PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 92(106) (2012), 165–176 DOI: 10.2298/PIM1206165H

SHARP FUNCTION INEQUALITIES AND BOUNDNESS

FOR TOEPLITZ TYPE OPERATOR RELATED TO GENERAL FRACTIONAL SINGULAR INTEGRAL OPERATOR

Chuangxia Huang and Lanzhe Liu

Communicated by Stevan Pilipović

Abstract. We establish some sharp maximal function inequalities for the Toeplitz type operator, which is related to certain fractional singular integral operator with general kernel. These results are helpful to investigate the boundedness of the operator on Lebesgue, Morrey and Triebel–Lizorkin spaces respectively.

1. Introduction

In recent decades, commutators have attracted a rapidly growing attention of the researchers in the field of harmonic analysis and have been widely studied by many authors [7, 19, 20]. In [4, 17, 18], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on L^{p}(R^{n}) for 1 < p < ∞. Chanillo proves a similar result when singular integral operators are replaced by the fractional integral operators in [2]. The boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel–Lizorkin and L^{p}(R^{n}) (1 < p < ∞) spaces are obtained in [3, 8, 14]. Some singular integral operators with general kernel are introduced, and the boundedness for the operators and their commutators generated by BMO and Lipschitz functions are obtained [1, 11]. In [9, 10], some Toeplitz type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by BMO and Lipschitz functions are established. The main purpose of this paper is to study the Toeplitz type operators generated by some fractional singular integral operators with general kernel and the Lipschitz and BMO functions. We will prove the sharp maximal inequalities for the Toeplitz type operator T_{δ}^{b}. These results are helpful to investigate the the L^{p}-norm inequality, the boundedness of the operator on Lebesgue, Morrey and Triebel–Lizorkin spaces respectively.

2010 Mathematics Subject Classification: Primary 42B20; Secondary 42B25.

This work was jointly supported by the National Natural Science Foundation of China (No. 11101053), the Key Project of Chinese Ministry of Education (No. 211118), the Excellent Youth Foundation of Educational Committee of Hunan Provincial (No. 10B002).

2. Preliminaries

At first, we should introduce some notations in the following. Throughout this paper, Q denotes a cube of R^{n} with sides parallel to the axes. For any locally integrable function f, the sharp maximal function of f is defined by

M^{#} (f)(x) = sup_{Q∋x} \frac{1}{|Q|} ∫_{Q} |f(y) − f_{Q}|dy,

where, f_{Q} = |Q|^{−1} ∫_{Q} f(x) dx. It is well known that [7, 19]

M^{#} (f)(x) ≈ sup_{Q∋x} inf_{c∈C} \frac{1}{|Q|} ∫_{Q} |f(y) − c|dy.

We say that f belongs to BMO(R^{n}) if M^{#} (f) belongs to L^{∞} (R^{n}) and define ∥f∥_{BMO} = ∥M^{#} (f)∥_{L^{∞}} . It is known [19] that ∥f − f_{2^{k} Q}∥_{BMO} ⩽ Ck∥f∥_{BMO} . Let

M(f)(x) = sup_{Q∋x} \frac{1}{|Q|} ∫_{Q} |f(y)|dy.

For η > 0 we denote M_{η}(f)(x) = M(|f|^{η})^{1/η} (x). For 0 < η < n and 1 ⩽ r < ∞ set

M_{η,r} (f)(x) = sup_{Q∋x} ( \frac{1}{|Q|^{1−rη/n}} ∫_{Q} |f(y)|^{r}dy) ^{1/r} .

According to [7], the A_{p} weight can be defined as follows, for 1 < p < ∞,

A_{p} = \{ w ∈ L_{loc}^{1}(R^{n}) : sup_{Q}( \frac{1}{|Q|} ∫_{Q} w(x) dx) ( \frac{1}{|Q|} ∫_{Q} w(x)^{−1/(p−1)} dx) ^{p−1} < ∞\}

and A_{1} = {w ∈ L_{loc}^{p}(R^{n}) : M(w)(x) ⩽ Cw(x), a.e.}.

For β > 0 and p > 1 let Ḟ_{p}^{β,∞} (R^{n}) be the homogeneous Triebel–Lizorkin space [14]. For β > 0 the Lipschitz space Lip_{β}(R^{n}) is the space of functions f such that

∥f∥_{Lip_{β}} = sup_{{x,y ∈R^{n} \atop x≠y}} \frac{|f(x) − f(y)|}{|x − y|^{β}} < ∞.

Definition 2.1. Let φ be a positive, increasing function on R^{+} and there exists a constant D > 0 such that φ(2t) ⩽ Dφ(t) for t ⩾ 0. Let f be a locally integrable function on R^{n}. Set, for 1 ⩽ p < ∞,

∥f∥_{L^{p,φ}} = sup_{x∈R^{n}, d>0} ( \frac{1}{φ(d)} ∫_{Q(x,d)} |f(y)|^{p}dy) ^{1/p},

where Q(x, d) = {y ∈ R^{n} : |x − y| < d}. As usual, the generalized Morrey space can be defined by L^{p,φ} (R^{n}) = {f ∈ L_{loc}^{1}(R^{n}) : ∥f∥_{L^{p,φ}} < ∞}.

If φ(d) = d^{δ}, δ > 0, then L^{p,φ} (R^{n}) = L^{p,δ}(R^{n}) are classical Morrey spaces [15, 16]. If φ(d) = 1, then L^{p,φ} (R^{n}) = L^{p}(R^{n}) are Lebesgue spaces [13].

Since Morrey spaces can be regarded as extensions of Lebesgue spaces, it is natural and important to study the boundedness of operators on the Morrey spaces [5, 6, 12, 13]. Here we study some singular integral operators, defined as follows [1].

Definition 2.2. Fix 0 < δ < n. Let T_{δ} : S → S′ be a linear operator such that T_{δ} is bounded on L^{2}(R^{n}) and has a kernel K, that is, there exists a locally integrable function K(x, y) on R^{n} × R^{n} ∖ {(x, y) ∈ R^{n} × R^{n} : x = y} such that T_{δ}(f)(x) = ∫_{R^{n}} K(x, y)f(y) dy for every bounded and compactly supported function f, where K satisfies: there is a sequence of positive constant numbers {C_{j} } such that for any j ⩾ 1,

∫_{2|y−z|<|x−y|} (|K(x, y) − K(x, z)| + |K(y, x) − K(z, x)|) dx ⩽ C,

(1) ( ∫_{2^{k} |z−y|⩽|x−y|<2^{k+1} |z−y|} (|K(x, y) − K(x, z)| + |K(y, x) − K(z, x)|)^{q}dy) ^{1/q}

⩽ C_{k}(2^{k}|z − y|)^{−n/q′+δ} ,

where 1 < q′ < 2 and 1/q + 1/q′ = 1. We write T_{δ} ∈ GSIO(δ).

Moreover, if b is a locally integrable function on R^{n}, then the Toeplitz type operator related to T_{δ} can be defined by T_{δ}^{b} = ∑ _{k=1}^{m} T_{δ}^{k,1}M_{b}T^{k,2}, where T_{δ}^{k,1} are T_{δ} or ±I (the identity operator), T^{k,2} are the linear operators, k = 1,..., m, M_{b}(f) = bf .

Remark 2.1. We should point out that the classical Calderón–Zygmund singular integral operator satisfies Definition 2.2 with C_{j} = 2^{−jδ} [7, 19].

Remark 2.2. It is obvious that the fractional integral operator with rough kernel satisfies Definition 2.2 [3], that is, for 0 < δ < n, let T_{δ} be the fractional integral operator with rough kernel defined by (see [3])

T_{δ}f(x) = ∫_{R^{n}} \frac{Ω(x − y)}{|x − y|^{n−δ}} f(y) dy,

where Ω is homogeneous of degree zero on R^{n}, ∫_{S^{n−1}} Ω(x′) dσ(x′) = 0 and Ω ∈ Lip_{ε}(S^{n−1} ) for some 0 < ε ⩽ 1, and there exists a constant M > 0 such that for any x, y ∈ S^{n−1} , |Ω(x) − Ω(y)| ⩽ M|x − y|^{ε}. When Ω ≡ 1, T_{δ} is the Riesz potential (fractional integral operator) [2].

Remark 2.3. One can obtain that the commutator [b, T_{δ}](f) = bT_{δ}(f)−T_{δ}(bf) is a particular operator of the Toeplitz type operator T_{b}. The Toeplitz type operators T_{δ}^{b} are nontrivial generalizations of commutators.

3. Some Lemmas

We begin with some preliminary lemmas.

Lemma 3.1. [1] Let T_{δ} be the singular integral operator as Definition 1.2. Then T_{δ} is bounded from L^{p}(R^{n}) to L^{r}(R^{n}) for 1 < p < n/δ and 1/r = 1/p − δ/n.

Lemma 3.2. [14] For 0 < β < 1 and 1 < p < ∞, we have

∥f∥Ḟ_{p}^{β,∞} ≈ ∥sup_{Q∋·} \frac{1}{|Q|^{1+β/n}} ∫_{Q}|f(x)−f_{Q}| dx∥_{L^{p}}≈ ∥sup_{Q∋·} inf_{c} \frac{1}{|Q|^{1+β/n}} ∫_{Q}|f(x)−c| dx∥_{L^{p}}.

Lemma 3.3. [7] Let 0 < p < ∞ and w ∈ ⋃ _{1⩽r<∞} A_{r}. Then, for any smooth function f for which the left-hand side is finite,

∫_{R^{n}} M(f)(x)^{p}w(x) dx ⩽ C ∫_{R^{n}} M^{#} (f)(x)^{p}w(x) dx.

Lemma 3.4. [2] Suppose that 0 < η < n, 1 ⩽ s < p < n/η and 1/r = 1/p−η/n. Then ∥M_{η,s} (f)∥_{L^{r}} ⩽ C∥f∥_{L^{p}} .

Lemma 3.5. Let 1 < p < ∞, 0 < D < 2^{n}. Then, for any smooth function f for which the left-hand side is finite, ∥M(f)∥_{L^{p,φ}} ⩽ C∥M^{#} (f)∥_{L^{p,φ}} .

Proof. For any cube Q = Q(x_{0}, d) in R^{n}, we know M(χ_{Q}) ∈ A_{1} for any cube Q = Q(x, d) by [7]. Noticing that M(χ_{Q}) ⩽ 1 and M(χ_{Q})(x) ⩽ d^{n}/(|x − x_{0}| − d)^{n} if x ∈ Q^{c}, by Lemma 3.3, we have, for f ∈ L^{p,φ} (R^{n}),

∫_{Q} M(f)(x)^{p}dx = ∫_{R^{n}} M(f)(x)^{p}χ_{Q}(x) dx

⩽ ∫_{R^{n}} M(f)(x)^{p}M(χ_{Q})(x) dx ⩽ C ∫_{R^{n}} M^{#} (f)(x)|^{p}M(χ_{Q})(x) dx

= C( ∫_{Q} M^{#} (f)(x)^{p}M(χ_{Q})(x) dx + ∑_{k=0}^{∞} ∫_{2^{k+1} Q∖2^{k} Q} M^{#} (f)(x)^{p}M(χ_{Q})(x) dx)

⩽ C( ∫_{Q} M^{#} (f)(x)^{p}dx + ∑_{k=0}^{∞} ∫_{2^{k+1} Q∖2^{k} Q} M^{#} (f)(x)^{p} \frac{|Q|}{|2^{k+1} Q|}dx)

⩽ C( ∫_{Q} M^{#} (f)(x)^{p}dx + ∑_{k=0}^{∞} ∫_{2^{k+1} Q} M^{#} (f)(x)^{p}2^{−kn} dy)

⩽ C∥M^{#} (f)∥_{L^{p,φ}}^{p} ∑_{k=0}^{∞} 2^{−kn} φ(2^{k+1} d) ⩽ C∥M^{#} (f)∥_{L^{p,φ}}^{p} ∑_{k=0}^{∞} (2^{−n} D)^{k}φ(d)

⩽ C∥M^{#} (f)∥_{L^{p,φ}}^{p} φ(d),

thus

( \frac{1}{φ(d)} ∫_{Q} M(f)(x)^{p}dx) ^{1/p} ⩽ C ( \frac{1}{φ(d)} ∫_{Q} M^{#} (f)(x)^{p}dx) ^{1/p}

and ∥M(f)∥_{L^{p,φ}} ⩽ C∥M^{#} (f)∥_{L^{p,φ}} . This finishes the proof. □

Lemma 3.6. Let 0 < D < 2^{n}, 1 ⩽ s < p < n/η and 1/r = 1/p − η/n. Then

∥M_{η,s} (f)∥_{L^{r,φ}} ⩽ C∥f∥_{L^{p,φ}} .

The proof of Lemma 3.6 is similar to that of Lemma 3.5 by Lemma 3.4, we omit the details.

4. Theorems and Their Proofs

We shall prove the following theorems.

Theorem 4.1. Let the sequence {C_{j} } ∈ l^{1}, 0 < β < 1, q′ ⩽ s < ∞ and b ∈ Lip_{β}(R^{n}). Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) for any p, r with 1 < p < n/δ and 1/r = 1/p − δ/n, and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞), then there exists a constant C > 0 such that, for any f ∈ C_{0}^{∞} (R^{n}) and x̃ ∈ R^{n},

M^{#} (T_{δ}^{b}(f))(x̃) ⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{β+δ,s} (T^{k,2}(f))(x̃).

Proof. It suffices to prove for f ∈ C_{0}^{∞} (R^{n}) and some constant C_{0}, the following inequality holds:

\frac{1}{|Q|} ∫_{Q} |T_{δ}^{b}(f)(x) − C_{0} | dx ⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{β+δ,s} (T^{k,2}(f))(x̃).

Without loss of generality, we may assume T_{δ}^{k,1} are T_{δ}(k = 1, …, m). Fix a cube Q = Q(x_{0}, d) and x̃ ∈ Q. Write, for f_{1} = fχ_{2Q} and f_{2} = fχ_{(2Q)^{c}} ,

T_{δ}^{b}(f)(x) = T_{δ}^{b−b_{Q}} (f)(x) = T_{δ}^{(b−b_{Q} )χ_{2Q}} (f)(x) + T_{δ}^{(b−b_{Q} )χ_{(2Q)^{c}}} (f)(x) = g(x) + h(x).

Then

\frac{1}{|Q|} ∫_{Q} |T_{δ}^{b}(f)(x) − h(x_{0})| dx ⩽ \frac{1}{|Q|} ∫_{Q} |g(x)|dx+ \frac{1}{|Q|} ∫_{Q} |h(x)−h(x_{0})|dx = I_{1}+I_{2}.

For I_{1}, choose 1 < r < ∞ such that 1/r = 1/s − δ/n, by (L^{s}, L^{r})-boundedness of T_{δ} and Hölder’s inequality, we obtain

\frac{1}{|Q|} ∫_{Q} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|dx

⩽ ( \frac{1}{|Q|} ∫_{R^{n}} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|^{r}dx) ^{1/r}

⩽ C|Q|^{−1/r} ( ∫_{R^{n}} |M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|^{s}dx) ^{1/s}

⩽ C|Q|^{−1/r} ( ∫_{2Q}(|b(x) − b_{Q}∥T^{k,2}(f)(x)|)^{s}dx) ^{1/s}

⩽ C|Q|^{−1/r} ∥b∥_{Lip_{β}} |2Q|^{β/n} |2Q|^{1/s−(β+δ)/n}

× ( \frac{1}{|2Q|^{1−s(β+δ)/n}} ∫_{2Q} |T^{k,2}(f)(x)|^{s}dx) ^{1/s}

⩽ C∥b∥_{Lip_{β}} M_{β+δ,s} (T^{k,2}(f))(x̃), thus

I_{1} ⩽ ∑_{k=1}^{m} \frac{1}{|Q|} ∫_{Q} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|dx

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{β+δ,s} (T^{k,2}(f))(x̃).

For I_{2}, recalling that s > q′, we get, for x ∈ Q,

|T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x) − T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x_{0})|

⩽ ∫_{(2Q)^{c}} |b(y) − b_{2Q}∥K(x, y) − K(x_{0}, y)∥T^{k,2}(f)(y)|dy

= ∑_{j=1}^{∞} ∫_{2^{j} d⩽|y−x_{0} |<2^{j+1} d} |b(y) − b_{2Q}∥K(x, y) − K(x_{0}, y)∥T^{k,2}(f)(y)|dy

⩽ C∥b∥_{Lip_{β}} ∑_{j=1}^{∞} |2^{j+1} Q|^{β/n} ( ∫_{2^{j} d⩽|y−x_{0} |<2^{j+1} d} |K(x, y) − K(x_{0}, y)|^{q}dy) ^{1/q}

× ( ∫_{2^{j+1} Q} |T^{k,2}(f)(y)|^{q′}dy) ^{1/q′}

⩽ C∥b∥_{Lip_{β}} ∑_{j=1}^{∞} |2^{j+1} Q|^{β/n} C_{j} (2^{j} d)^{−n/q′+δ} |2^{j+1} Q|^{1/q′−(β+δ)/n}

× ( \frac{1}{|2^{j+1} Q|^{1−s(β+δ)/n}} ∫_{2^{j+1} Q} |T^{k,2}(f)(y)|^{s}dy) ^{1/s}

⩽ C∥b∥_{Lip_{β}} M_{β+δ,s} (T^{k,2}(f))(x̃) ∑_{j=1}^{∞} C_{j} ⩽ C∥b∥_{Lip_{β}} M_{β+δ,s} (T^{k,2}(f))(x̃),

thus

I_{2} ⩽ \frac{1}{|Q|} ∫_{Q} ∑_{k=1}^{m} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x) − T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x_{0})|dx

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{β+δ,s} (T^{k,2}(f))(x̃).

This completes the proof of Theorem 4.1. □

Theorem 4.2. Let the sequence {2^{jβ} C_{j} } ∈ l^{1}, 0 < β < 1, q′ ⩽ s < ∞ and

b ∈ Lip_{β}(R^{n}). Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) for any p, r with 1 < p < n/δ and 1/r = 1/p − δ/n, and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞), then there exists a constant C > 0 such that, for any f ∈ C_{0}^{∞} (R^{n}) and x̃ ∈ R^{n},

sup_{Q∋x̃} \frac{1}{|Q|^{1+β/n}} ∫_{Q} |T_{δ}^{b}(f)(x) − C_{0} | dx ⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃).

Proof. It suffices to prove for f ∈ C_{0}^{∞} (R^{n}) and some constant C_{0}, the following inequality holds:

\frac{1}{|Q|^{1+β/n}} ∫_{Q} |T_{δ}^{b}(f)(x) − C_{0} | dx ⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃).

Without loss of generality, we may assume T_{δ}^{k,1} are T_{δ}(k = 1, …, m). Fix a cube Q = Q(x_{0}, d) and x̃ ∈ Q. For f_{1} = fχ_{2Q} and f_{2} = fχ_{(2Q)^{c}} , write

T_{δ}^{b}(f)(x) = T_{δ}^{b−b_{Q}} (f)(x) = T_{δ}^{(b−b_{Q} )χ_{2Q}} (f)(x) + T_{δ}^{(b−b_{Q} )χ_{(2Q)^{c}}} (f)(x) = g(x) + h(x)

and

\frac{1}{|Q|^{1+β/n}} ∫_{Q} |T_{δ}^{b}(f)(x) − h(x_{0})| dx ⩽ \frac{1}{|Q|^{1+β/n}} ∫_{Q} |g(x)|dx

+ \frac{1}{|Q|^{1+β/n}} ∫_{Q} |h(x) − h(x_{0})|dx = I_{3} + I_{4}.

By using the same argument as in the proof of Theorem 4.1, we get, for 1 < r < ∞ with 1/r = 1/s − δ/n,

I_{3} ⩽ ∑_{k=1}^{m} \frac{C}{|Q|^{1+β/n}} |Q|^{1−1/r} ( ∫_{2Q} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|^{r}dx) ^{1/r}

⩽ ∑_{k=1}^{m} \frac{C}{|Q|^{β/n}} |Q|^{−1/r} ( ∫_{2Q} |M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|^{s}dx) ^{1/s}

⩽ ∑_{k=1}^{m} \frac{C}{|Q|^{β/n}} |Q|^{−1/r} ∥b∥_{Lip_{β}} |2Q|^{β/n} ( ∫_{2Q} |T^{k,2}(f)(x)|^{s}dx) ^{1/s}

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} ( \frac{1}{|2Q|^{1−sδ/n}} ∫_{2Q} |T^{k,2}(f)(x)|^{s}dx) ^{1/s}

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃),

I_{4} ⩽ ∑_{k=1}^{m} \frac{1}{|Q|^{1+β/n}} ∫_{Q} ∑_{j=1}^{∞} ∫_{2^{j} d⩽|y−x_{0} |<2^{j+1} d} |b(y) − b_{2Q}|

× |K(x, y) − K(x_{0}, y)∥T^{k,2}(f)(y)|dydx

⩽ ∑_{k=1}^{m} \frac{C}{|Q|^{1+β/n}} ∫_{Q} ∑_{j=1}^{∞} ∥b∥_{Lip_{β}} |2^{j+1} Q|^{β/n} ( ∫_{2^{j+1} Q} |T^{k,2}(f)(y)|^{q′}dy) ^{1/q′}

× ( ∫_{2^{j} d⩽|y−x_{0} |<2^{j+1} d} |K(x, y) − K(x_{0}, y)|^{q}dy) ^{1/q}dx

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} |Q|^{−β/n} ∑_{j=1}^{∞} |2^{j+1} Q|^{β/n} C_{j} (2^{j} d)^{−n/q′+δ} |2^{j+1} Q|^{1/q′−δ/n}

× ( \frac{1}{|2^{j+1} Q|^{1−sδ/n}} ∫_{2^{j+1} Q} |T^{k,2}(f)(y)|^{s}dy) ^{1/s}

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃) ∑_{j=1}^{∞} 2^{jβ} C_{j}

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃).

This completes the proof of Theorem 4.2. □

Theorem 4.3. Let the sequence {jC_{j} } ∈ l^{1}, q′ ⩽ s < ∞ and b ∈ BMO(R^{n}).

Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) for any p, r with 1 < p < n/δ and 1/r = 1/p − δ/n, and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞), then there exists a constant C > 0 such that, for any f ∈ C_{0}^{∞} (R^{n}) and x̃ ∈ R^{n},

M^{#} (T_{δ}^{b}(f))(x̃) ⩽ C∥b∥_{BMO} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃).

Proof. It suffices to prove for f ∈ C_{0}^{∞} (R^{n}) and some constant C_{0}, the following inequality holds:

\frac{1}{|Q|} ∫_{Q} |T_{δ}^{b}(f)(x) − C_{0} | dx ⩽ C∥b∥_{BMO} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃).

Without loss of generality, we may assume T_{δ}^{k,1} are T_{δ} (k = 1, …, m). Fix a cube Q = Q(x_{0}, d) and x̃ ∈ Q. For f_{1} = fχ_{2Q} and f_{2} = fχ_{(2Q)^{c}} , similar to the proof of Theorem 4.1, we have

T_{δ}^{b}(f)(x) = T_{δ}^{b−b_{Q}} (f)(x) = T_{δ}^{(b−b_{Q} )χ_{2Q}} (f)(x) + T_{δ}^{(b−b_{Q} )χ_{(2Q)^{c}}} (f)(x) = g(x) + h(x),

\frac{1}{|Q|} ∫_{Q} |T_{δ}^{b}(f)(x)−h(x_{0})| dx ⩽ \frac{1}{|Q|} ∫_{Q} |g(x)|dx + \frac{1}{|Q|} ∫_{Q} |h(x)−h(x_{0})|dx = I_{5}+I_{6}.

For I_{5}, choose 1 < t < s, by Hölder’s inequality and the boundedness of T_{δ} with 1 < r < ∞ and 1/r = 1/t − δ/n, we obtain

\frac{1}{|Q|} ∫_{Q} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|dx

⩽ ( \frac{1}{|Q|} ∫_{R^{n}} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|^{r}dx) ^{1/r}

⩽ C|Q|^{−1/r} ( ∫_{R^{n}} |M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|^{t}dx) ^{1/t}

⩽ C|Q|^{−1/r} ( ∫_{2Q} |T^{k,2}(f)(x)|^{s}dx)^{1/s}( ∫_{2Q} |b(x) − b_{Q}|^{st/(s−t)} dx) ^{(s−t)/st}

⩽ C∥b∥_{BMO} ( \frac{1}{|2Q|^{1−sδ/n}} ∫_{2Q} |T^{k,2}(f)(x)|^{s}dx) ^{1/s}

⩽ C∥b∥_{BMO} M_{δ,s} (T^{k,2}(f))(x̃),

thus

I_{5} ⩽ ∑_{k=1}^{l} \frac{1}{|Q|} ∫_{Q} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{2Q}} T^{k,2}(f)(x)|dx ⩽ C∥b∥_{BMO} ∑_{k=1}^{m} M_{δ,s} (T^{k,2}(f))(x̃).

For I_{6}, recalling that s > q′, taking 1 < p < ∞ with 1/p + 1/q + 1/s = 1, we get, for x ∈ Q,

|T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x) − T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x_{0})|

⩽ ∑_{j=1}^{∞} ∫_{2^{j} d⩽|y−x_{0} |<2^{j+1} d} |K(x, y) − K(x_{0}, y)∥b(y) − b_{2Q}∥T^{k,2}(f)(y)|dy

⩽ ∑_{j=1}^{∞} ( ∫_{2^{j} d⩽|y−x_{0} |<2^{j+1} d} |K(x, y) − K(x_{0}, y)|^{q}dy) ^{1/q}

× ( ∫_{2^{j+1} Q} |b(y) − b_{Q}|^{p}dy) ^{1/p}( ∫_{2^{j+1} Q} |T^{k,2}(f)(y)|^{s}dy) ^{1/s}

⩽ C∥b∥_{BMO} ∑_{j=1}^{∞} C_{j} (2^{j} d)^{−n/q′+δ} j(2^{j} d)^{n/p} (2^{j} d)^{n/s−δ}

× ( \frac{1}{|2^{j+1} Q|^{1−sδ/n}} ∫_{2^{j+1} Q} |T^{k,2}(f)(y)|^{s}dy) ^{1/s}

⩽ C∥b∥_{BMO} M_{δ,s} (T^{k,2}(f))(x̃) ∑_{j=1}^{∞} jC_{j} ⩽ C∥b∥_{BMO} M_{δ,s} (T^{k,2}(f))(x̃),

thus

I_{6} ⩽ \frac{1}{|Q|} ∫_{Q} ∑_{k=1}^{l} |T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x) − T_{δ}^{k,1}M_{(b−b_{Q} )χ_{(2Q)^{c}}} T^{k,2}(f)(x_{0})|dx

⩽ C∥b∥_{BMO} ∑_{k=1}^{l} M_{δ,s} (T^{k,2}(f))(x̃).

This completes the proof of Theorem 4.3. □

Theorem 4.4. Let the sequence {C_{j} } ∈ l^{1}, 0 < β < min(1, n − δ), q′ < p <

n/(β + δ), 1/r = 1/p − (β + δ)/n and b ∈ Lip_{β}(R^{n}). Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞) and T^{k,2} are the bounded operators on L^{p}(R^{n}) for 1 < p < ∞, k = 1, …, m, then T_{δ}^{b} is bounded from L^{p}(R^{n}) to L^{r}(R^{n}).

Proof. Choose q′ < s < p in Theorem 4.1, we have, by Lemma 3.3 and 3.4,

∥T_{δ}^{b}(f)∥_{L^{r}} ⩽ ∥M(T_{δ}^{b}(f))∥_{L^{r}} ⩽ C∥M^{#} (T_{δ}^{b}(f))∥_{L^{r}}

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} ∥M_{β+δ,s} (T^{k,2}(f))∥_{L^{r}} ⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} ∥T^{k,2}(f)∥_{L^{p}}

⩽ C∥b∥_{Lip_{β}} ∥f∥_{L^{p}} .

This completes the proof. □

Theorem 4.5. Let the sequence {C_{j} } ∈ l^{1}, 0 < β < min(1, n − δ), q′ < p <

n/(β + δ), 1/r = 1/p − (β + δ)/n, 0 < D < 2^{n} and b ∈ Lip_{β}(R^{n}). Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞) and T^{k,2} are the bounded operators on L^{p,φ} (R^{n}) for 1 < p < ∞, k = 1, …, m, then T_{δ}^{b} is bounded from L^{p,φ} (R^{n}) to L^{r,φ} (R^{n}).

Proof. Choose q′ < s < p in Theorem 4.1, we have, by Lemma 3.5 and 3.6,

∥T_{δ}^{b}(f)∥_{L^{r,φ}} ⩽ ∥M(T_{δ}^{b}(f))∥_{L^{r,φ}} ⩽ C∥M^{#} (T_{δ}^{b}(f))∥_{L^{r,φ}}

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} ∥M_{δ,s} (T^{k,2}(f))∥_{L^{r,φ}} ⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} ∥T^{k,2}(f)∥_{L^{p,φ}}

⩽ C∥b∥_{Lip_{β}} ∥f∥_{L^{p,φ}} .

This completes the proof. □

Theorem 4.6. Let the sequence {2^{jβ} C_{j} } ∈ l^{1}, 0 < β < 1, q′ < p < n/δ,

1/r = 1/p − δ/n and b ∈ Lip_{β}(R^{n}). Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞) and T^{k,2} are the bounded operators on L^{p}(R^{n}) for 1 < p < ∞, k = 1, …, m, then T_{δ}^{b} is bounded from L^{p}(R^{n}) to Ḟ_{r}^{β,∞} (R^{n}).

Proof. Choose q′ < s < p in Theorem 4.2, we have, by Lemma 3.2 and 3.3,

∥T_{δ}^{b}(f)∥Ḟ_{r}^{β,∞} ⩽ C∥sup_{Q∋·} \frac{1}{|Q|^{1+β/n}} ∫_{Q} |T_{δ}^{b}(f)(x) − C_{0} | dx∥_{L^{r}}

⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} ∥M_{δ,s} (T^{k,2}(f))∥_{L^{r}} ⩽ C∥b∥_{Lip_{β}} ∑_{k=1}^{m} ∥T^{k,2}(f)∥_{L^{p}}

⩽ C∥b∥_{Lip_{β}} ∥f∥_{L^{p}} .

This completes the proof. □

Theorem 4.7. Let the sequence {jC_{j} } ∈ l^{1}, q′ < p < n/δ, 1/r = 1/p−δ/n and

b ∈ BMO(R^{n}). Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞) and T^{k,2} are the bounded operators on L^{p}(R^{n}) for 1 < p < ∞, k = 1, …, m, then T_{δ}^{b} is bounded from L^{p}(R^{n}) to L^{r}(R^{n}).

Proof. Choose q′ < s < p in Theorem 4.3, we have, by Lemma 3.3 and 3.4,

∥T_{δ}^{b}(f)∥_{L^{r}} ⩽ ∥M(T_{δ}^{b}(f))∥_{L^{p}} ⩽ C∥M^{#} (T_{δ}^{b}(f))∥_{L^{r}}

⩽ C∥b∥_{BMO} ∑_{k=1}^{m} ∥M_{δ,s} (T^{k,2}(f))∥_{L^{r}} ⩽ C∥b∥_{BMO} ∑_{k=1}^{m} ∥T^{k,2}(f)∥_{L^{p}}

⩽ C∥b∥_{BMO} ∥f∥_{L^{p}} .

This completes the proof. □

Theorem 4.8. Let the sequence {jC_{j} } ∈ l^{1}, q′ < p < n/δ, 1/r = 1/p−δ/n, 0 < D < 2^{n} and b ∈ BMO(R^{n}). Suppose T_{δ} is a bounded linear operator from L^{p}(R^{n}) to L^{r}(R^{n}) and has a kernel K satisfying (1). If T_{δ}^{1}(g) = 0 for any g ∈ L^{u}(R^{n}) (1 < u < ∞) and T^{k,2} are the bounded operators on L^{p,φ} (R^{n}) for 1 < p < ∞, k = 1, …, m, then T_{δ}^{b} is bounded from L^{p,φ} (R^{n}) to L^{r,φ} (R^{n}).

Proof. Choose q′ < s < p in Theorem 4.3, we have, by Lemma 3.5 and 3.6,

∥T_{δ}^{b}(f)∥_{L^{r,φ}} ⩽ ∥M(T_{δ}^{b}(f))∥_{L^{r,φ}} ⩽ C∥M^{#} (T_{δ}^{b}(f))∥_{L^{r,φ}}

⩽ C∥b∥_{BMO} ∑_{k=1}^{m} ∥M_{δ,s} (T^{k,2}(f))∥_{L^{r,φ}} ⩽ C∥b∥_{BMO} ∑_{k=1}^{m} ∥T^{k,2}(f)∥_{L^{p,φ}}

⩽ C∥b∥_{BMO} ∥f∥_{L^{p,φ}} .

This completes the proof. □

Corollary. Let T_{δ} ∈ GSIO(δ), that is T_{δ} is the singular integral operator as Definition 1.2. Then Theorems 4.1–4.8 hold for T_{δ}^{b}.

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Department of Mathematics (Received 23 07 2012) Changsha University of Science and Technology (Revised 15 09 2012) Changsha 410114 P. R. of China

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