Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 339–354

ON MULTIPLIERS FOR BMO_{ϕ} ON GENERAL DOMAINS

Yasuhiro Gotoh

Kyoto University, Faculty of Science, Department of Mathematics Kyoto 606, Japan

Abstract. We characterize pointwise multipliers for BMO_{ϕ,p,loc} on general domains in Euclidean space under certain conditions for ϕ, where BMO_{ϕ,p,loc} is the space of functions of locally bounded p mean oscillation with respect to ϕ.

1. Introduction

Let ϕ: R^{+} → R^{+} be a measurable function, D a domain lying in R^{n}, and BMO_{ϕ,p}(D), 1 ≤ p < ∞, the space of all L_{loc}^{p} functions f on D such that

sup_{Q} ϕ(l(Q))^{−1}(m(Q)^{−1} ∫_{Q} |f − f_{Q}|^{p} dm)^{1/p} < ∞

where the supremum is taken over all cubes in D whose sides are parallel to the coordinate axes. Let BMO_{ϕ,p,loc}(D) be its local version.

As a generalization of Janson’s result [4] for (pointwise) multipliers of BMO_{ϕ,p} space on the n-dimensional torus, Nakai–Yabuta [9] characterized the BMO_{ϕ,p}(R^{n}) multipliers under certain conditions for ϕ, and Nakai [8] extended this result to multipliers of weighted BMO spaces on R^{n}. See also Maz’ya– Shaposhnikova [7], where they characterized multipliers for BMO space on R^{n}, but their BMO is different from ours.

Here we investigate BMO_{ϕ,p}(D) and BMO_{ϕ,p,loc}(D) multipliers for general domains D in R^{n}. We give geometrically simple characterizations which partially extend Nakai’s result and also extend our former result [2] for multipliers of the standard BMO space on general domains D in R^{n}.

The author would like to thank the referee for his helpful comments and suggestions.

1991 Mathematics Subject Classification: Primary 42B15.

2. Notation and results

For a measurable function ϕ: R^{+} → R^{+} we consider the following conditions [A] and [B_{m}], m ∈ R:

[A] There exists a constant M > 0 such that

\begin{aligned}tϕ(l) & ≤ Mϕ(tl),  0 < t ≤ 1, l > 0, \\ M^{−1} ≤ ϕ(s)/ϕ(t) & ≤ M,  2^{−1} ≤ s/t ≤ 2. \end{aligned}

[B_{m}] There exists a constant M > 0 such that

\begin{aligned}tϕ(l) & ≤ Mϕ(tl),  0 < t ≤ 1, l > 0, \\ ∫_{0}^{l} ϕ(t)t^{m−1} dt & ≤ Ml^{m}ϕ(l),  l > 0. \end{aligned}

We have [B_{m}] ⇒ [B_{m′}] ⇒ [A] when m ≤ m′. The second inequality of [B_{0}] is called the Dini condition. In the following we say ϕ ∈ [A] (or [B_{m}]) when ϕ satisfies the condition [A] (or [B_{m}]). If ϕ ∈ [A] is non-decreasing, it satisfies [B_{m}] for every m > 0. In case of ϕ(t) = t^{α}, ϕ ∈ [B_{m}] if and only if −m < α ≤ 1.

In the following ‘cube’ means a closed cube in R^{n} whose sides are parallel to the coordinate axes, l(Q) denotes its side length, tQ, t > 0, denotes the cube having the same center as Q and tl(Q) as its side length, d(·, ·) denotes the Euclidean distance and dm denotes n-dimensional Lebesgue measure.

Let D be a domain in R^{n} and set λ = A\sqrt{n}, where A > 0 is a sufficiently large absolute constant. (For example A = 1000 suffices.) We say that a cube Q lying in D is admissible if it satisfies d(Q, ∂D) ≥ λl(Q) and we denote the set of all admissible cubes in D by 𝒜 (D). A sequence of admissible cubes Q_{0}, Q_{1}, …, Q_{k} in D satisfying the conditions

\begin{matrix}Q_{i} ∩ Q_{i+1} & ≠ ∅,  0 ≤ i ≤ k − 1, \\ 2^{−1} ≤ l(Q_{i+1})/l(Q_{i}) & ≤ 2,  0 ≤ i ≤ k − 1, \end{matrix}

is called an admissible chain. Let Q, Q′ be two admissible cubes in D. We define

δ_{D}^{ϕ}(Q, Q′) = inf{∑_{i=0}^{k} ϕ(l(Q_{i})) | Q = Q_{0}, Q_{1}, …, Q_{k} = Q′ is an admissible chain}.

Since we define δ_{D}^{ϕ} so that δ_{D}^{ϕ} > 0 for technical reasons, δ_{D}^{ϕ} is not a distance function; however, the triangle inequality still holds. Note that when ϕ ∈ [A], then δ_{D}^{ϕ} corresponds to the metric

k_{D}^{ϕ}(x, x′) = inf ∫_{γ} \frac{ϕ(d(y, ∂D))}{d(y, ∂D)} ds(y), x, x′ ∈ D,

On multipliers for BMO_{ϕ} on general domains 341

where the infimum is taken over all rectifiable curves γ ⊂ D joining x to x′.

Let ϕ ∈ [A] and 1 ≤ p < ∞. Let BMO_{ϕ,p}(D) be the space of all L_{loc}^{p} functions f on D such that

∥f∥_{∗,p,D} = ∥f∥_{∗,p} = sup_{Q} ϕ(l(Q))^{−1}M_{p}(f, Q) < ∞

where

M_{p}(f, Q) = (m(Q)^{−1} ∫_{Q} |f − f_{Q}|^{p} dm)^{1/p}

and the supremum is taken over all cubes Q lying in D. (See also [3] for two other versions of the definition of BMO_{ϕ,p}(D). We can rewrite Proposition 1 below for these versions.) Let BMO_{ϕ,p,loc}(D) be the space of all L_{loc}^{p} functions f on D defined in the same way by restricting the supremum to cubes in 𝒜 (D). Let ∥f∥_{∗,p,D,loc} = ∥f∥_{∗,p,loc} denote this supremum. The space BMO_{ϕ,p,loc}(D) is determined independent of the choice of λ (cf. [5, Lemma 2.3], [10, Hilfssatz 2, p. 4]). It holds that BMO_{ϕ,p}(D) ⊂ BMO_{ϕ,p,loc}(D) by definition. If ϕ ∈ [B_{1/p}], BMO_{ϕ,p,loc}(D) and BMO_{ϕ,p}(D) coincide for every domain D ⊂ R^{n} (see Proposition 1).

We say a measurable function g on D is a (pointwise) BMO_{ϕ,p}(D) (respectively BMO_{ϕ,p,loc}(D)) multiplier if gf ∈ BMO_{ϕ,p}(D) (BMO_{ϕ,p,loc}(D)) for every f ∈ BMO_{ϕ,p}(D) (BMO_{ϕ,p,loc}(D)). To consider BMO_{ϕ,p}(D) or BMO_{ϕ,p,loc}(D) multipliers it is convenient to introduce the norm

∥f∥_{∗∗,p,D,Q_{0}} = ∥f∥_{∗∗,p} = ∥f∥_{∗,p} + |f|_{Q_{0}}ϕ(l(Q_{0}))^{−1}, f ∈ BMO_{ϕ,p}(D),
∥f∥_{∗∗,p,D,Q_{0},loc} = ∥f∥_{∗∗,p,loc} = ∥f∥_{∗,p,loc} + |f|_{Q_{0}}ϕ(l(Q_{0}))^{−1}, f ∈ BMO_{ϕ,p,loc}(D),

where Q_{0} is a fixed cube in 𝒜 (D) and |f|_{Q_{0}} = m(Q_{0})^{−1}∫_{Q_{0}} |f| dm. Let g be a BMO_{ϕ,p}(D) (respectively BMO_{ϕ,p,loc}(D)) multiplier. Then the closed graph theorem shows that the operator T_{g}: f ↦ gf on BMO_{ϕ,p}(D) (BMO_{ϕ,p,loc}(D)) is bounded with respect to the norm ∥f∥_{∗∗,p} (∥f∥_{∗∗,p,loc} ). Let ∥T_{g}∥_{ϕ,p} (∥T_{g}∥_{ϕ,p,loc} ) denote its operator norm.

Let

Φ(t) = ∫_{1}^{t} ϕ(s)s^{−1} ds, t > 0

and set

ψ^{ϕ}(Q, Q′) = Φ(2(l(Q)+l(Q′)+d(Q, Q′)))−Φ(min{l(Q), l(Q′)}), Q, Q′ ⊂ R^{n}.

Nakai [8] characterized the multipliers of weighted BMO spaces on R^{n}. In particular for the case of BMO_{ϕ,p}(R^{n}) multipliers his result implies

Theorem 1 ([8]). Let 1 ≤ p < ∞ and ϕ ∈ [B_{n/p}]. Then a measurable function g on R^{n} is a BMO_{ϕ,p}(R^{n}) (= BMO_{ϕ,p,loc}(R^{n})) multiplier if and only if there exists a constant K ≥ 0 such that ∥g∥_{∞} ≤ K and

M_{p}(g, Q) ≤ K \frac{ϕ(l(Q))}{ψ^{ϕ}(Q, Q_{0})}, Q ⊂ R^{n}.

In this case ∥T_{g}∥_{ϕ,p} ≤ CK holds. Conversely if g is a BMO_{ϕ,p}(R^{n}) multiplier then we can choose the constant K so that K ≤ C′∥T_{g}∥_{ϕ,p} where C, C′ > 0 are constants depending only on n, p and the constant M in the condition [B_{n/p}].

We shall extend the above result as follows:

Theorem 2. Let D be a domain in R^{n}, 1 ≤ p < ∞ and ϕ ∈ [B_{n/p}]. Then a measurable function g on D is a BMO_{ϕ,p,loc}(D) multiplier if and only if there exists a constant K ≥ 0 such that ∥g∥_{∞} ≤ K and

M_{p}(g, Q) ≤ K \frac{ϕ(l(Q))}{δ_{D}^{ϕ}(Q, Q_{0})}, Q ∈ 𝒜 (D).

In this case ∥T_{g}∥_{ϕ,p,loc} ≤ CK holds. Conversely if g is a BMO_{ϕ,p,loc}(D) multiplier then we can choose the constant K so that K ≤ C′∥T_{g}∥_{ϕ,p,loc} , where C, C′ > 0 are constants depending only on n, p and the constant M in the condition [B_{n/p}].

Let E_{n,m} be the m-dimensional hypersurface { (x_{1}, x_{2}, …, x_{n}) ∈ R^{n} | x_{i} = 0, m + 1 ≤ i ≤ n } and H_{n} the upper half-space { (x_{1}, x_{2}, …, x_{n}) ∈ R^{n} | x_{n} > 0 }. We define the domains D_{n,m}, n ≥ 1, 0 ≤ m ≤ n − 1, in R^{n} as follows:

D_{n,m} = \{ \begin{matrix}H_{n}, & m = n − 1, \\ R^{n} \ E_{n,m}, & 1 ≤ m ≤ n − 2, \\ R^{n} \ \{0\}, & m = 0, n ≥ 2. \end{matrix}

Under various conditions on ϕ, we showed in [3] that BMO_{ϕ,p,loc}(D) and BMO_{ϕ,p}(D) coincide.

Proposition 1 ([3]).

(1) Let ϕ ∈ [B_{1/p}], then BMO_{ϕ,p,loc}(D) = BMO_{ϕ,p}(D) for every domain D in R^{n}.

(2) Let ϕ ∈ [B_{(n−m)/p}], then BMO_{ϕ,p,loc}(D_{n,m}) = BMO_{ϕ,p}(D_{n,m}).

(3) Let ϕ ∈ [A] and D a proper subdomain of R^{n}. Then BMO_{ϕ,p,loc}(D) = BMO_{ϕ,p}(D) if and only if k_{D}^{ϕ}(·, x_{0}), x_{0} ∈ D, belongs to BMO_{ϕ,p}(D) and its BMO_{ϕ,p}(D) norm is bounded.

Clearly, when any of the conditions (1)–(3) above hold, Theorem 2 provides a result for multipliers on BMO_{ϕ,p} as well.

In Section 3 we give the proof of the ‘if’ part of Theorem 2, which is much easier than that of the ‘only if’ part, and the ‘only if’ part is treated in Section 4.

Throughout Section 3 and 4 we assume that 1 ≤ p < ∞, ϕ ∈ [B_{n/p}] and C, C′, ⋯, C_{1}, C_{2}, ⋯ > 0 (respectively C_{α} > 0) denote constants depending only on n, p and the constant M in the condition [B_{n/p}] (and a given variable α), which may vary from place to place.

3. Proof of Theorem 2 (Part 1)

The following five lemmas are showed by elementary calculations.

Lemma 1. (1) Let l_{i} > 0, 1 ≤ i ≤ k, then ϕ(∑_{i=1}^{k} l_{i}) ≤ M ∑_{i=1}^{k} ϕ(l_{i}).

(2) Let 1/2 ≤ s/t ≤ 2, then C^{−1} ≤ ϕ(s)/ϕ(t) ≤ C.

Lemma 2. Let f ∈ L^{p}(Q), then

M_{p}(f, Q) ≤ 2 inf_{c}(m(Q)^{−1} ∫_{Q} |f − c|^{p} dm)^{1/p}.

Lemma 3. Let f ∈ BMO_{ϕ,p}(D), let Q, Q′ be cubes in D such that Q∪Q′ ⊂ Q̃ ⊂ D, with l(Q̃) ≤ α min{l(Q), l(Q′)} for some cube Q̃. Then |f_{Q} − f_{Q′}| ≤ C_{α}∥f∥_{∗,p}ϕ(l(Q)).

Since BMO_{ϕ,p,loc}(D) is independent of the choice of λ > 0, Lemma 3 yields

Lemma 4. Let f ∈ BMO_{ϕ,p,loc}(D), then

|f_{Q} − f_{Q′}| ≤ C∥f∥_{∗,p,loc}δ_{D}^{ϕ}(Q, Q′), Q, Q′ ∈ 𝒜 (D).

Lemma 5 ([9], [12]). Let f ∈ L^{p}(Q) and g ∈ L^{∞}(Q), then

| |f_{Q}|M_{p}(g, Q) − M_{p}(gf, Q) | ≤ 2∥g∥_{∞}M_{p}(f, Q).

The above lemma implies that the characterization of BMO_{ϕ,p,loc}(D) multipliers is almost equivalent to the estimation of the growth of |f_{Q}|, Q ∈ 𝒜 (D), f ∈ BMO_{ϕ,p,loc}(D). Hence the ‘if’ part of Theorem 2 follows from Lemma 4 which gives a one-sided estimation of |f_{Q}|.

Proof of ‘if’ part of Theorem 2. Let f ∈ BMO_{ϕ,p,loc}(D) and Q ∈ 𝒜 (D). By Lemma 4 and 5

M_{p}(gf, Q) ≤ |f_{Q}|M_{p}(g, Q) + 2∥g∥_{∞}M_{p}(f, Q)

≤ (|f|_{Q_{0}} + C_{1}∥f∥_{∗,p,loc}δ_{D}^{ϕ}(Q, Q_{0}))\frac{Kϕ(l(Q))}{δ_{D}^{ϕ}(Q, Q_{0})} + 2K∥f∥_{∗,p,loc}ϕ(l(Q))

≤ K∥f∥_{∗∗,p,loc}ϕ(l(Q_{0}))\frac{ϕ(l(Q))}{δ_{D}^{ϕ}(Q, Q_{0})} + C_{2}K∥f∥_{∗,p,loc}ϕ(l(Q))

≤ C_{3}K∥f∥_{∗∗,p,loc}ϕ(l(Q)), 344 Yasuhiro Gotoh also |fg|_{Q_{0}} ≤ ∥g∥_{∞}|f|_{Q_{0}} ≤ K∥f∥_{∗∗,p,loc}ϕ(l(Q_{0})), summarizing above we have ∥T_{g}∥_{ϕ,p,loc} ≤ C_{4}K.

We shall in the next section show that the estimation in Lemma 4 is best possible (Theorem 3).

4. Proof of Theorem 2 (Part 2)

We begin with two lemmas which are consequences of elementary calculations. Lemma 6. Let 0 < 2a ≤ b, 0 < 2a′ ≤ b′ and α^{−1} ≤ a′/a, b′/b ≤ α, then

C_{α}^{−1} ≤ ∫_{a′}^{b′} ϕ(t)t^{−1} dt/ ∫_{a}^{b} ϕ(t)t^{−1} dt ≤ C_{α}.

Lemma 7. Let f ∈ L^{p}(Q) and F: C → C satisfy |F (x) − F (y)| ≤ α|x − y|, then

M_{p}(F ∘ f, Q) ≤ C_{α}M_{p}(f, Q).

We set G(x) = Φ(|x|), x ∈ R^{n}. Lemma 8 ([8]). G belongs to BMO_{ϕ,p}(R^{n}) and ∥G∥_{∗,p} ≤ C. Proof. Let Q ⊂ R^{n}. First we assume d(Q, 0) ≥ l(Q). Then

M_{p}(G, Q) ≤ sup_{x,y∈Q} |G(x) − G(y)| ≤ C_{1} \frac{l(Q)}{d(Q, 0)}ϕ(d(Q, 0)) ≤ C_{2}ϕ(l(Q)).

Next let d(Q, 0) < l(Q). Let B be the smallest ball containing Q centered at the origin and let r be its radius. Applying the Minkowsky inequality, we have

M_{p}(G, Q) ≤ 2(m(Q)^{−1} ∫_{Q} |G − Φ(r)|^{p} dm)^{1/p}
≤ C_{3}(m(B)^{−1} ∫_{B} |G − Φ(r)|^{p} dm)^{1/p}
= C_{3}(nr^{−n} ∫_{0}^{r}(∫_{s}^{r} ϕ(t)t^{−1} dt)^{p}s^{n−1} ds)^{1/p}

≤ C_{3} ∫_{0}^{r}(∫_{0}^{t} nr^{−n}s^{n−1} ds)^{1/p}ϕ(t)t^{−1} dt = C_{3}r^{−n/p} ∫_{0}^{r} ϕ(t)t^{n/p−1} dt,

which implies the assertion since ϕ ∈ [B_{n/p}].

Lemma 9. Let Q be a cube centered at the origin and let B be its inscribed ball. Then there exists a constant C > 1 such that

Φ(l(Q)/C) ≤ G_{B} ≤ G_{Q} ≤ Φ(\sqrt{n}l(Q)/2).

Proof. It suffices to prove the first inequality. We set ε = C^{−1} . Let r (= l(Q)/2) be the radius of B. Then

G_{B} − Φ(εl(Q)) = nr^{−n} ∫_{0}^{r}(Φ(t) − Φ(εl(Q)))t^{n−1} dt
= r^{−n}(∫_{εl(Q)}^{r}(r^{n} − t^{n})ϕ(t)t^{−1} dt − ∫_{0}^{εl(Q)} t^{n−1}ϕ(t) dt)
= r^{−n}(I_{1} − I_{2}).

Let C ≥ 8, then 2εl(Q) ≤ r/2. Hence

I_{1} ≥ ∫_{εl(Q)}^{2εl(Q)}(r^{n} − t^{n})ϕ(t)t^{−1} dt ≥ C_{1}l(Q)^{n}ϕ(εl(Q)).

On the other hand since ϕ ∈ [B_{n/p}] ⊂ [B_{n}] we have I_{2} ≤ C_{2}ε^{n}l(Q)^{n}ϕ(εl(Q)). Hence the required inequality holds if we choose the constant C ≥ 8 so that C_{2} ≤ C^{n}C_{1} .

Lemma 10. There exist constants C, C′ > 0 satisfying the following condition: Let Q, Q′ be cubes in R^{n} such that Cl(Q) ≤ l(Q)+l(Q′)+d(Q, Q′), l(Q) ≤ l(Q′) and assume the center of Q is the origin, then ψ^{ϕ}(Q, Q′) ≤ C′(G_{Q′} − G_{Q}).

Proof. First, assume l(Q′) ≥ d(Q, Q′). Then l(Q′) ≥ (C − 1)l(Q)/2. Let B be a ball centered at the origin such that m(B) = m(Q′), then we have G_{Q′} ≥ G_{B} ≥ Φ(l(Q′)/C_{1}), G_{Q} ≤ Φ(\sqrt{n} l(Q)/2) by Lemma 9. And so if we choose the constant C so that (C − 1)/2C_{1} ≥ \sqrt{n} Lemma 6 shows that

\begin{aligned}G_{Q′} − G_{Q} & ≥ ∫_{\sqrt{n} l(Q)/2}^{l(Q′)/C_{1}} ϕ(t)t^{−1} dt \\ & ≥ C_{2} ∫_{l(Q)}^{2(l(Q)+l(Q′)+d(Q,Q′))} ϕ(t)t^{−1} dt = C_{2}ψ^{ϕ}(Q, Q′). \end{aligned}

Next, assume l(Q′) < d(Q, Q′). Then d(Q, Q′) > (C − 1)l(Q)/2. Since G_{Q′} ≥ Φ(d(Q, Q′)) and G_{Q} ≤ Φ(\sqrt{n} l(Q)/2), if we choose the constant C so that (C − 1)/2 ≥ \sqrt{n} Lemma 2 shows that

\begin{aligned}G_{Q′} − G_{Q} & ≥ ∫_{\sqrt{n} l(Q)/2}^{d(Q,Q′)} ϕ(t)t^{−1} dt \\ & ≥ C_{3} ∫_{l(Q)}^{2(l(Q)+l(Q′)+d(Q,Q′))} ϕ(t)t^{−1} dt = C_{3}ψ^{ϕ}(Q, Q′). \end{aligned}

We set

ϱ_{D,p}^{ϕ}(Q, Q′) = sup |f_{Q} − f_{Q′}| + ϕ(l(Q)) + ϕ(l(Q′)), Q, Q′ ∈ 𝒜 (D),

where the supremum is taken over all f ∈ BMO_{ϕ,p,loc}(D) such that ∥f∥_{∗,p,loc} ≤ 1.

Lemma 11. Let Q, Q′ ∈ 𝒜 (D), then ϱ_{D,p}^{ϕ}(Q, Q′) ≤ C(sup |f_{Q} − f_{Q′}| + ϕ(l(Q))) where the supremum is taken over all f ∈ BMO_{ϕ,p,loc}(D) such that ∥f∥_{∗,p,loc} ≤ 1.

Proof. It suffices to show ϕ(l(Q′)) ≤ C_{1}(sup |f_{Q} − f_{Q′}| + ϕ(l(Q))). We may assume C_{2}l(Q) ≤ l(Q′), where C_{2} ≥ 1 is the constant ‘C ’ in Lemma 10. Let f(x) = G(x − x_{0}) where x_{0} is the center of Q. Then Lemma 8 and 10 shows that ∥f∥_{∗,p,loc} ≤ C_{3} and |f_{Q} − f_{Q′}| ≥ C_{4}ψ^{ϕ}(Q, Q′) ≥ C_{5}ϕ(l(Q′)).

Since the next lemma is almost trivial, we omit its proof.

Lemma 12. Let Q, Q′ be cubes in D. Assume there exists a cube Q̃ such that Q ∪ Q′ ⊂ Q̃ ⊂ D and l(Q̃) ≤ α min{l(Q), l(Q′)}. Let Q̂ = λ^{−1}Q, Q̂′ = λ^{−1}Q′. Then Q̂, Q̂′ ∈ 𝒜 (D) and δ_{D}^{ϕ}(Q̂, Q̂′) ≤ C_{α}ϕ(l(Q)).

Lemma 13. Let Q, Q′ ∈ 𝒜 (D). Then C^{−1}ψ^{ϕ}(Q, Q′) ≤ ϱ_{D,p}^{ϕ}(Q, Q′) ≤ C′δ_{D}^{ϕ}(Q, Q′). Moreover, if there exists a cube Q̃ such that Q ∪ Q′ ⊂ Q̃ ⊂ D then these three values are comparable to each other.

Proof. First we show ψ^{ϕ}(Q, Q′) ≤ Cϱ_{D,p}^{ϕ}(Q, Q′). Because of Lemma 10, we can assume that C_{1} min{l(Q), l(Q′)} > l(Q) + l(Q′) + d(Q, Q′) where C_{1} is the constant ‘C ’ in Lemma 10. Then l(Q), l(Q′) and l(Q) + l(Q′) + d(Q, Q′) are comparable, and so ψ^{ϕ}(Q, Q′) ≤ C_{2}ϕ(l(Q)) ≤ C_{3}ϱ_{D,p}^{ϕ}(Q, Q′).

Next, the inequality ϱ_{D,p}^{ϕ}(Q, Q′) ≤ C′δ_{D}^{ϕ}(Q, Q′) is a consequence of Lemma 4. Finally let Q̃ be a cube such that Q ∪ Q′ ⊂ Q̃ ⊂ D. We may assume

l(Q̃) ≤ l(Q) + l(Q′) + d(Q, Q′). Then there exists an admissible chain Q = Q_{0} ⊂ Q_{1} ⊂ ⋯ ⊂ Q_{k} = Q̃ with respect to R^{n} such that l(Q_{i}) = 2^{i}l(Q), 0 ≤ i ≤ k − 1, and l(Q_{k−1}) ≤ l(Q_{k}) ≤ 2l(Q_{k−1}). Hence

∑_{i=0}^{k} ϕ(l(Q_{i})) ≤ C_{4} ∑_{i=0}^{k} ϕ(2^{i}l(Q)) ≤ C_{5} ∫_{0}^{k+1} ϕ(2^{t}l(Q)) dt

≤ C_{6} ∫_{l(Q)}^{2^{k+1}l(Q)} ϕ(t)t^{−1} dt

≤ C_{7} ∫_{l(Q)}^{2(l(Q)+l(Q′)+d(Q,Q′))} ϕ(t)t^{−1} dt = C_{7}ψ^{ϕ}(Q, Q′).

Let Q̂_{i} = λ^{−1}Q_{i} and Q′′ = λ^{−1}Q̃ (= Q̂_{k}) then Q̂_{i} ∈ 𝒜 (D) and δ_{D}^{ϕ}(Q̂_{i}, Q̂_{i+1}) ≤ C_{8}ϕ(l(Q_{i})) by Lemma 12. And so

\begin{aligned}δ_{D}^{ϕ}(Q, Q′′) & ≤ δ_{D}^{ϕ}(Q, Q̂_{0}) + ∑_{i=0}^{k−1} δ_{D}^{ϕ}(Q̂_{i}, Q̂_{i+1}) \\ & ≤ C_{9}ϕ(l(Q)) + C_{8} ∑_{i=0}^{k−1} ϕ(l(Q_{i})) ≤ C_{10}ψ^{ϕ}(Q, Q′). \end{aligned}

Similarly, we have δ_{D}^{ϕ}(Q′, Q′′) ≤ C_{10}ψ^{ϕ}(Q, Q′) and hence by the triangle inequality δ_{D}^{ϕ}(Q, Q′) ≤ C_{11}ψ^{ϕ}(Q, Q′).

Remark. Let f ∈ BMO_{ϕ,p,loc}(D) and Q, Q′ ∈ 𝒜 (D) such that Q ∪ Q′ ⊂ Q̃ ⊂ D for some Q̃. The above Lemma shows that

|f_{Q} − f_{Q′}| ≤ C∥f∥_{∗,p,loc} ∫_{min{l(Q),l(Q′)}}^{2(l(Q)+l(Q′)+d(Q,Q′))} ϕ(t)t^{−1} dt,

so when ∫_{0}^{ε} ϕ(t)t^{−1} dt < ∞ we have

|f(x) − f(x′)| ≤ C∥f∥_{∗,p,loc} ∫_{0}^{|x−x′|} ϕ(t)t^{−1} dt

by Q → x, Q′ → x′, where x, x′ ∈ Q̃ are Lebesgue points of f. Hence f is a continuous function on each cube Q̃ ⊂ D (modulo a null set) with modulus of continuity C∥f∥_{∗,p,loc} ∫_{0}^{t} ϕ(t)t^{−1} dt (cf. [11]).

We say a cube Q ⊂ R^{n} is ‘dyadic’ if can be represented in the following form

{ x ∈ R^{n} | s_{i}2^{k} ≤ x_{i} ≤ (s_{i} + 1)2^{k}, s_{i}, k ∈ Z, 1 ≤ i ≤ n }.

Lemma 14 (cf. [13]). Let D be a proper subdomain of R^{n}. Then there exists a decomposition of D into a family of dyadic cubes 𝒟(D) = {Q_{i}}, Q_{i}^{∘} ∩ Q_{j}^{∘} = ∅, (i ≠ j), ∪_{i}Q_{i} = D for each α > \sqrt{n} such that

\begin{aligned}α & ≤ d(Q_{i}, ∂D)/l(Q_{i}) ≤ 2α + \sqrt{n}, \\ 2^{−1} & ≤ l(Q_{i})/l(Q_{j}) ≤ 2, if Q_{i} ∩ Q_{j} ≠ ∅. \end{aligned}

Proof. We decompose R^{n} into a family of dyadic cubes with side length 1. If there exists a cube Q in this family such that d(Q, ∂D) < αl(Q), then we decompose Q into 2^{n} congruent subcubes. Let Q′ be one such subcube. Then

d(Q′, ∂D) ≤ 2(d(Q, ∂D) + 2^{−1}\sqrt{n} l(Q)) < (2α + \sqrt{n})l(Q′).

Hence by repeating the above process, we can decompose Q into a family of dyadic cubes Q′′ satisfying the condition α ≤ d(Q′′, ∂D)/l(Q′′) ≤ 2α + \sqrt{n}. Next, there exists a dyadic cube Q such that 2α+\sqrt{n} < d(Q, ∂D)/l(Q). Let Q′ be the dyadic cube containing Q such that l(Q′) = 2l(Q). Then

d(Q′, ∂D) ≥ d(Q, ∂D) − \sqrt{n} l(Q) ≥ (2α + \sqrt{n} − \sqrt{n})l(Q) > αl(Q′).

Hence by repeating the above process, we obtain a dyadic cube Q′′ ⊃ Q satisfying the first inequality.

Next, for two such cubes Q, Q′ such that Q ∩ Q′ ≠ ∅ we have

l(Q′) ≤ α^{−1}d(Q′, ∂D) ≤ α^{−1}(d(Q, ∂D) + \sqrt{n} l(Q)) ≤ (2 + 2\sqrt{n}/α)l(Q) < 4l(Q),

so that l(Q′) ≤ 2l(Q).

In the following, 𝒟(D) denotes the family obtained by the above method with α = λ, which we call the Whitney decomposition of D. We say that a sequence Q_{0}, Q_{1}, …, Q_{n} ∈ 𝒟(D) is a Whitney chain if Q_{i} ∩ Q_{i+1} ≠ ∅. Since 𝒟(D) ⊂ 𝒜 (D), every Whitney chain is admissible. We set

W_{D}^{ϕ}(Q, Q′) = inf{∑_{k=0}^{n} ϕ(l(Q_{k})) | Q = Q_{0}, Q_{1}, …, Q_{n} = Q′ is a Whitney chain}.

It holds that δ_{D}^{ϕ}(Q, Q′) ≤ W_{D}^{ϕ}(Q, Q′), Q, Q′ ∈ 𝒟(D), by definition.

We fix a cube Q ∈ 𝒟(D) and set

f(x) = W_{D}^{ϕ}(Q, Q′), x ∈ Q′ ∈ 𝒟(D).

Then |f(x) − f(y)| ≤ Cϕ(l(Q′)), x, y ∈ 2Q′, for every Q′ ∈ 𝒟(D). Hence there exists a C^{1} modification F_{D,Q}^{ϕ} of f such that

\begin{aligned}|F_{D,Q}^{ϕ}(x) − W_{D}^{ϕ}(Q, Q′)| & ≤ Cϕ(l(Q′)),  x ∈ Q′ ∈ 𝒟(D), \\ |∇F_{D,Q}^{ϕ}(x)| & ≤ C′ϕ(l(Q′))/l(Q′),  x ∈ Q′ ∈ 𝒟(D). \end{aligned}

For example ∑_{Q′∈𝒟(D)} W_{D}^{ϕ}(Q, Q′)φ_{Q′} is one such function, where {φ_{Q′} }_{Q′∈𝒟(D)} is the partition of unity associated with the Whitney decomposition 𝒟(D) (cf. [13, p. 172]).

Lemma 15. Let D be a proper subdomain of R^{n}. Then ∥F_{D,Q}^{ϕ}∥_{ϕ,p,loc} ≤ C.

Proof. Let Q′ ∈ 𝒜 (D). Let Q̃′ be a cube in 𝒟(D) containing the center of Q′. Since l(Q′) ≤ C_{1}l(Q̃′) we have

\begin{aligned}M_{p}(F_{D,Q}^{ϕ}, Q′) & ≤ sup_{x,y∈Q′} |F_{D,Q}^{ϕ}(x) − F_{D,Q}^{ϕ}(y)| \\ & ≤ C_{2}ϕ(l(Q̃′))l(Q′)/l(Q̃′) ≤ C_{3}ϕ(l(Q′)). \end{aligned}

Lemma 16. Let D be a proper subdomain of R^{n}, let Q ∈ 𝒜 (D) with x_{0} as its center. Let Q̃ be a cube in 𝒟(D) containing x_{0} . Let f(x) = G(x − x_{0}). Then ψ^{ϕ}(Q, Q̃) ≤ C(fQ̃ − f_{Q}) + C′ϕ(l(Q)).

Proof. Because of Lemma 10 we can assume that l(Q) > l(Q̃) or C_{1}l(Q) > l(Q) + l(Q̃) + d(Q, Q̃) where C_{1} is the constant ‘C ’ in Lemma 10. Then since l(Q) and l(Q̃) are comparable we have |fQ̃ − f_{Q}| ≤ C_{2}ϕ(l(Q)) by Lemma 3 and ψ^{ϕ}(Q, Q̃) ≤ C_{3}ϕ(l(Q)). Therefore if we choose C_{4} so that C_{4} − C_{2} (=: C_{5}) > 0 we have (fQ̃ − f_{Q}) + C_{4}ϕ(l(Q)) ≥ C_{5}ϕ(l(Q)) ≥ C_{6}ψ^{ϕ}(Q, Q̃).

Let Q, Q′ ∈ 𝒜 (D). We set

σ_{D}^{ϕ}(Q, Q′) = \{ \begin{matrix}ψ^{ϕ}(Q, Q′),  if Q ∪ Q′ ⊂ Q′′ ⊂ D for some Q′′, \\ ψ^{ϕ}(Q, Q̃) + W_{D}^{ϕ}(Q̃, Q̃′) + ψ^{ϕ}(Q′, Q̃′),  any other case, \end{matrix}

where Q̃, Q̃′ are the cubes in 𝒟(D) containing the center of Q, Q′ respectively. In the case D = R^{n}, σ_{D}^{ϕ} reduces to ψ^{ϕ}.

Theorem 3. δ_{D}^{ϕ}, ϱ_{D,p}^{ϕ} and σ_{D}^{ϕ} are comparable on 𝒜 (D) × 𝒜 (D).

Proof. Let Q, Q′ ∈ 𝒜 (D). Because of Lemma 13 we can assume that D ≠ R^{n} and there exists no cube Q′′ such that Q ∪ Q′ ⊂ Q′′ ⊂ D. The inequality ϱ_{D,p}^{ϕ}(Q, Q′) ≤ C_{1}δ_{D}^{ϕ}(Q, Q′) has been already proved in Lemma 13. Let Q̃, Q̃′ be cubes in 𝒟(D) containing the centers of Q, Q′ respectively. By Lemma 13 we have,

\begin{aligned}δ_{D}^{ϕ}(Q, Q′) & ≤ δ_{D}^{ϕ}(Q, Q̃) + δ_{D}^{ϕ}(Q̃, Q̃′) + δ_{D}^{ϕ}(Q̃′, Q′) \\ & ≤ C_{2}ψ^{ϕ}(Q, Q̃) + W_{D}^{ϕ}(Q̃, Q̃′) + C_{2}ψ^{ϕ}(Q′, Q̃′) ≤ C_{2}σ_{D}^{ϕ}(Q, Q′). \end{aligned}

We shall show the remaining inequality σ_{D}^{ϕ}(Q, Q′) ≤ C_{3}ϱ_{D,p}^{ϕ}(Q, Q′). Let f_{1}(x) = G(x − x_{0}), f_{2}(x) = −G(x − x_{1}), where x_{0} , x_{1} are the centers of Q, Q′ respectively. Since (f_{1})Q̃ ≤ (f_{1})_{Q′}, (f_{2})_{Q} ≤ (f_{2})Q̃′, Lemma 16 shows

\begin{aligned}ψ^{ϕ}(Q, Q̃) & ≤ C_{4}((f_{1})_{Q′} − (f_{1})_{Q}) + C_{5}ϕ(l(Q)), \\ ψ^{ϕ}(Q′, Q̃′) & ≤ C_{4}((f_{2})_{Q′} − (f_{2})_{Q}) + C_{5}ϕ(l(Q′)). \end{aligned}

Let f_{3} = F_{D,Q̃}^{ϕ}. Since ϕ(l(Q̃)) ≤ C_{6}ψ^{ϕ}(Q, Q̃), ϕ(l(Q̃′)) ≤ C_{6}ψ^{ϕ}(Q′, Q̃′) we have

W_{D}^{ϕ}(Q̃, Q̃′) ≤ (W_{D}^{ϕ}(Q̃, Q̃′) − (f_{3})_{Q′}) + ((f_{3})_{Q′} − (f_{3})_{Q}) + (f_{3})_{Q}
≤ ((f_{3})_{Q′} − (f_{3})_{Q}) + C_{7}ϕ(l(Q̃)) + C_{8}ϕ(l(Q̃′))

≤ ((f_{3})_{Q′} − (f_{3})_{Q}) + C_{9}((f_{1})_{Q′} − (f_{1})_{Q}) + C_{10}((f_{2})_{Q′} − (f_{2})_{Q})

+ C_{11}ϕ(l(Q)) + C_{12}ϕ(l(Q′)).

Therefore if we set f = f_{3} + (C_{4} + C_{9})f_{1} + (C_{4} + C_{10})f_{2} , we have ∥f∥_{ϕ,p,loc} ≤ C_{13} and

σ_{D}^{ϕ}(Q, Q′) ≤ (f_{Q′} − f_{Q}) + C_{14}ϕ(l(Q)) + C_{15}ϕ(l(Q′)).

which implies the assertion.

Now we are in position to prove the ‘only if’ part of of Theorem 2. Lemma 17 and 18 below complete the proof of Theorem 2. Recall that we defined

∥f∥_{∗∗,p,loc} = ∥f∥_{∗,p,loc} + |f|_{Q_{0}}ϕ(l(Q_{0}))^{−1}, f ∈ BMO_{ϕ,p,loc}(D),

where Q_{0} is a fixed cube in 𝒜 (D).

Lemma 17 (cf. [12], [9]). Let g be a BMO_{ϕ,p,loc}(D) multiplier. Then g ∈ L^{∞}(D) and ∥g∥_{∞} ≤ C∥T_{g}∥_{ϕ,p,loc} .

Proof. Let x_{0} ∈ D and let Q a cube in 𝒜 (D) centered at x_{0} such that l(Q) (=: l) ≤ l(Q_{0}). Let B = \{ x ∈ R^{n} | |x − x_{0}| ≤ l/4 \}, so that (1/2\sqrt{n})Q ⊂ B ⊂ (1/2)Q. Let

f(x) = \{ \begin{matrix}G(l/4) − G(x − x_{0}), & x ∈ B, \\ 0, & x ∉ B. \end{matrix}

Then f ≥ 0 and ∥f∥_{∗,p,loc} ≤ C_{1} by Lemma 7 and 8.

First we shall show that ∥f∥_{∗∗,p,loc} ≤ C_{2} . Since l ≤ l(Q_{0}) there exists a cube Q′_{0} ∈ 𝒜 (D) such that Q_{0} ∩ Q′_{0} ≠ ∅, Q′_{0} ∩ Q = ∅, 1/2 ≤ l(Q′_{0})/l(Q_{0}) ≤ 2. Since f = 0 on Q′_{0} , Lemma 4 shows that |f|_{Q_{0}} = (f_{Q_{0}} − f_{Q′_{0}}) + f_{Q′_{0}} ≤ C_{3}ϕ(l(Q_{0})).

Hence ∥f∥_{∗∗,p,loc} = ∥f∥_{∗,p,loc} + |f|_{Q_{0}}ϕ(l(Q_{0}))^{−1} ≤ C_{4} .

Let g be a BMO_{ϕ,p,loc}(D) multiplier and set c = (gf)_{Q}, Q′ = (1/4\sqrt{n})Q. Since ∥gf∥_{∗∗,p,loc} ≤ C_{5}∥T_{g}∥_{ϕ,p,loc} we have,

C_{5}∥T_{g}∥_{ϕ,p,loc}m(Q)ϕ(l) ≥ m(Q)M_{p}(gf, Q) ≥ m(Q)M_{1}(gf, Q)
≥ ∫_{Q\(1/2)Q} |c| dm + m(Q)^{−1} ∫_{Q′} |gf − c| dm
≥ ∫_{Q′}(|c| + |gf − c|) dm ≥ ∫_{Q′} |gf| dm.

Since f ≥ C_{6}ϕ(l) on Q′, we have ∫_{Q′} |g| dm ≤ C_{7}∥T_{g}∥_{ϕ,p,loc}m(Q). Letting l → 0, we obtain ∥g∥_{∞} ≤ C_{8}∥T_{g}∥_{ϕ,p,loc} .

Lemma 18. Let g be a BMO_{ϕ,p,loc}(D) multiplier then

M_{p}(g, Q) ≤ C∥T_{g}∥_{ϕ,p,loc} \frac{ϕ(l(Q))}{δ_{D}^{ϕ}(Q, Q_{0})}, Q ∈ 𝒜 (D).

Proof. Let g be a BMO_{ϕ,p,loc}(D) multiplier and Q ∈ 𝒜 (D). By Theorem 3 and Lemma 11 there exists a function f such that ∥f∥_{∗,p,loc} ≤ C_{1} and

δ_{D}^{ϕ}(Q, Q_{0}) ≤ C_{2}(f_{Q} − f_{Q_{0}}) + C_{3}ϕ(l(Q_{0})).

By considering f − f_{Q_{0}} instead of f, we can assume f_{Q_{0}} = 0. Then

|f|_{Q_{0}} = M_{1}(f, Q_{0}) ≤ ∥f∥_{∗,p,loc}ϕ(l(Q_{0})) ≤ C_{1}ϕ(l(Q_{0}))

hence ∥f∥_{∗∗,p,loc} ≤ C_{4} . Therefore, by Lemma 17 and 5 we have

|f|_{Q}M_{p}(g, Q) ≤ M_{p}(gf, Q) + 2∥g∥_{∞}M_{p}(f, Q)

≤ ∥T_{g}∥_{ϕ,p,loc}∥f∥_{∗∗,p,loc}ϕ(l(Q)) + C_{5}∥T_{g}∥_{ϕ,p,loc}∥f∥_{∗∗,p,loc}ϕ(l(Q))

≤ C_{6}∥T_{g}∥_{ϕ,p,loc}ϕ(l(Q)).

Since δ_{D}^{ϕ}(Q, Q_{0}) ≤ C_{2}|f|_{Q}+C_{3}ϕ(l(Q_{0})) and ∥T_{g}1∥_{∗∗,p,loc} ≤ ∥T_{g}∥_{ϕ,p,loc}ϕ(l(Q_{0}))^{−1} we have

\begin{aligned}δ_{D}^{ϕ}(Q, Q_{0})M_{p}(g, Q) & ≤ C_{7}∥T_{g}∥_{ϕ,p,loc}ϕ(l(Q)) + C_{8}ϕ(l(Q_{0}))M_{p}(g, Q) \\ & ≤ C_{9}∥T_{g}∥_{ϕ,p,loc}ϕ(l(Q)). \end{aligned}

Finally in this section, we give a remark on uniform domains. We say a proper subdomain D of R^{n} is uniform when W_{D}^{1} ≤ Cψ^{1} on 𝒟(D) × 𝒟(D) for some C > 0, where W_{D}^{1} and ψ^{1} are W_{D}^{ϕ} and ψ^{ϕ} with ϕ = 1 respectively. In this case W_{D}^{1} and ψ^{1} are comparable to each other. The uniform domains are precisely the domains with BMO extension property (cf. [5]).

Corollary 1. Let D be a uniform domain then δ_{D}^{ϕ} and ψ^{ϕ} are comparable on 𝒜 (D) × 𝒜 (D). Especially W_{D}^{ϕ} and ψ^{ϕ} are comparable on 𝒟(D) × 𝒟(D).

Proof. It suffices to show W_{D}^{ϕ} ≤ C_{1}ψ^{ϕ} on 𝒟(D) × 𝒟(D). (See Theorem 3 and its proof.) Let D be a uniform domain and Q, Q′ ∈ 𝒟(D). Let Q = Q_{0}, Q_{1}, …, Q_{m} = Q′ be a Whitney chain minimizing its length m. Let l = l(Q), l′ = l(Q′) and L = 2^{M}l = 2^{N}l′ be the maximum side length of Q_{i}, 0 ≤ i ≤ m. We can assume l ≤ l′. Let

\begin{aligned}i_{k} & = min\{ i | l(Q_{i}) = 2^{k}l \},  0 ≤ k ≤ M, \\ i′_{k} & = max\{ i | l(Q_{i}) = 2^{k}l′ \},  0 ≤ k ≤ N, \end{aligned}

then i_{k+1} − i_{k} ≤ C_{2} , i′_{k} − i′_{k+1} ≤ C_{2} , j_{M} − j′_{N} ≤ C_{3} and C_{4}^{−1} ≤ (l(Q) + l(Q′) + d(Q, Q′))/L ≤ C_{4} (cf. [5], [1]). Hence

W_{D}^{ϕ}(Q, Q′) ≤ ∑_{k=0}^{M−1} ∑_{i=i_{k}}^{i_{k+1}−1} ϕ(l(Q_{i})) + ∑_{i=i_{M}}^{i′_{N}} ϕ(l(Q_{i})) + ∑_{k=0}^{N−1} ∑_{i=i′_{k+1}+1}^{i′_{k}} ϕ(l(Q_{i}))

≤ C_{5} ∑_{k=0}^{M−1} ϕ(2^{k}l) + C_{6}ϕ(L) + C_{5} ∑_{k=0}^{N−1} ϕ(2^{k}l′)

≤ C_{7} ∑_{k=0}^{M} ϕ(2^{k}l) ≤ C_{8} ∫_{l}^{2L} ϕ(t)t^{−1} dt ≤ C_{9}ψ^{ϕ}(Q, Q′).

5. Remarks on Λ_{ϕ} multipliers

Let ϕ ∈ [A] be a non-decreasing function such that lim_{t→+0} ϕ(t) = 0 and Λ_{ϕ}(D), the space of all continuous functions f on D, such that

sup_{x,y∈D} ϕ(|x − y|)^{−1}|f(x) − f(y)| < ∞.

Let Λ_{ϕ,loc}(D) be the space of all continuous functions f on D defined in the same way by restricting the supremum to points x, y such that x, y ∈ Q ⊂ D for some Q, and Λ_{ϕ,loc,loc}(D) the space obtained by restricting the supremum to points x, y such that x, y ∈ Q ⊂ D for some Q such that d(Q, ∂D) ≥ λl(Q), where λ > 0 is a given constant. Then we have

(1) Λ_{ϕ}(D) = Λ_{ϕ}(R^{n})|D for every D.

(2) Let 1 ≤ p < ∞. If ϕ ∈ [B_{0}] then Λ_{ϕ,loc}(D) = Λ_{ϕ,loc,loc}(D) = BMO_{ϕ,p}(D) = BMO_{ϕ,p,loc}(D).

(3) Conversely if there exists a proper subdomain D of R^{n} containing an arbitrary large cube such that Λ_{ϕ,loc,loc}(D) = Λ_{ϕ,loc}(D), then ϕ ∈ [B_{0}].

(cf. [11], [6], see also Remark under Lemma 13.) Note that when ϕ ∈ [B_{0}] we can assume ϕ is non-decreasing.

We set

\begin{aligned}δ̂_{D}^{ϕ}(x, y) & = lim_{{𝒜 (D)∋Q→x \atop 𝒜 (D)∋Q→y}} δ_{D}^{ϕ}(Q, Q′), \\ ϱ̂_{D}^{ϕ}(x, y) & = sup |f(x) − f(y)|, \end{aligned}

where the supremum is taken over all f such that ∥f∥_{Λ_{ϕ,loc,loc}(D)} ≤ 1, and

σ̂_{D}^{ϕ}(x, y) = \{ \begin{matrix}ϕ(|x − y|), & if x, y ∈ Q ⊂ D for some Q, \\ k_{D}^{ϕ}(x, y), & any other case. \end{matrix}

Then Theorem 3 with Q → x, Q′ → y shows that Lemma 19. Let ϕ ∈ [B_{0}]. Then δ̂_{D}^{ϕ}, ϱ̂_{D}^{ϕ} and σ̂_{D}^{ϕ} are comparable on D × D.

Recall that the characterization of multipliers for Lipschitz space on general metric spaces is well known.

Proposition 2. Let L(X, d) be the space of all Lipschitz continuous functions on a metric space (X, d), then a function g on X is a (pointwise) L(X, d) multiplier if and only if g is bounded and

|g(x) − g(y)| ≤ C \frac{d(x, y)}{1 + d(x, y_{0})}, x, y ∈ X,

where y_{0} is a fixed point on X.

Since we can isometrically identify Λ_{ϕ,loc,loc}(D) with the space of all Lipschitz continuous functions on the metric space (D, ϱ̂_{D}^{ϕ}), we have the following, which is another version of Theorem 2 in the case when ϕ ∈ [B_{0}]. We note that when ϕ ∈ [B_{0}] we can assume ϕ(t) is non-decreasing and ϕ(t)/t is non-increasing, then σ̂_{R^{n}}^{ϕ}(x, y) = ϕ(|x − y|) defines a metric on R^{n}. Hence Λ_{ϕ,loc,loc}(R^{n}) = L(R^{n}, σ̂_{R^{n}}^{ϕ}).

Lemma 20. Let ϕ ∈ [B_{0}]. Then a function g on D is a Λ_{ϕ,loc,loc}(D) multiplier if and only if g is bounded and

|g(x) − g(y)| ≤ C \frac{δ̂_{D}^{ϕ}(x, y)}{1 + δ̂_{D}^{ϕ}(x, y_{0})}, x, y ∈ D,

where y_{0} is a fixed point in D, or equivalently g is bounded and

|g(x) − g(y)| ≤ C \frac{ϕ(|x − y|)}{1 + σ̂_{D}^{ϕ}(x, y_{0})},

for all x, y ∈ D such that x, y ∈ Q ⊂ D for some Q.

References

[1] Gotoh, Y.: BMO extension theorem for relative uniform domains. - J. Math. Kyoto Univ. 33, 1993, 171–193.

[2] Gotoh, Y.: Remarks on multipliers for BMO on general domains. - Kodai Math. J. 16, 1993, 79–89.

[3] Gotoh, Y.: On localization theorem for BMO_{ϕ} on general domains. - To appear.

[4] Janson, S.: On functions with conditions on the mean oscillation. - Ark. Mat. 14, 1976, 189–196.

[5] Jones, P.: Extension theorems for BMO. - Indiana Univ. Math. J. 29, 1980, 41–66.

[6] Lappalainen, V.: Lip_{h} -extension domains. - Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 56, 1985, 1–52.

354 Yasuhiro Gotoh

[7] Maz’ya, V.G. and T.O. Shaposhnikova: Theory of multipliers in spaces of differentiable functions. - Pitman Publishing, 1985.

[8] Nakai, E.: Pointwise multipliers for functions of weighted bounded mean oscillation. - Studia Math. 105, 1993, 105–119.

[9] Nakai, E., and K. Yabuta: Pointwise multipliers for functions of bounded mean oscillation. - J. Math. Soc. Japan 37, 1985, 207–218.

[10] Reimann, H.M., and T. Rychener: Funktionen beschränkter mittelerer Oszillation. - Lecture Notes in Math. 489, Springer-Verlag, 1975.

[11] Spanne, S.: Some function spaces defined using the mean oscillation over cubes. - Ann. Scuola Norm. Sup. Piza 19, 1965, 593–608.

[12] Stegenga, D.A.: Bounded Toeplitz operators on H^{1} and applications of duality between H^{1} and the functions of bounded mean oscillation. - Amer. J. Math. 98, 1976, 573– 589.

[13] Stein, E.M.: Singular integrals and differentiability properties of functions. - Princeton Univ. Press, 1970.

Received 25 February 1993