Lobachevskii Journal of Mathematics Vol. 12, 2003, 63 – 71 \begin{matrix}c \\ ◯ \end{matrix}L. Lanzhe

L. Lanzhe

BOUNDEDNESS FOR COMMUTATORS OF LITTLEWOOD-PALEY OPERATORS ON SOME HARDY SPACES

(submitted by F. G. Avkhadiev)

Abstract. In the present paper the (H_{b}^{p}, L^{p})-type and (H_{b}^{p,∞}, L^{p,∞})- type boundedness for the commutators associated with the Littlewood-Paley operators and b ∈ BMO(R^{n}) are obtained, where H_{b}^{p} and H_{b}^{p,∞} are, respectively, variants of the standard Hardy spaces and weak Hardy spaces, and n/(n + ε) < p ≤ 1.

1. Introduction

Let b ∈ BMO(R^{n}) and T be the Calderon-Zygmund 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 [3] proved that the commutator [b, T] is bounded on L^{p}(R^{n}) (1 < p < ∞). However, it was observed that [b, T] is not bounded, in general, from H^{p}(R^{n}) to L^{p}(R^{n}) and from L^{1}(R^{n}) to L^{1,∞} (R^{n}) and from H^{p,∞}(R^{n}) to L^{p,∞}(R^{n}) for p ≤ 1. But, if H^{p}(R^{n}) is replaced by a suitable atom space H_{b}^{p}(R^{n}) and H^{p,∞}(R^{n}) by H_{b}^{p,∞}(R^{n}) (see [1][6]), then [b, T] maps continuously H_{b}^{p}(R^{n}) into L^{p}(R^{n}) and H_{b}^{p,∞}(R^{n}) into L^{p,∞}(R^{n}) for p ∈ (n/(n + 1), 1]. In addition, we easily know that H_{b}^{p}(R^{n}) ⊂ H^{p}(R^{n}) and H_{b}^{p,∞}(R^{n}) ⊂ H^{p,∞} (R^{n}). The main purpose of this paper is to consider the boundedness of the commutators related to Littlewood-Paley operators and BMO(R^{n}) functions from H_{b}^{p}(R^{n}) to L^{p}(R^{n}) and from H_{b}^{p,∞}(R^{n}) to L^{p,∞}(R^{n}) for p ≤ 1. In fact, we prove that the commutators of Littlewood-Paley operators are bounded from H_{b}^{p}(R^{n}) to L^{p}(R^{n}) and from H_{b}^{p,∞} (R^{n}) to L^{p,∞}(R^{n}) for p ∈ (n/(n + ε), 1]. We will work on R^{n}, n > 2. Let us first introduce some definitions (see [1] [4] [6] [9]).

Key words and phrases. Littlewood-Paley operator, Commutator, BMO(R^{n}), Hardy space, Weak Hardy space.

2000 Mathematical Subject Classification. 42B25, 42B20.

Definition 1. Let b be a locally integrable function and 0 < p ≤ 1. A bounded measurable function a on R^{n} is said to be a (p, b) atom if

i) supp a ⊂ B = B(x_{0}, r), ii) ||a||_{L^{∞}} ≤ |B|^{−1/p}, iii) ∫ a(y)dy = ∫ a(y)b(y)dy = 0; A temperate distribution f is said to belong to H_{b}^{p}(R^{n}), if, in the Schwartz

distributional sense, it can be written as

f(x) = ∑_{j=0}^{∞} λ_{j}a_{j}(x),

where a′_{j}s are (p, b) atoms, λ_{j} ∈ C and ∑_{j=0}^{∞} |λ_{j}|^{p} < ∞. Moreover, ||f||_{H_{b}^{p}(R^{n})} ∼

(∑_{j=0}^{∞} |λ_{j}|^{p})^{1/p}.

Definition 2. Let b be a locally integrable function and 0 < p ≤ 1. A temperate distribution f is said to belong to the space H_{b}^{p,∞}(R^{n}) if there exists a sequence of functions {f_{k}}_{k=−∞}^{∞} ⊂ L^{∞}(R^{n}) such that

a) f(x) = ∑_{k=−∞}^{∞} f_{k}(x) in the Schwartz distribution sense;

b) Each f_{k} can be decomposed into f_{k} = ∑_{j=1}^{∞} b_{j}^{k} in L^{∞}(R^{n}) ∩ H^{p}(R^{n}), where b_{j}^{k} satisfy the following properties:

b)_{1} Supp b_{j}^{k} ⊂ B_{j}^{k}, B_{j}^{k}’s are the balls with

sup_{k} ∑_{j=1}^{∞} χ_{B_{j}^{k}}(x) < ∞, sup_{k} 2^{kp} ∑_{j=1}^{∞} |B_{j}^{k}| < ∞,

where, and in what follows, χ_{E} denotes the characteristic function of the set E,

b)_{2} there exists a constant C = C(n, p) > 0 such that

||b_{j}^{k}||_{L^{∞}} ≤ C2^{k} for every k, j,

b)_{3} ∫_{R^{n}} b_{j}^{k}(x)dx = ∫_{R^{n}} b_{j}^{k}(x)b(x)dx = 0; The quasinorm on the space H_{b}^{p,∞}(R^{n}) is defined by

||f||_{H_{b}^{p,∞}(R^{n})}^{p} = inf ; ∑_{k=−∞}^{∞} ∑_{j=1}^{∞} b_{j}^{k}=f sup_{k∈Z} 2^{kp} ∑_{j=1}^{∞} |B_{j}^{k}|,

where the infimum is taken over all decompositions of f.

Definition 3. Let ε > 0, fixed a function ψ satisfied the following properties:

(1) ∫ ψ(x)dx = 0,

(2) |ψ(x)| ≤ C(1 + |x|)^{−(n+ε)},

(3) |ψ(x + y) − ψ(x)| ≤ C|y|^{ε}(1 + |x|)^{−(n+1+ε)} when 2|y| < |x|; Let b b a locally integrable function. The commutator of Littlewood-Paley

operator is defined by

g_{μ,b}^{∗}(f)(x) = [∫ ∫_{R_{+}^{n+1}} (\frac{t}{t + |x − y|})^{nμ} |F_{b,t}(x, y)|^{2} \frac{dydt}{t^{1+n}}]^{1/2} , μ > 1,

where

F_{b,t}(x, y) = ∫_{R^{n}} ψ_{t}(y − z)f(z)(b(x) − b(z))dz,

and ψ_{t}(x) = t^{−n}ψ(x/t) for t > 0. We also define

g_{μ}^{∗}(f)(x) = [∫ ∫_{R_{+}^{n+1}} (\frac{t}{t + |x − y|})^{nμ} |f ∗ ψ_{t}(y)|^{2} \frac{dydt}{t^{1+n}}]^{1/2} ,

which is the Littlewood-Paley operator (see [8]).

2. Theorems and Proofs

Theorem 1. Let b ∈ BMO(R^{n}) and 1 ≥ p > n/(n + ε), and μ > n/2. Then the commutator g_{μ,b}^{∗} is bounded from H_{b}^{p}(R^{n}) to L^{p}(R^{n}).

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

||g_{μ,b}^{∗}(a)||_{L^{p}} ≤ C.

Let a be a (p, b) atom supported on a ball B = B(x_{0}, r). We write

∫_{R^{n}}[g_{μ,b}^{∗}(a)(x)]^{p}dx
= ∫_{|x−x_{0}|≤2r}[g_{μ,b}^{∗}(a)(x)]^{p}dx + ∫_{|x−x_{0}|>2r}[g_{μ,b}^{∗}(a)(x)]^{p}dx
≡I + II,

For I, taking q > 2, by Hölder’s inequality and the L^{q}-boundedness of g_{μ,b}^{∗} when μ > max(n/2, n/q) (see [2]), we see that

I ≤C||g_{μ,b}^{∗}(a)||_{L^{q}}^{p} · |B(x_{0}, 2r)|^{1−p/q}
≤C||a||_{L^{q}}^{p}|B|^{1−p/q} ≤ C.

For II, by the vanishing moment of a and Definition 3(3), we have

g_{μ,b}^{∗}(a)(x) ≤[∫ ∫_{R_{+}^{n+1}} (\frac{t}{t + |x − y|})^{nμ} (∫_{B} |ψ_{t}(y − z)

− ψ_{t}(y − x_{0})||a(z)||b(x) − b(z)|dz)^{2} \frac{dydt}{t^{1+n}}]^{1/2}

≤C[∫ ∫_{R_{+}^{n+1}} (\frac{t}{t + |x − y|})^{nμ} (∫_{B} t^{−n}|a(z)||b(x) − b(z)|

\frac{(|x_{0} − z|/t)^{ε}}{(1 + |x_{0} − y|/t)^{n+1+ε}} dz)^{2} \frac{dydt}{t^{n+1}}]^{1/2}

66 L. LANZHE

≤C|B|^{ε/n−1/p}[∫ ∫_{R_{+}^{n+1}} (\frac{t}{t + |x − y|})^{nμ} \frac{t^{2}}{(t + |x_{0} − y|)^{2(n+1+ε)}}
(∫_{B} |b(x) − b(z)|dz)^{2} \frac{dydt}{t^{1+n}}]^{1/2}
≤C|B|^{ε/n−1/p+1}[∫ ∫_{R_{+}^{n+1}} (\frac{t}{t + |x − y|})^{nμ} \frac{t^{1−n}}{(t + |x_{0} − y|)^{2(n+1+ε)}}

(\frac{1}{|B|} ∫_{B}(|b(x) − b_{0}| + |b_{0} − b(z)|)dz)^{2})dydt]^{1/2}

≤C|B|^{ε/n−1/p+1}(|b(x) − b_{0}| + ||b||_{BMO})[∫_{0}^{∞} t(t^{−n} ∫_{R^{n}} (\frac{t}{t + |x − y|})^{nμ}

\frac{dy}{(t + |x_{0} − y|)^{2(n+1+ε)}} )dt]^{1/2},

where b_{0} = |B(x_{0}, r)|^{−1} ∫_{B(x_{0},r)} b(y)dy, notice that

t^{−n} ∫_{R^{n}} (\frac{t}{t + |x − y|})^{nμ} \frac{dy}{(t + |x_{0} − y|)^{2(n+1+ε)}} ≤ CM (\frac{1}{(t + |x_{0} − x|)^{2(n+1+ε)}} )

≤C \frac{1}{(t + |x − z|)^{2(n+1+ε)}}

(where Mg denotes the Hardy-Littlewood maximal function of g) and

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

Then, we deduce

g_{μ,b}^{∗}(a)(x) ≤ C|B|^{ε/n−1/p+1}(|b(x) − b_{0}| + ||b||_{BMO})|x − x_{0}|^{−(n+ε)}, thus, by Hölder’s inequality,

II ≤C ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p} [∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} g_{μ,b}^{∗}(a)(x)dx]^{1/p}
≤C ∑_{k=1}^{∞} |B(x_{0}, 2^{k+1}r)|^{1−p}|B|^{(ε/n−1/p+1)p}||b||_{BMO}^{p}
(∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} |x − x_{0}|^{−(n+ε)}dx)^{p}

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

(∫_{2^{k+1}r≥|x−x_{0}|>2^{k}r} |b(x) − b_{0}||x − x_{0}|^{−(n+ε)}dx)^{p}
≡II_{1} + II_{2};

For II_{2}, using the properties of BMO(R^{n}) (see [7]), we obtain

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

≤C||b||_{BMO}^{p} ∑_{k=0}^{∞} k^{p}2^{kn(1−(n+ε)p/n)}
≤C||b||_{BMO}^{p};

For II_{1}, similar to the estimate of II_{2}, we obtain II_{1} ≤ C||b||_{BMO}^{p}; Combining the estimates of II_{1} with II_{2}, we gain

II ≤ C||b||_{BMO}^{p} ≤ C.

This finishes the proof of Theorem 1.

Theorem 2. Let b ∈ L^{∞}(R^{n}) and p = n/(n + ε) and μ > 1. Then g_{μ,b}^{∗} is bounded from H_{b}^{p}(R^{n}) to L^{p,∞}(R^{n}).

To prove the theorem, we recall the following lemma (see [8]): Lemma. Let {f_{k}} be a sequence of measurable functions and p ∈ (0, 1).

Assume that

|{x ∈ R^{n} : |f_{k}(x)| > λ}| ≤ cλ^{−p} for any k and λ > 0.

Then, for every p-summable numerical sequence {C_{k}} we have

|\{x ∈ R^{n} : |∑_{k} C_{k}f_{k}(x)| > λ\}| ≤ \frac{2 − p}{1 − p} \frac{C}{λ^{p}} ∑_{k} |C_{k}|^{p}.

Proof of Theorem 2. It suffices to show that there exists a constant C > 0, such that for each (p, b) atom a and any λ > 0, we have

λ^{p}|{x ∈ R^{n} : g_{μ,b}^{∗}(a)(x) > λ}| ≤ c||b||_{L^{∞}}^{p}.

We write

λ^{p}|{x ∈ R^{n} : g_{μ,b}^{∗}(a)(x) > λ}|

≤λ^{p} |{x ∈ R^{n} : |b(x)|g_{μ}^{∗}(a)(x) > λ/2}| + λ^{p} |{x ∈ R^{n} : g_{μ}^{∗}(ab)(x) > λ/2}|
≡I + II.

For I, by a ∈ H^{p}(R^{n}) and the boundedness of g_{μ}^{∗} from H^{p}(R^{n}) to L^{p}(R^{n}) (0 < p ≤ 1)(see[5]), we have

I ≤Cλ^{p} |{x ∈ R^{n} : g_{μ}^{∗}(a)(x) > λ/(2||b||_{L^{∞}})}|
≤C||b||_{L^{∞}}^{p};

For II, note that ab/||b||_{∞} is also a (p, ∞) atom in the space H^{p}(R^{n}), thus, we gain

II ≤ λ^{p} |{x ∈ R^{n} : g_{μ}^{∗}(ab/||b||_{L^{∞}})(x) > λ/(2||b||_{L^{∞}})}| ≤ C||b||_{L^{∞}}^{p}.

This finishes the proof of Theorem 2.

Theorem 3. Let b ∈ BMO(R^{n}) , 1 ≥ p > n/(n + ε) and μ > n/2. Then g_{μ,b}^{∗} is bounded from H_{b}^{p,∞}(R^{n}) to L^{p,∞}(R^{n}).

Proof. Given f ∈ H_{b}^{p,∞}(R^{n}), let f = ∑_{k=−∞}^{∞} f_{k} = ∑_{k=−∞}^{∞} ∑_{j=1}^{∞} b_{j}^{k} be an atomic decomposition as in Definition 2. By a limiting argument, it suffices to show that

sup_{λ>0} λ^{p}|{x ∈ R^{n} : g_{μ,b}^{∗} ( ∑_{k=−N}^{N} f_{k}) (x) > λ}| ≤ CC_{1}

fore every N = 0, 1, 2, ⋯, where C_{1} = sup_{k∈Z} 2^{kp} ∑_{j=1}^{∞} |B_{j}^{k}|.

Given λ > 0, we taking k_{0} ∈ Z such that 2^{k_{0}} ≤ λ < 2^{k_{0}+1}. Let

∑_{k=−N}^{N} f_{k} = ∑_{k=−N}^{k_{0}} f_{k} + ∑_{k=k_{0}+1}^{N} f_{k} ≡ F_{1} + F_{2}.

Note that

|F_{1}(x)| ≤C ∑_{k=−N}^{k_{0}} 2^{k} ∑_{j=1}^{∞} |b_{j}^{k}(x)| ≤ C ∑_{k=−N}^{k_{0}} 2^{k} ∑_{j=1}^{∞} χ_{B_{j}^{k}}(x)

≤C ∑_{k=−N}^{k_{0}} 2^{k}χ_{∪_{j=1}^{∞}B_{j}^{k}}(x),

thus

||F_{1}||_{L^{q}} ≤C ∑_{k=−N}^{k_{0}} 2^{k} |⋃_{j=1}^{∞} B_{j}^{k}|^{1/q} ≤ C ∑_{k=−N}^{k_{0}} 2^{k} (∑_{j=1}^{∞} |B_{j}^{k}|)^{1/q}

≤CC_{1}^{1/q} ∑_{k=−N}^{k_{0}} 2^{k−kp/q} ≤ CC_{1}^{1/q}2^{k_{0}−k_{0}p/q}
≤CC_{1}^{1/q}λ^{1−p/q}, for any 1 < q < ∞.

Since g_{μ,b}^{∗} is bounded on L^{q}(R^{n}) for 1 < q < ∞ (see [2]), we have

λ^{p} |{x ∈ R^{n} : g_{μ,b}^{∗}(F_{1})(x) > λ/2}| ≤ Cλ^{p−q}||F_{1}||_{L^{q}}^{q} ≤ CC_{1};

For F_{2}, let A_{k}B_{j}^{k} be the ball with the same center as B_{j}^{k} and A_{k} times the radius of B_{j}^{k}, where A_{k} ≥ 2^{n+1} is a positive number to be determined later. For brevity, let

B_{k_{0},N} = ⋃_{k_{0}+1≤k≤N,j≥1} A_{k}B_{j}^{k}.

We write

λ^{p}|{x ∈ R^{n} : g_{μ,b}^{∗}(F_{2})(x) > λ/2}|

=λ^{p}|{x ∈ B_{k_{0},N} : g_{μ,b}^{∗}}(F_{2})(x) > λ}| + λ^{p}|{x ∉ B_{k_{0},N} : g_{μ,b}^{∗}(F_{2})(x) > λ}|
≡I + II.

Let C_{j}^{k} = |B_{j}^{k}|^{−1} ∫_{B_{j}^{k}} b(y)dy, first, we have COMMUTATORS OF LITTLEWOOD-PALEY OPERATORS 69

II ≤λ^{p} |{x ∉ B_{k_{0},N} : |b(x) − C_{j}^{k}|g_{μ}^{∗}(F_{2})(x) > λ/2}|
+ λ^{p} |{x ∉ B_{k_{0},N} : g_{μ}^{∗}((b − C_{j}^{k})F_{2})(x) > λ/2}|
≡II_{1} + II_{2},
II_{1} ≤C ∫_{(B_{k_{0},N})^{c}} |b(x) − C_{j}^{k}|^{p}(g_{μ}^{∗}(F_{2})(x))^{p}dx
≤C ∑_{k=k_{0}+1}^{N} ∑_{j=1}^{∞} ∫_{(A_{k}B_{j}^{k})^{c}} |b(x) − C_{j}^{k}|(g_{μ}^{∗}(b_{j}^{k})(x))^{p}dx,

let us now fix j and k, then ∫_{(A_{k}B_{j}^{k})^{c}} |b(x) − C_{j}^{k}|(g_{μ}^{∗}(b_{j}^{k})(x))^{p}dx

= ∑_{l=0}^{∞} ∫_{2^{l+1}A_{k}B_{j}^{k}\2^{l}A_{k}B_{j}^{k}} |b(x) − C_{j}^{k}|^{p}(g_{μ}^{∗}(b_{j}^{k})(x))^{p}dx
≤ ∑_{l=0}^{∞} (∫_{2^{l+1}A_{k}B_{j}^{k}\2^{l}A_{k}B_{j}^{k}}(g_{μ}^{∗}(b_{j}^{k})(x))^{p/(1−p)}dx)^{1−p} (∫_{2^{l+1}A_{k}B_{j}^{k}\2^{l}A_{k}B_{j}^{k}} |b(x) − C_{j}^{k}|dx)^{p}

(where usual modification is made when p = 1)

≡L,

let B_{l} = |2^{l+1}A_{k}B_{j}^{k}|^{−1} ∫_{2^{l+1}A_{k}B_{j}^{k}} |b(x) − C_{j}^{k}|dx, then

L ≤ ∑_{l=0}^{∞} B_{l}^{p}|2^{l+1}A_{k}B_{j}^{k}|^{p} (∫_{2^{l+1}A_{k}B_{j}^{k}\2^{l}A_{k}B_{j}^{k}}(g_{μ}^{∗}(b_{j}^{k})(x))^{p/(1−p)}dx)^{1−p} ,

by |x| > 2|y|, similar to the proof of Theorem 1, we have

g_{μ}^{∗}(b_{j}^{k})(x) ≤ C2^{k}|x|^{−(n+ε)}|B_{j}^{k}|^{1+ε/n},

thus

L ≤C ∑_{l=0}^{∞} B_{l}^{p}|2^{l+1}A_{k}B_{j}^{k}|^{p}2^{kp}|B_{j}^{k}|^{(1+ε/n)p}|2^{l+1}A_{k}B_{j}^{k}|^{1−p(n+ε)/n−p}
=C2^{kp}A_{k}^{n(1−(n+ε)p/n)}|B_{j}^{k}| ∑_{l=0}^{∞} B_{l}^{p}2^{(l+1)(n−(n+ε)p)}
≤C||b||_{BMO}^{p}2^{kp}A_{k}^{n−p(n+ε)}|B_{j}^{k}|,

now we take A_{k} = A2^{(k−k_{0})/(n+ε)}, where A is fixed and large enough, then

II_{1} ≤C||b||_{BMO}^{p} ∑_{j=1}^{∞} ∑_{k=k_{0}+1}^{N} 2^{kp}|B_{j}^{k}|A_{k}^{n−p(n+ε)}

≤CC_{1} ∑_{k=k_{0}+1}^{N} A_{k}^{n−p(n+ε)}
≤CC_{1}, since p > n/(n + ε).

Now, let us estimate II_{2}, by the estimate of g_{μ}^{∗} similar to above, we reduce

II_{2} ≤λ^{p} |{x ∉ B_{k_{0},N} : ∑_{k=k_{0}+1}^{N} ∑_{j=1}^{∞} g_{μ}^{∗}((b − C_{j}^{k})b_{j}^{k})(x) > λ/2}|

≤λ^{p}|{x ∉ B_{k_{0},N} : ∑_{k=k_{0}+1}^{N} ∑_{j=1}^{∞} C|x|^{−(n+ε)}2^{k}|B_{j}^{k}|^{1+ε/n} (|B_{j}^{k}|^{−1} ∫_{B_{j}^{k}} |b(y) − c_{j}^{k}|dy)

> λ/2}|

≤Cλ^{p} ∑_{k=k_{0}+1}^{N} ∑_{j=1}^{∞} [2^{k}|B_{j}^{k}|^{1+ε/k}||b||_{BMO}]^{n/(n+ε)} λ^{−n/(n+ε)}

≤Cλ^{p−n/(n+ε)}||b||_{BMO}^{n/(n+ε)} · C_{1} ∑_{k=k_{0}+1}^{N} 2^{k(n/(n+ε)−p)}

≤CC_{1}λ^{p−n/(n+ε)}2^{k_{0}(n/(n+ε)−p)}
≤CC_{1}, since λ ≤ 2^{k_{0}+1},

combining the estimates of II_{1} with II_{2}, we obtain

II ≤ CC_{1}.

Finally, let us estimate I. We have

I ≤λ^{p} ∑_{k=k_{0}+1}^{N} ∑_{j=1}^{∞} A_{k}^{n}|B_{j}^{k}| ≤ Cλ^{p}C_{1} ∑_{k=k_{0}+1}^{N} 2^{n(k−k_{0})/(n+ε)}2^{−kp}

≤CC_{1}λ^{p}2^{−k_{0}p} ∑_{k=k_{0}+1}^{N} 2^{(k−k_{0})(n/(n+ε)−p)} ≤ CC_{1}.

Now, let us combine the estimates of I with II, and let N → ∞, then, we have obtained the conclusion of Theorem 3.

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Department of Applied Mathematics Hunan University, Changsha 410082, P.R. of China E-mail address: lanzheliu@263.net

Received October 13, 2002