{k}}(z_{0})|^{−1} ∀(z ∈ u(z_{0}, ρ_{2^{k}}(z_{0})), k ≥ 1).

Hence, taking into account (2.15), we obtain

|P_{2^{n}}(ζ) − P_{2^{n}}(z)| = |(P_{1}(ζ) − P_{1}(z)) +

+ ∑_{k=1}^{n} (Π_{2^{k}}(ζ) − Π_{2^{k}}(z))| ≼ |ζ − z| + ∑_{k=1}^{n} ∫_{[ζ,z]} |Π′_{2^{k}}(ξ)||dξ| ≼

≼ |ζ − z|(1 + ∑_{k=1}^{n} |ρ_{2^{k}}(z_{0})|^{−1}) ≼ (ρ_{2^{n}}(z_{0}) + ∑_{k=1}^{n} \frac{ρ_{2^{n}}(z_{0})}{ρ_{2^{k}}(z_{0})}) ≼

≼ (ρ_{2^{n}}(z_{0}) + ∑_{k=1}^{n} (\frac{1}{2^{n−k}} )^{β}) ≤ const

for all z, ζ ∈ Q ∩ G.

This means that I′′_{2} ≤ const. But then, taking into account (3.6) and (3.5), we get

I_{2} ≤ const μ^{1/p}(z_{0}, ρ_{n}(z_{0})),

which, with regard to (3.4), completes the proof of Theorem 1.

Proof of Theorem 2. Let us prove first that (b)⇒(a). Let ω ∈ A_{p}(F (G)) (1 < p < ∞), f ∈ BMO_{p}(G, ω) and let us show that f ∈ BMO(G).

Indeed,

\frac{1}{|Q ∩ G|} ∫∫_{Q∩G} |f − f_{Q∩G}|dσ ≤

≤ \frac{2}{|Q ∩ G|}( ∫∫_{Q∩G} |f − f_{μ,Q∩G}|^{p}wdσ)^{\frac{1}{p}} ( ∫∫_{Q∩G} w^{− \frac{1}{p−1}} dσ)^{\frac{p−1}{p}} ≼

≼ (\frac{1}{μ(Q ∩ G)} ∫∫_{Q∩G} |f − f_{μ,Q∩G}|^{p}dμ)^{\frac{1}{p}} ×

×(\frac{1}{|Q ∩ G|} ∫∫_{Q∩G} w)^{\frac{1}{p}} (\frac{1}{|Q ∩ G|} ∫∫_{Q∩G} w^{− \frac{1}{p−1}} )^{\frac{p−1}{p}} ≤ const ∥f∥_{BMO(G)}.

It remains to prove that (a)⇒(b).

Let f ∈ BMO(G) be an analytic function in the domain G, let f^{∗} be a function defined by the equality (2.9), ω ∈ A_{P} (F (G)) (1 < p < ∞), and let μ^{∗} be the measure defined by (2.3). We prove first that for all Q ∈ F (G) and λ > 0 the relation

\frac{1}{μ^{∗}(Q)}μ^{∗}\{z ∈ Q : |f^{∗}(z) − f_{Q}^{∗}| > λ\} ≤ C exp (\frac{−cλ}{∥f∥_{BMO(G)}} ) (3.7)

holds, where C and c are the constants independent of f^{∗}, Q, and λ.

Choose a square Q ∈ F (G). Let δ be the number from Lemma 1, and let c^{∗} be the constant from (2.10). Without loss of generality we can assume that δ < 1/c^{∗}. In this case we can apply Lemma 4 to the function f^{∗} and α = (1/δ)∥f∥_{BMO(G)}. Hence we get a family of disjoint squares A_{1} = {Q_{j}^{1} : Q_{j}^{1} ∈ F (G, k)} such that

|f^{∗}(z) − f_{Q}^{∗}| ≤ 12α

for all z ∈ Q\ ∪_{A_{1}} Q_{j}^{1},

|f_{Q_{j}^{1}}^{∗} − f_{Q}^{∗}| < 4α  (3.8)

according to (2.14), and by (2.15) we have

| ∪_{A_{1}} Q_{j}^{1}| = ∑_{A_{1}} |Q_{j}^{1}| ≤ \frac{1}{α}∥f∥_{BMO(G)} · |Q|.

Since (1/α)∥f∥_{BMO(G)} = δ, by virtue of Lemma 1 we obtain

μ^{∗}{ ∪_{A_{1}}Q_{j}^{1}} ≤ εμ^{∗}(Q).  (3.9)

Applying again Lemma 4 to the function f^{∗} and α = (1/δ)∥f∥_{BMO(G)}, for every Q_{j}^{1} we obtain a family of nonintersecting squares A_{2} = {Q_{j}^{2} : Q_{j}^{2} ∈ F (G, k)} such that each of these squares is contained in one of the Q_{j}^{1}. Thus, by (3.8) and (2.13) the relation

|f^{∗} − f_{Q}^{∗}| ≤ |f^{∗} − f_{Q_{j}^{1}}^{∗}| + |f_{Q_{j}^{1}}^{∗} − f_{Q}^{∗}| < 12α + 4α < 2 · 12α

is fulfilled on Q\ ∪_{A_{2}} Q_{j}^{2}, while owing to (2.14) and (3.8) we have that

|f_{Q_{j}^{2}}^{∗} − f_{Q}^{∗}| ≤ |f_{Q_{j}^{2}}^{∗} − f_{Q_{j}^{1}}^{∗}| + |f_{Q_{j}^{1}}^{∗} − f_{Q}^{∗}| < 4α + 4α < 2 · 12α.

Finally, according to (2.15), we have

| ∪_{Q_{j}^{2}⊂Q_{j}^{1}} Q_{j}^{2}| = ∑_{Q_{j}^{2}⊂Q_{j}^{1}} |Q_{j}^{2}| ≤ \frac{1}{α}∥f∥_{BMO(G)} · |Q_{j}^{1}|

for every Q_{j}^{1}.

Then again, by virtue of Lemma 1 and (3.9), we obtain

μ^{∗}{ ∪_{A_{2}}Q_{j}^{2}} = ∑_{Q_{j}^{1}∈A_{1}} μ^{∗}{ ∪_{Q_{j}^{2}⊂Q_{j}^{1}}Q_{j}^{2}} ≤ ∑_{Q_{j}^{1}∈A_{1}} εμ^{∗}(Q_{j}^{1}) ≤ ε^{2}μ^{∗}(Q).

Continuing this process ad infinitum, we obtain at the step n a family of intersecting squares A_{=}{Q_{j}^{n}} such that

|f^{∗} − f_{Q}^{∗}| ≤ 12α · n a.e. in Q\ ∪_{A_{n}} Q_{j}^{n} and μ^{∗}{ ∪_{A_{n}}Q_{j}^{n}} ≤ ε^{n}μ^{∗}(Q).

Assume now that λ > 12α. Let n ≥ 1 be a natural number such that 12αn < λ ≤ 12αn + 12α. Then, obviously, we shall have

μ^{∗}{z ∈ Q : |f^{∗}(z) − f_{Q}^{∗}| > λ} ≤ μ^{∗}{z ∈ Q : |f^{∗}(z) − f_{Q}^{∗}| > 12αn} ≤

≤ μ^{∗}\{ ∪_{A_{n}} Q_{j}^{n}\} ≤ ε^{n}μ^{∗}(Q) ≤ ε^{\frac{λ}{12α} −1}μ^{∗}(Q) = \frac{1}{ε} exp (\frac{−cλ}{∥f∥_{BMO(G)}} )μ^{∗}(Q)

for c = (1/12)δ · ln(1/ε).

Hence, estimate (3.7) is valid for all λ > 12α. But for all 0 < λ ≤ 12α we, obviously, have

μ^{∗}\{z ∈ Q : |f^{∗}(z) − f_{Q}^{∗}| > λ\} ≤ μ^{∗}(Q) = exp (\frac{cλ}{∥f∥_{BMO(G)}} ) ×

× exp (\frac{−cλ}{∥f∥_{BMO(G)}} )μ^{∗}(Q) ≤ exp (\frac{12c}{δ}) exp (\frac{−cλ}{∥f∥_{BMO(G)}} )μ^{∗}(Q).

Consequently, assuming C = max \{\frac{1}{ε}, exp (\frac{12c}{δ})\}, we get estimate (3.7) for all λ > 0.

Relation (3.7) with regard to (2.8) implies that

\frac{1}{μ(Q ∩ G)}μ\{z ∈ Q ∩ G : |f(z) − f_{Q}^{∗}| > λ\} ≼ exp (\frac{−cλ}{∥f∥_{BMO(G)}} ).(3.10)

The latter relation allows us to complete the proof of Theorem 2. Indeed, using first the Minkowsky inequality and then writing the corresponding integral in terms of a distribution function, applying estimate (3.10), we obtain

(\frac{1}{μ(Q ∩ G)} ∫∫_{Q∩G}|f −f_{μ,Q∩G}|^{p}dμ)≤(\frac{2}{μ(Q ∩ G)} ∫∫_{Q∩G}|f −f_{Q}^{∗}|^{p}dμ)^{\frac{1}{p}} =

= (2p ∫_{0}^{∞}λ^{p−1} \frac{1}{μ(Q ∩ G)}μ\{z ∈ Q ∩ G : |f(z) − f_{Q}^{∗}| > λ\}dλ)^{\frac{1}{p}} ≼

≼ (2p ∫_{0}^{∞}λ^{p−1} exp (\frac{−cλ}{∥f∥_{BMO(G)}} )dλ)^{\frac{1}{p}} ≼ c(p)(∥f∥_{BMO(G)})^{\frac{1}{p}} ,

which implies that f ∈ BMO_{p}(G, w).

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(Received 21.09.1994)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Alexidze St., Tbilisi 380093 Republic of Georgia