Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 22, 2006, 7–18 \begin{matrix}c \\ ◯ \end{matrix} Changhong Wu and Lanzhe Liu

Changhong Wu and Lanzhe Liu

BOUNDEDNESS FOR MULTILINEAR COMMUTATOR OF

LITTLEWOOD-PALEY OPERATOR ON HARDY AND HERZ-HARDY SPACES

(submitted by D. Kh. Mushtari)

Abstract. In this paper, the (Hp ; b⃗, L^{p}) and (HK̇_{q,b⃗}^{α,p}, K̇_{q}^{α,p}) type boundedness for the multilinear commutator associated with the Littlewood-Paley operator are obtained.

1. Introduction and definition

Let L_{loc}^{q}(R^{n}) = {f^{q} is locally integrable on R^{n}}. Suppose f ∈ L_{loc}^{1}(R^{n}), B = B(x_{0}, r) = {x ∈ R^{n} : |x − x_{0}| < r} denotes a ball of R^{n} centered at x_{0} and having radius r, write f_{B} = |B|^{−1} ∫_{B} f(x)dx and f^{#}(x) = sup_{x∈B} |B|^{−1} ∫_{B} |f(x) − f_{B}|dx < ∞. The function f is said to belong to BMO(R^{n}) if f^{#} ∈ L^{∞}(R^{n}), and define ||f||_{BMO} = ||f^{#}||_{L^{∞}}.

Let T be a linear operator and K be a function on R^{n} × R^{n},

T (f)(x) = ∫_{R^{n}} K(x, y)f(y)dy for f ∈ C_{0}^{∞},

where K satisfies:

(1) |K(x, y + h) − K(x, y)| ≤ C · |h|^{α} · |x − y|^{−n−α} for 2|h| < |x − y|, 0 < α ≤ 1;

(2) ||T (f)||_{L^{p_{0}}} ≤ C||f||_{L^{p_{0}}} for some 1 < p_{0} ≤ ∞; Then we call T is the Calderón-Zygmund singular integral operator. Let b ∈ BMO(R^{n}) and T be the Calderón-Zygmund singular integral

2000 Mathematical Subject Classification. 42B20, 42B25.

Key words and phrases. Littlewood-Paley operator, Multilinear commutator, BMO, Hardy space, Herz-Hardy space.

operator. The commutator [b, T] generated by b and T is defined by

[b, T](f)(x) = b(x)T (f)(x) − T (bf)(x).

A classical result of Coifman, Rochberg and Weiss (see[2]) proved that the commutator [b, T] is bounded on L^{p}(R^{n}) (1 < p < ∞). However, it was observed that the [b, T] is not bounded, in general, from H^{p}(R^{n}) to L^{p}(R^{n}). But if H^{p}(R^{n}) is replaced by a suitable atomic space Hp ; b⃗(R^{n}) and HK̇_{q,b⃗}^{α,p}(R^{n})(see[1][7][12]), then [b, T] maps continuously Hp ; b⃗(R^{n}) into L^{p}(R^{n}) and HK̇_{q,b⃗}^{α,p}(R^{n}) into K̇_{q}^{α,p}. In addition we have easily known that

Hp ; b⃗(R^{n}) ⊂ H^{p}(R^{n}), K̇_{q,b⃗}^{α,p}(R^{n}) ⊂ HK̇_{q}^{α,p}(R^{n}). The main purpose of this

paper is to consider the continuity of the multilinear commutators related to the Littlewood-Paley operators and BMO(R^{n}) functions on certain Hardy and Herz-Hardy spaces. Let us first introduce some definitions(see [1][3-10][12][13]).

Given a positive integer m and 1 ≤ j ≤ m, we denote by C_{j}^{m} the family of all finite subsets σ = {σ(1), ···, σ(j)} of {1, ···, m} of j different elements. For σ ∈ C_{j}^{m}, set σ^{c} = {1, ⋯, m} \ σ. For b⃗ = (b_{1}, ⋯, b_{m}) and σ = {σ(1), ⋯, σ(j)} ∈ C_{j}^{m}, set b⃗_{σ} = (b_{σ(1)}, ⋯, b_{σ(j)}), b_{σ} = b_{σ(1)} ⋯ b_{σ(j)} and ||b⃗_{σ}||_{BMO} = ||b_{σ(1)}||_{BMO} ⋯ ||b_{σ(j)}||_{BMO}.

Definition 1. Let b_{i} (i = 1, ⋯, m) be a locally integrable functions and 0 < p ≤ 1. A bounded measurable function a on R^{n} is called a (p, b⃗) atom, if

(1) supp a ⊂ B = B(x_{0}, r)

(2) ||a||_{L^{∞}} ≤ |B(x_{0}, r)|^{−1/p}

(3) ∫_{B} a(y)dy = ∫_{B} a(y) ∏_{l∈σ} b_{l}(y)dy = 0 for any σ ∈ C_{j}^{m} , 1 ≤ j ≤ m.

A temperate distribution(see[14][15]) f is said to belong to Hp ; b⃗(R^{n}), if, in the Schwartz distribution sense, it can be written as

f(x) = ∑_{j=1}^{∞} λ_{j}a_{j}(x).

where a′_{j}s are (p, b⃗) atoms, λ_{j} ∈ C and ∑_{j=1}^{∞} |λ_{j}|^{p} < ∞. Moreover,

||f||_{Hp ; b⃗} ≈ (∑_{j=1}^{∞} |λ_{j}|^{p})^{1/p}.

Definition 2. Let 0 < p, q < ∞, α ∈ R. For k ∈ Z, set B_{k} = {x ∈ R^{n} : |x| ≤ 2^{k}} and C_{k} = B_{k} \ B_{k−1},andχ_{k} = χ_{C_{k}} for k ∈ Z,where χ_{C_{k}} is the characteristic function of set C_{k}. Denote the characteristic function of B_{0} by χ_{0}.

(1) The homogeneous Herz space is defined by

K̇_{q}^{α,p}(R^{n}) = {f ∈ L_{loc}^{q}(R^{n} \ {0}) : ||f||K̇_{q}^{α,p} < ∞} .

where

||f||K̇_{q}^{α,p} = [ ∑_{k=−∞}^{∞} 2^{kαp}||fχ_{k}||_{L^{q}}^{p}]^{1/p} .

(2) The nonhomogeneous Herz space is defined by

K_{q}^{α,p}(R^{n}) = {f ∈ L_{loc}^{q}(R^{n}) : ||f||_{K_{q}^{α,p}} < ∞} .

where

||f||_{K_{q}^{α,p}} = [∑_{k=1}^{∞} 2^{kαp}||fχ_{k}||_{L^{q}}^{p} + ||fχ_{0}||_{L^{q}}^{p}]^{1/p} .

Definition 3. Let α ∈ R^{n}, 1 < q < ∞, α ≥ n(1 − 1/q), b_{i} ∈ BMO(R^{n}), 1 ≤ i ≤ m. A function a(x) is called a central (α, q, b⃗) -atom (or a central (α, q, b⃗)-atom of restrict type ), if

(1) supp a ⊂ B = B(x_{0}, r)(or for some r ≥ 1),

(2) ||a||_{L^{q}} ≤ |B(x_{0}, r)|^{−α/n},

(3) ∫_{B} a(x)x^{β}dx = ∫_{B} a(x)x^{β} ∏_{i∈σ} b_{i}(x)dx = 0 for any σ ∈ C_{j}^{m} , 1 ≤ j ≤ m, 0 ≤ |β| ≤ α, where β = (β_{1}, ..., β_{n}) is the multi-indices with β_{i} ∈ N for 1 ≤ i ≤ n and |β| = ∑_{i=1}^{n} β_{i}.

A temperate distribution f is said to belong to HK̇_{q,b⃗}^{α,p}(R^{n})(or HK_{q,b⃗}^{α,p}(R^{n})), if it can be written as f = ∑_{j=−∞}^{∞} λ_{j}a_{j} (or f = ∑_{j=0}^{∞} λ_{j}a_{j}), in the S′(R^{n}) sense, where a_{j} is a central (α, q, b⃗)-atom(or a central (α, q, b⃗)- atom of restrict type ) supported on B(0, 2^{j}) and ∑_{−∞}^{∞} |λ_{j}|^{p} < ∞(or ∑_{j=0}^{∞} |λ_{j}| < ∞). Moreover,

||f||_{HK̇_{q,b⃗}^{α,p}}( or ||f||_{HK_{q,b⃗}^{α,p}}) = inf(∑_{j} |λ_{j}|^{p})^{1/p},

where the infimum are taken over all the decompositions of f as above.

Definition 4. Let ε > 0 and ψ be a fixed function which satisfies the following properties:

(1) ∫_{R^{n}} ψ(x)dx = 0,

(2) |ψ(x)| ≤ C(1 + |x|)^{−(n+1)},

(3) |ψ(x + y) − ψ(x)| ≤ C|y|^{ε}(1 + |x|)^{−(n+1+ε)} when 2|y| < |x|; The Littlewood-Paley multilinear commutator is defined by

Sb⃗ ; ψ(f)(x) = [∫ ∫_{Γ(x)} |F b⃗ ; t(f)(x, y)|^{2} \frac{dydt}{t^{n+1}}]^{1/2} ,

where

F b⃗ ; t(f)(x, y) = ∫_{R^{n}} [∏_{j=1}^{m}(b_{j}(x) − b_{j}(z))] ψ_{t}(y − z)f(z)dz,

Γ(x) = {(y, t) ∈ R_{+}^{n+1} : |x − y| < t} and ψ_{t}(x) = t^{−n}ψ(x/t) for t > 0. Set F_{t}(f)(y) = ∫_{R^{n}} ψ_{t}(y − x)f(x)dx. We also define that

S_{ψ}(f)(x) = (∫ ∫_{Γ(x)} |F_{t}(f)(x, y)|^{2} \frac{dydt}{t^{n+1}})^{1/2} ,

which is the Littlewood-Paley operator(see [15]).

2.Theorems and Proofs

Theorem 1. Let ε > 0,b_{i} ∈ BMO, 1 ≤ i ≤ m, b⃗ = (b_{1}, ⋯, b_{m}), n/(n+ ε) < p ≤ 1. Then the multilinear commutator Sb⃗ ; ψ is bounded from

Hp ; b⃗(R^{n}) to L^{p}(R^{n}).

Proof. It suffices to show that there exist a constant C > 0, such that for every (p, b⃗) atom a,

||Sb⃗ ; ψ(a)||_{L^{p}} ≤ C.

Let a be a (p, b⃗) atom supported on a ball B = B(x_{0}, r). When m = 1 see [7], and now we prove m > 1. Write

\begin{aligned}_{∫R^{n}} & |Sb⃗ ; ψ(a)(x)|^{p}dx \\ & = ∫_{|x−x_{0}|≤2r} |Sb⃗ ; ψ(a)(x)|^{p}dx + ∫_{|x−x_{0}|>2r} |Sb⃗ ; ψ(a)(x)|^{p}dx = I + II. \end{aligned}

For I, taking q > 1, by Hölder’s inequality and the L^{q}-boundedness of

Sb⃗ ; ψ, we have

I ≤ (∫_{|x−x_{0}|≤2r} |Sb⃗ ; ψ(a)(x)|^{q}dx)^{p/q} · |B(x_{0}, 2r)|^{1−p/q}
≤ C||Sb⃗ ; ψ(a)(x)||_{L^{q}}^{p} · |B(x_{0}, 2r)|^{1−p/q}
≤ C||b⃗||_{BMO}^{p}||a||_{L^{q}}^{p}|B|^{1−p/q} ≤ C||b⃗||_{BMO}^{p}.

BOUNDEDNESS FOR MULTILINEAR COMMUTATOR 11

For II, denoting λ = (λ_{1}, ⋯, λ_{m}) with λ_{i} = (b_{i})_{B}, 1 ≤ i ≤ m, where (b_{i})_{B} = |B(x_{0}, r)|^{−1} ∫_{B(x_{0},r)} b_{i}(x)dx, by Hölder’s inequality and the vanishing moment of a, we get

II = ∑_{k=1}^{∞} ∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} |Sb⃗ ; ψ(a)(x)|^{p}dx

≤ C ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p} (∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} |Sb⃗ ; ψ(a)(x)|dx)^{p}

≤ C ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p} ×_{[∫2^{k+1}r≥|x−x_{0}|>2^{k}r}

(∫ ∫_{Γ(x)} | ∫_{B} ∏_{j=1}^{m}(b_{j}(x) − b_{j}(z))ψ_{t}(y − z)a(z)dz|^{2} \frac{dydt}{t^{n+1}} )^{1/2} dx]^{p}

≤ C ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p} × [∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} _{(∫ ∫Γ(x)}

(∫_{B} |ψ_{t}(y − z) − ψ_{t}(y − x_{0})| ∏_{j=1}^{m} |(b_{j}(x) − b_{j}(z))||a(z)|dz)^{2} \frac{dydt}{t^{n+1}} )^{1/2} dx]^{p};

noting that z ∈ B, x ∈ B(x_{0}, 2^{k+1}r) \ B(x_{0}, 2^{k}r), then

Sb⃗ ; ψ(a)(x)

= [∫ ∫_{Γ(x)} (∫_{B} |ψ_{t}(y − z) − ψ_{t}(y − x_{0})| ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)||a(z)|dz)^{2} \frac{dydt}{t^{n+1}} ]^{1/2}

≤ C [∫ ∫_{Γ(x)} (∫_{B} t^{−n}|a(z)| ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)|\frac{(|x_{0} − z|/t)^{ε}}{(1 + |x_{0} − y|/t)^{n+1+ε}} dz)^{2} \frac{dydt}{t^{n+1}} ]^{1/2}

≤ C (∫ ∫_{Γ(x)} \frac{t^{1−n}}{(t + |x_{0} − y|)^{2(n+1+ε)}} dydt)^{1/2}

× ∫_{B} ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)||x_{0} − z|^{ε}|a(z)|dz

≤ C (∫ ∫_{Γ(x)} \frac{t^{1−n}2^{2(n+1+ε)}}{(2t + 2|x_{0} − y|)^{2(n+1+ε)}} dydt)^{1/2}

× ∫_{B} ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)||x_{0} − z|^{ε}|a(z)|dz;

12 CHANGHONG WU AND LANZHE LIU Notice that 2t + |x_{0} − y| > 2t + |x_{0} − x| − |x − y| > t + |x_{0} − x| when |x − y| < t, and it is easy to calculate that

∫_{0}^{∞} \frac{tdt}{(t + |x − x_{0}|)^{2(n+1+ε)}} = C|x − x_{0}|^{−2(n+ε)};

then, we deduce

Sb⃗ ; ψ(a)(x)

≤ C (∫ ∫_{Γ(x)} \frac{t^{1−n}}{(2t + |x_{0} − y|)^{2(n+1+ε)}} dydt)^{1/2}

× ∫_{B} ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)||x_{0} − z|^{ε}|a(z)|dz

≤ C (∫ ∫_{Γ(x)} \frac{t^{1−n}}{(t + |x − x_{0}|)^{2(n+1+ε)}} dydt)^{1/2}

× ∫_{B} ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)||x_{0} − z|^{ε}|a(z)|dz

≤ C (∫_{0}^{∞} \frac{tdt}{(t + |x − x_{0}|)^{2(n+1+ε)}} )^{1/2}

× ∫_{B} ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)||x_{0} − z|^{ε}|a(z)|dz

≤ C|B|^{ε/n−1/p}|x − x_{0}|^{−(n+ε)} ∫_{B} ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)|dz.

So

II ≤ C|B|^{ε/n−1/p} ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p}

× [∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} |x − x_{0}|^{−(n+ε)} (∫_{B} ∏_{j=1}^{m} |b_{j}(x) − b_{j}(z)|dz) dx]^{p}

≤C|B|^{ε/n−1/p} ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p}
× [∑_{j=0}^{m} ∑_{σ∈C_{j}^{m}} ∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} |x − x_{0}|^{−(n+ε)}|(b⃗(x) − λ)_{σ}|dx ∫_{B} |(b⃗(z) − λ)_{σ^{c}}|dz]^{p}

≤C|B|^{ε/n−1/p} ∑_{j=0}^{m} ∑_{σ∈C_{j}^{m}} (∫_{B} |(b⃗(z) − λ)_{σ^{c}}|dz)^{p} ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p}

× [∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} |x − x_{0}|^{−(n+ε)}|(b⃗(x) − λ)_{σ}|dx]^{p}

≤C ∑_{j=0}^{m} ∑_{σ∈C_{j}^{m}} ||b⃗_{σ^{c}}||_{BMO}^{p}

· ||b⃗_{σ}||_{BMO}^{p} ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−(n+ε)p/n}k^{p}|B|^{(1+ε/n−1/p)p}

≤C||b⃗||_{BMO}^{p} ∑_{k=1}^{∞} k^{p} · 2^{kn(1−p(n+ε)/n)} ≤ C||b⃗||_{BMO}.

This finishes the proof of Theorem 1.

Theorem 2. Let 0 < p < ∞, 1 < q < ∞, n(1 − 1/q) ≤ α < n(1 − 1/q) + ε, ε > 0,and b_{i} ∈ BMO(R^{n}), 1 ≤ i ≤ m, b⃗ = (b_{1}, ⋯, b_{m}). Then Sb⃗ ; ψ is bounded from HK̇_{q,b⃗}^{α,p}(R^{n}) to K̇_{q}^{α,p}(R^{n}).

Proof. Let f ∈ HK̇_{q,b⃗}^{α,p}(R^{n}) and f(x) = ∑_{j=−∞}^{∞} λ_{j}a_{j}(x) be the atomic decomposition for f as in Definition 3, we write

||Sb⃗ ; ψ(f)(x)||K̇_{q}^{α,p} = ( ∑_{k=−∞}^{∞} 2^{kαp}||Sb⃗ ; ψ(f)χ_{k}||_{L_{q}}^{p})^{1/p}

≤C [ ∑_{k=−∞}^{∞} 2^{kαp}( ∑_{j=−∞}^{∞} |λ_{j}|||Sb⃗ ; ψ(a_{j})χ_{k}||_{L^{q}})^{p}]^{1/p}

≤C [ ∑_{k=−∞}^{∞} 2^{kαp}( ∑_{j=−∞}^{k−3} |λ_{j}|||Sb⃗ ; ψ(a_{j})χ_{k}||_{L^{q}})^{p}]^{1/p}

+ C [ ∑_{k=−∞}^{∞} 2^{kαp}( ∑_{j=k−2}^{∞} |λ_{j}|||Sb⃗ ; ψ(a_{j})χ_{k}||_{L^{q}})^{p}]^{1/p}

=I + II.

For II, by the boundedness of Sb⃗ ; ψ on L^{q} and the Hölder’s inequality, we have

II ≤ C [ ∑_{k=−∞}^{∞} 2^{kαp} ( ∑_{j=k−2}^{∞} |λ_{j}|||Sb⃗ ; ψ(a_{j})χ_{j}||_{L^{q}})^{p}]^{1/p}

14 CHANGHONG WU AND LANZHE LIU

≤ C [∑_{−∞}^{∞} 2^{kαp} ( ∑_{j=k−2}^{∞} |λ_{j}|||a_{j}||_{L^{q}})^{p}]^{1/p}
≤ C [ ∑_{k=−∞}^{∞} 2^{kαp} ( ∑_{j=k−2}^{∞} |λ_{j}| · 2^{−jα})^{p}]^{1/p}
≤ C \{ \begin{matrix}[∑_{j=−∞}^{∞} |λ_{j}|^{p} ∑_{k=−∞}^{j+2} 2^{(k−j)αp}]^{1/p} , 0 < p ≤ 1 \\ [∑_{j=−∞}^{∞} |λ_{j}|^{p} ∑_{k=−∞}^{j+2} 2^{(k−j)p/2}]^{1/p} , 1 < p < ∞ \end{matrix}
≤ C ( ∑_{j=−∞}^{∞} |λ_{j}|^{p})^{1/p}
≤ C||f||_{HK̇_{q,b⃗}^{α,p}}.

For I, when m=1, let b_{j}^{1} = |B_{j}|^{−1} ∫_{B_{j}} b_{1}(x)dx. We have

S_{ψ}^{b_{1}}(a_{j})(x)

= [∫ ∫_{Γ(x)} | ∫_{B_{j}}(b_{1}(x) − b_{1}(z))ψ_{t}(y − z)a_{j}(z)dz|^{2} \frac{dydt}{t^{n+1}}]^{1/2}

≤ [∫ ∫_{Γ(x)} (∫_{B_{j}} |ψ_{t}(y − z) − ψ_{t}(y − x_{0})||b_{1}(y) − b_{1}(z)||a_{j}(z)|dz)^{2} \frac{dydt}{t^{n+1}} ]^{1/2}

≤ [∫ ∫_{Γ(x)} (∫_{B_{j}} t^{−n}|a_{j}(z)||b_{1}(x) − b_{1}(z)|\frac{(|x_{0} − z|/t)^{ε}}{(1 + |x_{0} − y|/t)^{n+1+ε}} dy)^{2} \frac{dydt}{t^{n+1}} ]^{1/2}

≤ C (∫ ∫_{Γ(x)} \frac{t^{1−n}}{(t + |x_{0} − y|)^{2(n+1+ε)}} dydt)^{1/2}

∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)||b_{1}(x) − b_{1}(z)|dz

≤ C (∫ ∫_{Γ(x)} \frac{t^{1−n}}{(2t + |x_{0} − y|)^{2(n+1+ε)}} dydt)^{1/2}

∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)||b_{1}(x) − b_{1}(z)|dz

≤ C (∫ ∫_{Γ(x)} \frac{t^{1−n}}{(t + |x − x_{0}|)^{2(n+1+ε)}} dydt)^{1/2}

∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)||b_{1}(x) − b_{1}(z)|dz

BOUNDEDNESS FOR MULTILINEAR COMMUTATOR 15

≤ C (∫_{0}^{∞} \frac{tdt}{(t + |x − x_{0}|)^{2(n+1+ε)}} )^{1/2} (∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)||b_{1}(x) − b_{1}(z)|dz)

≤ C|x − x_{0}|^{−(n+ε)} ∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)|b_{1}(x) − b_{1}(z)|dz ≤ C|x − x_{0}|^{−(n+ε)} ∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)||b_{1}(x) − b_{j}^{1}|dz + C|x − x_{0}|^{−(n+ε)} ∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)||b_{1}(z) − b_{j}^{1}|dz ≤ C|x − x_{0}|^{−(n+ε)} (|b_{1}(x) − b_{j}^{1}|2^{j(ε+n(1−1/q)−α)} + 2^{j(ε+n(1−1/q)−α)}||b_{1}||_{BMO}) . So

||S_{ψ}^{b_{1}}(a_{j})χ_{k}||_{L_{q}}
≤ C2^{j(ε+n(1−1/q)−α)} [(∫_{B_{k}} |b_{1}(x) − b_{j}^{1}||x − x_{0}|^{−q(n+ε)dx})^{1/q}
+ (∫_{B_{k}} |x − x_{0}|^{−q(n+ε)}dx)^{1/q} ||b_{1}||_{BMO}]
≤ C2^{j(ε+n(1−1/q)−α)} [2^{−k(n+ε)} · |B_{k}|^{1/q}||b_{1}||_{BMO}
+2^{−k(n+ε)} · |B_{k}|^{1/q}||b_{1}||_{BMO}]
≤ C||b_{1}||_{BMO}2^{[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]};

thus I =

C [ ∑_{j=−∞}^{∞} 2^{kαp} ( ∑_{j=−∞}^{k−3} |λ_{j}|||S_{ψ}^{b_{1}}(a_{j})χ_{k}||_{L^{q}})^{p}]^{1/p}

≤ C||b_{1}||_{BMO} [ ∑_{k=−∞}^{∞} 2^{kαp} ( ∑_{j=−∞}^{k−3} |λ_{j}|2^{[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]})^{p}]^{1/p}

≤ C||b_{1}||_{BMO}

× \{ \begin{matrix}[∑_{k=−∞}^{∞} 2^{kαp} ∑_{j=−∞}^{k−3} |λ_{j}|^{p}2^{[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]p}]^{1/p} , \\ 0 < p ≤ 1 \\ [∑_{k=−∞}^{∞} 2^{kαp} (∑_{j=−∞}^{k−3} |λ_{j}|^{p}2^{p[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]/2} ) \\ × (∑_{j=−∞}^{k−3} 2^{p′[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]/2})^{p/p′}]^{1/p} , \\ 1 < p < ∞ \end{matrix}

16 CHANGHONG WU AND LANZHE LIU ≤ C||b_{1}||_{BMO}

× \{ \begin{matrix}[∑_{j=−∞}^{∞} |λ_{j}|^{p} ∑_{k=j+3}^{∞} 2^{(j−k)(ε+n(1−1/q)−α)p}]^{1/p} , 0 < p ≤ 1 \\ [∑_{j=−∞}^{∞} |λ_{j}|^{p} ∑_{k=j+3}^{∞} 2^{(j−k)(ε+n(1−1/q)−α)p/2}]^{1/p} , 1 < p < ∞ \end{matrix}

≤ C||b_{1}||_{BMO} ( ∑_{j=−∞}^{∞} |λ_{j}|^{p})^{1/p} ≤ C||f||_{HK̇_{q,b⃗}^{α,p}}.

When m > 1, Let b_{j}^{i} = |B_{j}|^{−1} ∫_{B_{j}} b_{i}(x)dx, 1 ≤ i ≤ m, b⃗′ = (b_{j}^{1}, ⋯, b_{j}^{m}). We have

Sb⃗ ; ψ(a_{j})(x)

= [∫ ∫_{Γ(x)} | ∫_{B_{j}} ∏_{i=1}^{m}(b_{i}(x) − b_{i}(z))ψ_{t}(y − z)a_{j}(z)dz|^{2} \frac{dydt}{t^{n+1}}]^{1/2}

≤ _{[∫ ∫Γ(x)}

(∫_{B_{j}} ∏_{i=1}^{m} |b_{i}(x) − b_{i}(z)||ψ_{t}(y − z) − ψ_{t}(y − x_{0})||a_{j}(z)|dz)^{2} \frac{dydt}{t^{n+1}} ]^{1/2}

≤ C|x − x_{0}|^{−(n+ε)} ∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)| ∏_{i=1}^{m} |b_{i}(x) − b_{i}(z)|dz

≤ C|x − x_{0}|^{−(n+ε)} ∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)| ∏_{i=1}^{m} |b_{i}(x) − b_{i}(y)|dz

≤ C|x − x_{0}|^{−(n+ε)}

× ∑_{i=0}^{m} ∑_{σ∈C_{i}^{m}} |(b⃗(x) − b⃗)_{σ}| ∫_{B_{j}} |x_{0} − z|^{ε}|a_{j}(z)||(b⃗(y) − b⃗)_{σ^{c}}|dz

≤ C|x − x_{0}|^{−(n+ε)} ∑_{i=0}^{m} ∑_{σ∈C_{i}^{m}} |(b⃗(x) − b⃗)_{σ}|2^{jε} · 2^{−jα} · 2^{jn(1−1/q)}||b⃗_{σ^{c}}||_{BMO}

≤ C|x − x_{0}|^{−(n+ε)} · 2^{j(ε+n(1−1/q)−α)} ∑_{i=0}^{m} ∑_{σ∈C_{i}^{m}} |(b⃗(x) − b⃗)_{σ}|||b⃗_{σ^{c}}||_{BMO};

So

\begin{matrix}||Sb⃗ ; ψ(a_{j})χ_{k}||_{L^{q}} ≤ C2^{j(ε+n(1−1/q)−α)}||b⃗_{σ^{c}}||_{BMO} \\ × [∫_{B_{k}} (|x|^{−(n+ε)} ∑_{i=0}^{m} ∑_{σ∈C_{i}^{m}} |(b⃗(x) − b⃗)_{σ}|)^{q} dx]^{1/q} \end{matrix}

\begin{matrix}≤ C||b⃗_{σ^{c}}||_{BMO}2^{j(ε+n(1−1/q)−α)} · 2^{−k(n+ε)+kn/q} \\ ≤ C||b⃗||_{BMO}; \end{matrix}

then

I = C [ ∑_{j=−∞}^{∞} 2^{kαp} ( ∑_{j=−∞}^{k−3} |λ_{j}|||Sb⃗ ; ψ(a_{j})χ_{k}||_{L^{q}})^{p}]^{1/p}

≤ C||b⃗||_{BMO}

[ ∑_{k=−∞}^{∞} 2^{kαp} ( ∑_{j=−∞}^{k−3} |λ_{j}|2^{[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]})^{p}]^{1/p}

≤ C||b⃗||_{BMO}

× \{ \begin{matrix}[∑_{k=−∞}^{∞} 2^{kαp} ∑_{j=−∞}^{k−3} |λ_{j}|^{p}2^{[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]p}]^{1/p} , \\ 0 < p ≤ 1 \\ [∑_{k=−∞}^{∞} 2^{kαp} (∑_{j=−∞}^{k−3} |λ_{j}|^{p}2^{p[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]/2}) \\ × (∑_{j=−∞}^{k−3} 2^{p′[j(ε+n(1−1/q)−α)−k(n+ε)+kn/q]})^{p/p′}]^{1/p} , \\ 1 < p < ∞ \end{matrix}

≤ C||b⃗||_{BMO}

\{ \begin{matrix}[∑_{j=−∞}^{∞} |λ_{j}|^{p} ∑_{k=j+3}^{∞} 2^{(j−k)(ε+n(1−1/q)−α)p}]^{1/p} , \\ 0 < p ≤ 1 \\ [∑_{j=−∞}^{∞} |λ_{j}|^{p} ∑_{k=j+3}^{∞} 2^{(j−k)(ε+n(1−1/q)−α)p/2}]^{1/p} , \\ 1 < p < ∞ \end{matrix}

≤ C||b⃗||_{BMO} ( ∑_{j=−∞}^{∞} |λ_{j}|^{p})^{1/p}

≤ C||f||_{HK̇_{q,b⃗}^{α,p}}.

Remark. Theorem 2 also hold for nonhomogeneous Herz-type spaces, we omit the details.

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18 CHANGHONG WU AND LANZHE LIU

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Department of Mathematics, Changsha University of Science and Technology, Changsha, 410076, P.R.of China E-mail address: acrossbow@tom.com E-mail address: lanzheliu@163.com

Received February 26, 2006