In this paper, we establish sharp maximal function inequalities for the Toeplitz type operator related to some general fractional integral operators

Banach J. Math. Anal. 7 (2013), no. 1, 142–159

B

J

M

A

www.emis.de/journals/BJMA/

SHARP MAXIMAL FUNCTION INEQUALITIES AND BOUNDEDNESS OF TOEPLITZ TYPE OPERATOR RELATED TO GENERAL FRACTIONAL INTEGRAL

OPERATORS

LANZHE LIU

Communicated by L. P. Castro

Abstract. In this paper, we establish sharp maximal function inequalities for the Toeplitz type operator related to some general fractional integral operators.

As an application, we obtain the boundedness of the operator on Lebesgue, Morrey and Triebel-Lizorkin spaces. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

1. Introduction and definitions

As the development of singular integral operators (see [6],[22]), their commu- tators have been well studied. In [3],[20],[21], the authors prove that the com- mutators generated by the singular integral operators and BM O functions are bounded onL^p(Rⁿ) for 1< p <∞. Chanillo (see [2]) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [7],[17], the boundedness of the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and L^p(Rⁿ)(1 < p < ∞) spaces is obtained. In [1], some singular integral operators with general kernel are introduced, and the boundedness of the operators and their commutators gener- ated by BM O and Lipschitz functions is obtained (see [1],[10]). In [8],[9], some Toeplitz type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness of the operators

Date: Received: 3 June 2012; Accepted: 30 August 2012.

2010Mathematics Subject Classification. Primary 47A10; Secondary 42B20, 42B25.

Key words and phrases. Toeplitz type operator, sharp maximal function, Morrey space, BM O, Lipschitz function.

142

generated by BM O and Lipschitz functions is obtained. In this paper, we will study the Toeplitz type operator generated by some general fractional integral operators with the Lipschitz and BM O functions.

First, let us introduce some notations. Throughout this paper, C denotes a positive constant, which is not necessarily the same at each occurrence,Qdenotes a cube of Rⁿ 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

Q3x

1

|Q|

Z

Q

|f(y)−f_Q|dy, where, and in what follows, fQ = |Q|⁻¹R

Qf(x)dx. It is well-known that (see [6],[22])

M^#(f)(x)≈sup

Q3x c∈infC

1

|Q|

Z

Q

|f(y)−c|dy.

We say that f belongs to BM O(Rⁿ) if M^#(f) belongs to L^∞(Rⁿ) and define

||f||_{BM O} =||M^#(f)||_L^∞. It has been known that (see [22])

||f−f₂^k_Q||_{BM O} ≤Ck||f||_{BM O}. Let

M(f)(x) = sup

Q3x

1

|Q|

Z

Q

|f(y)|dy.

Forη >0, let Mη(f)(x) =M(|f|^η)^1/η(x).

For 0< η < n and 1≤r <∞, set Mη,r(f)(x) = sup

Q3x

1

|Q|^1−rη/n Z

Q

|f(y)|^rdy 1/r

.

The A_p weight is defined by (see [6]), for 1< p <∞, A_p =

(

w∈L¹_loc(Rⁿ) : sup

Q

1

|Q|

Z

Q

w(x)dx 1

|Q|

Z

Q

w(x)^−1/(p−1)dx p−1

<∞ )

and

A₁ ={w∈L^p_loc(Rⁿ) :M(w)(x)≤Cw(x), a.e.}.

Forβ >0 andp > 1, let ˙F_p^β,∞(Rⁿ) be the homogeneous Triebel-Lizorkin space (see [17]).

Forβ >0, the Lipschitz space Lip_β(Rⁿ) is the space of functions f such that

||f||_Lipβ = sup

x,y∈Rn x6=y

|f(x)−f(y)|

|x−y|^β <∞.

Definition 1.1. Let ϕ be a positive, increasing function on R⁺ for which there exists a constant D >0 such that

ϕ(2t)≤Dϕ(t) for t≥0.

Letf be a locally integrable function onRⁿ. Set, for 1≤p < ∞,

||f||_L^p,ϕ = sup

x∈Rⁿ, d>0

1 ϕ(d)

Z

Q(x,d)

|f(y)|^pdy 1/p

,

where Q(x, d) denotes a cube of Rⁿ with sides parallel to the axes, whose center isx and side length is d. The generalized Morrey space is defined by

L^p,ϕ(Rⁿ) = {f ∈L¹_loc(Rⁿ) :||f||_L^p,ϕ <∞}.

If ϕ(d) = d^δ, δ > 0, then L^p,ϕ(Rⁿ) = L^p,δ(Rⁿ), which is the classical Morrey spaces (see [18],[19]). Ifϕ(d) = 1, then L^p,ϕ(Rⁿ) =L^p(Rⁿ), which is the Lebesgue spaces (see [6]).

Since the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [4],[5],[11],[16]).

In this paper, we will study some singular integral operators as following (see [1]).

Definition 1.2. LetF_t(x, y) be defined onRⁿ×Rⁿ×[0,+∞) andb be a locally integrable function onRⁿ. Set

F_t(f)(x) = Z

Rⁿ

F_t(x, y)f(y)dy

for every bounded and compactly supported function f. And F_t satisfies: there is a sequence of positive constant numbers{C_j}such that for any j ≥1,

Z

2|y−z|<|x−y|

(||F_t(x, y)−F_t(x, z)||+||F_t(y, x)−F_t(z, x)||)dx≤C, and

Z

2^j|z−y|≤|x−y|<2^j+1|z−y|

(||F_t(x, y)−F_t(x, z)||+||F_t(y, x)−F_t(z, x)||)^qdy 1/q

≤ C_j(2^j|z−y|)^−n/q0,

where 1< q⁰ <2 and 1/q+ 1/q⁰ = 1.

Let H be the Banach space H = {h : ||h|| < ∞}. For each fixed x ∈ Rⁿ, it is viewF_t(f)(x) as the mapping from [0,+∞) to H. Set

T(f)(x) =||F_t(f)(x)||,

whichT is bounded onL²(Rⁿ). Letbbe a locally integrable function onRⁿ. The Toeplitz type operator related toT is defined by

T^b(f) = ||F_t^b(f)||, where

F_t^b(f) =

m

X

k=1

(F_t^k,1M_bI_αF_t^k,2+F_t^k,3I_αM_bF_t^k,4),

moreover, F_t^k,1(f) are F_t(f) or ±I(the identity operator), F_t^k,2(f), F_t^k,3(f) and F_t^k,4(f) are the linear operators with T^k,2(f) = ||F_t^k,2(f)||, T^k,4(f) = ||F_t^k,4(f)||

and T^k,3 = ±I, k = 1,· · · , m, M_b(f) = bf and I_α is the fractional integral operator(0< α < n) (see [2]).

Note that the commutator [b, T](f) = bT(f)−T(bf) is a particular operator of the Toeplitz type operators T^b. The Toeplitz type operators T^b are the non- trivial generalizations of the commutator. It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [20],[21]). The main purpose of this paper is to prove sharp maximal inequalities for the Toeplitz type operatorsT^b. As the application, we obtain the the L^p-norm inequality and the boundedness of T^b on Triebel-Lizorkin spaces.

The operators include Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

2. Some Preliminary Lemmas We begin with some preliminary lemmas.

Lemma 2.1. (see [1]) Let T be the integral operator as Definition 1.2. Then T is bounded on L^p(Rⁿ) for 1< p <∞.

Lemma 2.2. (see [17]). For 0< β <1 and 1< p <∞, we have

||f||_F˙^β,∞

p ≈

sup

Q3·

1

|Q|^1+β/n Z

Q

|f(x)−f_Q|dx L^p

≈

sup

Q3·

infc

1

|Q|^1+β/n Z

Q

|f(x)−c|dx L^p

.

Lemma 2.3. (see[6]). Let0< p <∞andw∈ ∪1≤r<∞A_r. Then, for any smooth function f for which the left-hand side is finite,

Z

Rⁿ

M(f)(x)^pw(x)dx ≤C Z

Rⁿ

M^#(f)(x)^pw(x)dx.

Lemma 2.4. (see [2],[6]). Suppose that 0 < α < n, 1 ≤ s < p < n/α and 1/r= 1/p−α/n. Then

||I_α(f)||_L^r ≤C||f||_L^p and

||M_α,s(f)||_L^r ≤C||f||_L^p.

Lemma 2.5. Let 1< p <∞, 0< D < 2ⁿ. 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(x₀, d) in Rⁿ, we know M(χ_Q) ∈ A₁ for any cube Q=Q(x, d) by [6]. Noticing thatM(χ_Q)≤1 andM(χ_Q)(x)≤dⁿ/(|x−x₀| −d)ⁿ

if x∈Q^c. By Lemma 2.3, we have, for f ∈L^p,ϕ(Rⁿ), Z

Q

M(f)(x)^pdx = Z

Rⁿ

M(f)(x)^pχ_Q(x)dx

≤ Z

Rⁿ

M(f)(x)^pM(χ_Q)(x)dx≤C Z

Rⁿ

M^#(f)(x)|^pM(χ_Q)(x)dx

= C

Z

Q

M^#(f)(x)^pM(χ_Q)(x)dx+

∞

X

k=0

Z

2^k+1Q\2^kQ

M^#(f)(x)^pM(χ_Q)(x)dx

!

≤ C

Z

Q

M^#(f)(x)^pdx+

∞

X

k=0

Z

2^k+1Q\2^kQ

M^#(f)(x)^p |Q|

|2^k+1Q|dx

!

≤ C

Z

Q

M^#(f)(x)^pdx+

∞

X

k=0

Z

2^k+1Q

M^#(f)(x)^p2^−kndy

!

≤ C||M^#(f)||^p_Lp,ϕ

∞

X

k=0

2^−knϕ(2^k+1d)

≤ C||M^#(f)||^p_Lp,ϕ

∞

X

k=0

(2⁻ⁿD)^kϕ(d)

≤ C||M^#(f)||^p_Lp,ϕϕ(d), thus

1 ϕ(d)

Z

Q

M(f)(x)^pdx 1/p

≤C 1

ϕ(d) Z

Q

M^#(f)(x)^pdx 1/p

and

||M(f)||L^p,ϕ ≤C||M^#(f)||L^p,ϕ.

This finishes the proof.

Lemma 2.6. Let0< α < n, 0< D < 2ⁿ,1≤s < p < n/αand1/r = 1/p−α/n.

Then

||I_α(f)||_L^r,ϕ ≤C||f||_L^p,ϕ and

||M_α,s(f)||_L^r,ϕ ≤C||f||_L^p,ϕ.

The proof of the Lemma is similar to that of Lemma 2.5 by Lemma 2.4, we omit the details.

3. Theorems And Proofs

Now we are in the position to prove the following theorems. Suppose thatT is the integral operator as Definition 1.2 in the following theorems.

Theorem 3.1. Let 0 < β < 1, q⁰ ≤ s < ∞, b ∈ Lip_β(Rⁿ) and the sequence {C_j} ∈ l¹. If F_t¹(g) = 0 for any g ∈ L^u(Rⁿ)(1 < u < ∞), then there exists a constant C >0 such that for any f ∈C₀^∞(Rⁿ) and x˜∈Rⁿ,

M^#(T^b(f))(˜x)≤C||b||_Lipβ

m

X

k=1

(M_β,s(I_αT^k,2(f))(˜x) +M_β+α,s(T^k,4(f))(˜x)).

Proof. It suffices to prove that for f ∈ C₀^∞(Rⁿ) and some constant C₀, the fol- lowing inequality holds:

1

|Q|

Z

Q

T^b(f)(x)−C₀

dx≤C||b||_Lipβ

m

X

k=1

(M_β,s(I_αT^k,2(f))(˜x)+M_β+α,s(T^k,4(f))(˜x)).

Without loss of generality, we may assumeT^k,1 are T(k= 1,· · ·, m). Fix a cube Q=Q(x₀, d) and ˜x∈Q. Write, for f₁ =f χ_2Q and f₂ =f χ_(2Q)^c,

F_t^b(f)(x) =

m

X

k=1

F_t^k,1M_bI_αF_t^k,2(f)(x) +

m

X

k=1

F_t^k,3I_αM_bF_t^k,4(f)(x) =A_b(x) +B_b(x)

= Ab−b_Q(x) +Bb−b_Q(x), where

Ab−b_Q(x) =

m

X

k=1

F_t^k,1M(b−b_Q)χ2QI_αF_t^k,2(f)(x) +

m

X

k=1

F_t^k,1M(b−b_Q)χ_(2Q)cI_αF_t^k,2(f)(x)

= A₁(x) +A₂(x) and

Bb−b_Q(x) =

m

X

k=1

F_t^k,3IαM(b−b_Q)χ2QF_t^k,4(f)(x) +

m

X

k=1

F_t^k,3IαM(b−b_Q)χ(2Q)cF_t^k,4(f)(x)

= B₁(x) +B₂(x).

Then

1

|Q|

Z

Q

Ab−b_Q(f)(x)−A₂(x₀) dx

≤ 1

|Q|

Z

Q

|A₁(x)|dx+ 1

|Q|

Z

Q

|A₂(x)−A₂(x₀)|dx=I₁+I₂ and

1

|Q|

Z

Q

Bb−b_Q(f)(x)−B₂(x₀) dx

≤ 1

|Q|

Z

Q

|B₁(x)|dx+ 1

|Q|

Z

Q

|B₂(x)−B₂(x₀)|dx=I₃+I₄.

ForI₁, by H¨older’s inequality and Lemma 2.1, we obtain, for 1/r = 1/s−α/n, 1

|Q|

Z

Q

||F_t^k,1M(b−b_Q)χ2QI_αF_t^k,2(f)(x)||dx

= 1

|Q|

Z

Q

|T^k,1M(b−b_Q)χ2QIαT^k,2(f)(x)|dx

≤ 1

|Q|

Z

Rⁿ

|T^k,1M(b−b_Q)χ2QI_αT^k,2(f)(x)|^sdx 1/s

≤ C|Q|^−1/s Z

Rⁿ

|M(b−b_Q)χ2QI_αT^k,2(f)(x)|^sdx 1/s

≤ C|Q|^−1/s Z

2Q

(|b(x)−b_Q||I_αT^k,2(f)(x)|)^sdx 1/s

≤ C|Q|^−1/s||b||Lip_β|2Q|^β/n|2Q|^1/s−β/n

1

|2Q|^1−sβ/n Z

2Q

|IαT^k,2(f)(x)|^sdx 1/s

≤ C||b||Lip_βMβ,s(IαT^k,2(f))(˜x), thus

I₁ ≤

m

X

k=1

1

|Q|

Z

Rⁿ

||F_t^k,1M(b−b_Q)χ2QI_αF_t^k,2(f)(x)||dx≤C||b||_Lipβ

m

X

k=1

M_β,s(I_αT^k,2(f))(˜x).

ForI₂, by the boundedness of T and recalling that s > q⁰, we get, for x∈Q,

||F_t^k,1M(b−b_Q)χ_(2Q)cI_αF_t^k,2(f)(x)−F_t^k,1M(b−b_Q)χ_(2Q)cI_αF_t^k,2(f)(x₀)||

≤ Z

(2Q)^c

|b(y)−b_2Q|||F_t(x, y)−F_t(x₀, y)|||I_αT^k,2(f)(y)|dy

=

∞

X

j=1

Z

2^jd≤|y−x0|<2^j+1d

|b(y)−b_2Q|||F_t(x, y)−F_t(x₀, y)|||I_αT^k,2(f)(y)|dy

≤ C||b||_Lipβ

∞

X

j=1

|2^j+1Q|^β/n Z

2^jd≤|y−x0|<2^j+1d

||F_t(x, y)−F_t(x₀, y)||^qdy 1/q

× Z

2^j+1Q

|I_αT^k,2(f)(y)|^q0dy 1/q⁰

≤ C||b||Lip_β

∞

X

j=1

|2^j+1Q|^β/nCj(2^jd)^−n/q0|2^j+1Q|^1/q0−β/n

×

1

|2^j+1Q|^1−sβ/n Z

2^j+1Q

|I_αT^k,2(f)(y)|^sdy 1/s

≤ C||b||_LipβM_β,s(I_αT^k,2(f))(˜x)

∞

X

j=1

C_j

≤ C||b||_LipβM_β,s(I_αT^k,2(f))(˜x),

thus

I₂ ≤ 1

|Q|

Z

Q m

X

k=1

||F_t^k,1M(b−b_Q)χ_(2Q)cI_αF_t^k,2(f)(x)

−F_t^k,1M(b−b_Q)χ(2Q)cI_αF_t^k,2(f)(x₀)||dx

≤ C||b||_Lipβ

m

X

k=1

M_β,s(I_αT^k,2(f))(˜x).

Similarly I₃ ≤

m

X

k=1

1

|Q|

Z

Rⁿ

||I_αM(b−b_Q)χ2QF_t^k,4(f)(x)||^rdx 1/r

≤C

m

X

k=1

|Q|^−1/r Z

2Q

(|b(x)−b_Q||T^k,4(f)(x)|)^sdx 1/s

≤C||b||_Lipβ

m

X

k=1

|Q|^−1/r|2Q|^β/n|2Q|1/s−(β+α)/n

×

1

|2Q|^{1−s(β+α)/n} Z

2Q

|T^k,4(f)(x)|^sdx 1/s

≤C||b||_Lipβ

m

X

k=1

M_β+α,s(T^k,6(f))(˜x),

I₄ ≤

m

X

k=1

1

|Q|

Z

Q

Z

(2Q)^c

|b(y)−b_2Q|

1

|x−y|^n−α − 1

|x₀ −y|^n−α

||F_t^k,4(f)(y)||dydx

≤C

m

X

k=1

∞

X

j=1

||b||Lip_β|2^j+1Q|^β/n Z

2^jd≤|y−x₀|<2^j+1d

d

|x₀−y|^n−α+1|T^k,4(f)(y)|dy

≤C||b||_Lipβ

m

X

k=1

∞

X

j=1

(2^jd)^βd(2^jd)^−n+α−1(2^jd)^n(1−1/s)(2^jd)^{n/s−β−α}

×

1

|2^j+1Q|^{1−s(β+α)/n} Z

2^j+1Q

|T^k,4(f)(y)|^sdy 1/s

≤C||b||_Lipβ

m

X

k=1

M_β+α,s(T^k,4(f))(˜x)

∞

X

j=1

2^−j

≤C||b||_Lipβ

m

X

k=1

M_β+α,s(T^k,4(f))(˜x).

This completes the proof of Theorem3.1.

Theorem 3.2. Let 0 < β < 1, q⁰ ≤ s < ∞, b ∈ Lip_β(Rⁿ) and the sequence {2^jβC_j} ∈l¹. If F_t¹(g) = 0 for any g ∈ L^u(Rⁿ)(1 < u < ∞), then there exists a

constant C >0 such that for any f ∈C₀^∞(Rⁿ) and x˜∈Rⁿ,

sup

Q3˜x c∈infC

1

|Q|^1+β/n Z

Q

T^b(f)(x)−c

dx≤C||b||Lip_β m

X

k=1

(Ms(IαT^k,2(f))(˜x) +Mα,s(T^k,4(f))(˜x)).

Proof. It suffices to prove for f ∈ C₀^∞(Rⁿ) and some constant C₀, the following inequality holds:

sup

Q3˜x

1

|Q|^1+β/n Z

Q

T^b(f)(x)−C0

dx≤C||b||Lip_β m

X

k=1

(Ms(IαT^k,2(f))(˜x) +Mα,s(T^k,4(f))(˜x)).

Without loss of generality, we may assume T^k,1 are T(k = 1,· · · , m). Fix a cubeQ=Q(x₀, d) and ˜x∈Q. Similar to the proof of Theorem 3.1, we have, for f₁ =f χ_2Q and f₂ =f χ_(2Q)^c,

1

|Q|^1+β/n Z

Q

T^b(f)(x)−A₂(x₀)−B₂(x₀) dx

≤ 1

|Q|^1+β/n Z

Q

|A₁(x)|dx+ 1

|Q|^1+β/n Z

Q

|A₂(x)−A₂(x₀)|dx

+ 1

|Q|^1+β/n Z

Q

|B₁(x)|dx+ 1

|Q|^1+β/n Z

Q

|B₂(x)−B₂(x₀)|dx

= I5+I6+I7+I8.

By using the same argument as in the proof of Theorem 3.1, we get, for 1/r = 1/s−α/n,

I₅ ≤ |Q|^−β/n

m

X

k=1

1

|Q|

Z

Rⁿ

|T^k,1M(b−b_Q)χ2QI_αT^k,2(f)(x)|^sdx 1/s

≤ C|Q|^−β/n

m

X

k=1

|Q|^−1/s Z

2Q

(|b(x)−b_Q||I_αT^k,2(f)(x)|)^sdx 1/s

≤ C|Q|^−β/n

m

X

k=1

|Q|^−1/s||b||_Lipβ|2Q|^β/n|Q|^1/s 1

|Q|

Z

2Q

|I_αT^k,2(f)(x)|^sdx 1/s

≤ C||b||_Lipβ

m

X

k=1

M_s(I_αT^k,2(f))(˜x),

I6 ≤ |Q|^−β/n

m

X

k=1

1

|Q|

Z

Q

∞

X

j=1

Z

2^jd≤|y−x₀|<2^j+1d

|b(y)−b2Q|

× ||F_t(x, y)−F_t(x₀, y)|||I_αT^k,2(f)(y)|dydx

≤ |Q|^−β/n

m

X

k=1

C

|Q|

Z

Q

∞

X

j=1

||b||_Lipβ|2^j+1Q|^β/n Z

2^j+1Q

|I_αT^k,2(f)(y)|^q0dy 1/q⁰

× Z

2^jd≤|y−x0|<2^j+1d

||F_t(x, y)−F_t(x₀, y)||^qdy 1/q

dx

≤C||b||_Lipβ|Q|^−β/n

m

X

k=1

∞

X

j=1

|2^j+1Q|^β/nC_j(2^jd)^−n/q0|2^j+1Q|^1/q0

×

1

|2^j+1Q|

Z

2^j+1Q

|I_αT^k,2(f)(y)|^sdy 1/s

≤C||b||Lip_β m

X

k=1

Ms(IαT^k,2(f))(˜x)

∞

X

j=1

2^jβCj

≤C||b||Lip_β m

X

k=1

Ms(IαT^k,2(f))(˜x),

I₇ ≤ |Q|^−β/n

m

X

k=1

1

|Q|

Z

Rⁿ

|I_αM_(b−bQ)χ2QT^k,4(f)(x)|^rdx 1/r

≤C|Q|^{−β/n−1/r}

m

X

k=1

Z

2Q

(|b(x)−b_Q||T^k,4(f)(x)|)^sdx 1/s

≤C||b||_Lipβ

m

X

k=1

|Q|^{−β/n−1/r}|2Q|^β/n|Q|^1/s−α/n

1

|Q|^1−sα/n Z

2Q

|T^k,4(f)(x)|^sdx ¹_s

≤C||b||_Lipβ

m

X

k=1

M_α,s(T^k,4(f))(˜x),

I₈ ≤ |Q|^−β/n−1

m

X

k=1

Z

Q

Z

(2Q)^c

|b(y)−b_2Q|

1

|x−y|^n−α − 1

|x₀−y|^n−α

|T^k,4(f)(y)|dydx

≤C|Q|^−β/n

m

X

k=1

∞

X

j=1

||b||_Lipβ|2^j+1Q|^β/n

× Z

2^jd≤|y−x0|<2^j+1d

d

|x0−y|^n−α+1|T^k,4(f)(y)|dy

≤C||b||_Lipβ

m

X

k=1

∞

X

j=1

d^−β(2^jd)^βd(2^jd)^−n+α−1(2^jd)^n(1−1/s)(2^jd)^n/s−α

×

1

|2^j+1Q|^1−sα/n Z

2^j+1Q

|T^k,4(f)(y)|^sdy 1/s

≤C||b||_Lipβ

m

X

k=1

M_α,s(T^k,4(f))(˜x)

∞

X

j=1

2^j(β−1) ≤C||b||_Lipβ

m

X

k=1

M_α,s(T^k,4(f))(˜x).

This completes the proof of Theorem3.2.

Theorem 3.3. Let q⁰ ≤ s <∞, b ∈BM O(Rⁿ) and the sequence {jC_j} ∈l¹. If F_t¹(g) = 0 for any g ∈ L^u(Rⁿ)(1 < u < ∞), then there exists a constant C > 0 such that for any f ∈C₀^∞(Rⁿ) and x˜∈Rⁿ,

M^#(T^b(f))(˜x)≤C||b||_{BM O}

m

X

k=1

(M_s(I_αT^k,2(f))(˜x) +M_α,s(T^k,4(f))(˜x)).

Proof. It suffices to prove for f ∈ C₀^∞(Rⁿ) and some constant C0, the following inequality holds:

1

|Q|

Z

Q

T^b(f)(x)−C₀

dx≤C||b||_{BM O}

m

X

k=1

(M_s(I_αT^k,2(f))(˜x)+M_α,s(T^k,4(f))(˜x)).

Without loss of generality, we may assumeT^k,1 are T(k= 1,· · ·, m). Fix a cube Q = Q(x₀, d) and ˜x ∈ Q. Similar to the proof of Theorem 3.1, we have, for f₁ =f χ_2Q and f₂ =f χ_(2Q)^c,

1

|Q|

Z

Q

T^b(f)(x)−A₂(x₀)−B₂(x₀) dx

≤ 1

|Q|

Z

Q

|A₁(x)|dx+ 1

|Q|

Z

Q

|A₂(x)−A₂(x₀)|dx

+ 1

|Q|

Z

Q

|B₁(x)|dx+ 1

|Q|

Z

Q

|B₂(x)−B₂(x₀)|dx=I₉ +I₁₀+I₁₁+I₁₂. By using the same argument as in the proof of Theorem3.1, we get, for 1< r1 < s, 1< p <∞,1< r2 < swith 1/p+1/q+1/r2 = 1, 1/r3 = 1/p−α/nwith 1< p < s,

I₉ ≤

m

X

k=1

1

|Q|

Z

Rⁿ

|T^k,1M(b−b_Q)χ2QI_αT^k,2(f)(x)|^r1dx 1/r1

≤C

m

X

k=1

|Q|^−1/r Z

2Q

(|b(x)−b_Q||I_αT^k,2(f)(x)|)^r1dx 1/r1

≤C

m

X

k=1

1

|Q|

Z

2Q

|I_αT^k,2(f)(x)|^sdx 1/s

1

|Q|

Z

2Q

|b(x)−b_Q|^sr1/(s−r1)dx ^s−r_sr¹

1

≤C||b||_{BM O}

m

X

k=1

M_s(I_αT^k,2(f))(˜x),

I10 ≤

m

X

k=1

1

|Q|

Z

Q

∞

X

j=1

Z

2^jd≤|y−x₀|<2^j+1d

|b(y)−b2Q|

× ||F_t(x, y)−F_t(x₀, y)|||I_αT^k,2(f)(y)|dydx

≤

m

X

k=1

C

|Q|

Z

Q

∞

X

j=1

Z

2^jd≤|y−x0|<2^j+1d

||F_t(x, y)−F_t(x₀, y)||^qdy 1/q

× Z

2^j+1Q

|b(y)−b_Q|^pdy

1/pZ

2^j+1Q

|I_αT^k,2(f)(y)|^r2dy 1/r2

dx

≤ C||b||_{BM O}

m

X

k=1

∞

X

j=1

C_j(2^jd)^−n/q0j(2^jd)^n/p(2^jd)^n/s

×

1

|2^j+1Q|

Z

2^j+1Q

|IαT^k,2(f)(y)|^sdy 1/s

≤ C||b||_{BM O}

m

X

k=1

M_s(I_αT^k,2(f))(˜x)

∞

X

j=1

jC_j

≤ C||b||_{BM O}

m

X

k=1

M_s(I_αT^k,2(f))(˜x),

I₁₁ ≤

m

X

k=1

1

|Q|

Z

Rⁿ

|I_αM(b−b_Q)χ2QT^k,4(f)(x)|^r3dx 1/r3

≤ C|Q|^−1/r3

m

X

k=1

Z

2Q

(|b(x)−b_Q||T^k,4(f)(x)|)^pdx 1/p

≤ C

m

X

k=1

1

|Q|

Z

2Q

|b(x)−b_Q|^ps/(s−p)dx

(s−p)/ps

×

1

|Q|^1−sα/n Z

2Q

|T^k,4(f)(x)|^sdx 1/s

≤ C||b||_{BM O}

m

X

k=1

M_α,s(T^k,4(f))(˜x),

I₁₂ ≤ |Q|⁻¹

m

X

k=1

Z

Q

Z

(2Q)^c

|b(y)−b_2Q|

×

1

|x−y|^n−α − 1

|x₀−y|^n−α

|T^k,4(f)(y)|dydx

≤ C

m

X

k=1

∞

X

j=1

Z

2^jd≤|y−x0|<2^j+1d

|b(y)−b_2Q| d

|x₀ −y|^n−α+1|T^k,4(f)(y)|dy

≤ C

m

X

k=1

∞

X

j=1

d

(2^jd)^n−α+1(2^jd)^n(1−1/s)

1

|2^j+1Q|

Z

2^j+1Q

|b(y)−bQ|^s0dy 1/s⁰

× (2^jd)^n/s−α

1

|2^j+1Q|^1−sα/n Z

2^j+1Q

|T^k,4(f)(y)|^sdy 1/s

≤ C||b||_{BM O}

m

X

k=1

M_α,s(T^k,4(f))(˜x)

∞

X

j=1

j2^−j

≤ C||b||_{BM O}

m

X

k=1

M_α,s(T^k,4(f))(˜x).

This completes the proof of Theorem3.3

Theorem 3.4. Let 0 < β < min(1, n/q⁰), q⁰ < p < n/(α+β), 1/r = 1/p− (α+β)/n, b ∈ Lip_β(Rⁿ) and the sequence {Cj} ∈ l¹. If F_t¹(g) = 0 for any g ∈ L^u(Rⁿ)(1 < u < ∞) and T^k,4(f) = ||F_t^k,4(f)|| are the bounded operators on L^p(Rⁿ) for 1< p <∞(1≤k ≤m), then T^b is bounded from L^p(Rⁿ) to L^r(Rⁿ).

Proof. Choose q⁰ < s < p in Theorem 3.1 and set 1/v = 1/p−β/n. We have, by Lemmas2.3 and 2.4,

||T^b(f)||_L^r ≤ kM(T^b(f))k_L^r ≤CkM^#(T^b(f))k_L^r

≤ C||b||_Lipβ

m

X

k=1

(kM_β,s(I_αT^k,2(f))k_L^r +kM_β+α,s(T^k,4(f))k_L^r)

≤ C||b||_Lipβ

m

X

k=1

(kI_αT^k,2(f)k_L^v +kT^k,4(f)k_L^p)

≤ C||b||_Lipβ

m

X

k=1

(kT^k,2(f)k_L^p+kfk_L^p)

≤ C||b||Lip_βkfkL^p.

This completes the proof of the theorem.

Theorem 3.5. Let 0< β < min(1, n/q⁰), q⁰ < p < n/(α+β), 1/r= 1/p−(α+ β)/n, 0 < D < 2ⁿ, b ∈ Lip_β(Rⁿ) and the sequence {Cj} ∈ l¹. If F_t¹(g) = 0 for any g ∈L^u(Rⁿ)(1 < u <∞) and T^k,4(f) =||F_t^k,4(f)|| are the bounded operators on L^p,ϕ(Rⁿ) for 1 < p < ∞(1 ≤ k ≤ m), then T^b is bounded from L^p,ϕ(Rⁿ) to L^r,ϕ(Rⁿ).

Proof. Choose q⁰ < s < p in Theorem 3.1 and set 1/v = 1/p−β/n. We have, by Lemmas2.5 and 2.6,

||T^b(f)||_L^r,ϕ ≤ kM(T^b(f))k_L^r,ϕ ≤CkM^#(T^b(f))k_L^r,ϕ

≤ C||b||Lip_β m

X

k=1

(kMβ,s(IαT^k,2(f))kL^r,ϕ+kMβ+α,s(T^k,4(f))kL^r,ϕ)

≤ C||b||Lip_β m

X

k=1

(kIαT^k,2(f)kL^v,ϕ+kT^k,4(f)kL^p,ϕ)

≤ C||b||Lip_β m

X

k=1

(kT^k,2(f)kL^p,ϕ +kfkL^p,ϕ)

≤ C||b||_Lipβkfk_L^p,ϕ.

This completes the proof of the theorem.

Theorem 3.6. Let 0 < β < min(1, n/q⁰), q⁰ < p < n/α, 1/r = 1/p−α/n, b∈Lip_β(Rⁿ)and the sequence{2^jβCj} ∈l¹. IfF_t¹(g) = 0for anyg ∈L^u(Rⁿ)(1 <

u < ∞) and T^k,4(f) = ||F_t^k,4(f)|| are the bounded operators on L^p(Rⁿ) for 1 <

p <∞(1≤k ≤m), then T^b is bounded from L^p(Rⁿ) to F˙_r^β,∞(Rⁿ).

Proof. Choose q⁰ < s < p in Theorem 3.2. We have, by Lemmas 2.2,2.3 and 2.4,

||T^b(f)||_F˙^β,∞

r ≤C

sup

Q3·

1

|Q|^1+β/n Z

Q

T^b(f)(x)−C₀ dx

L^r

≤ C||b||_Lipβ

m

X

k=1

(kM_s(I_αT^k,2(f))k_L^r +kM_α,s(T^k,4(f))k_L^r)

≤ C||b||_Lipβ

m

X

k=1

(kI_αT^k,2(f)k_L^r +kT^k,4(f)k_L^p)

≤ C||b||_Lipβ

m

X

k=1

(kT^k,2(f)k_L^p+kfk_L^p)

≤ C||b||_Lipβ||f||_L^p.

This completes the proof of the theorem.

Theorem 3.7. Let q⁰ < p < n/α, 1/r = 1/p−α/n, b ∈ BM O(Rⁿ) and the sequence {jC_j} ∈ l¹. If F_t¹(g) = 0 for any g ∈ L^u(Rⁿ)(1 < u < ∞) and T^k,4(f) = ||F_t^k,4(f)|| are the bounded operators on L^p(Rⁿ) for 1 < p < ∞(1 ≤ k ≤m), then T^b is bounded from L^p(Rⁿ) to L^r(Rⁿ).

Proof. Choose q⁰ < s < p in Theorem 3.3, we have, by Lemmas 2.3 and 2.4,

||T^b(f)||_L^r ≤ kM(T^b(f))k_L^r ≤CkM^#(T^b(f))k_L^r

≤ C||b||_{BM O}

m

X

k=1

(kM_s(I_αT^k,2(f))k_L^r +kM_α,s(T^k,4(f))k_L^r)

≤ C||b||_{BM O}

m

X

k=1

(kI_αT^k,2(f)k_L^r +kT^k,4(f)k_L^p)

≤ C||b||BM OkfkL^p.

This completes the proof of the theorem.

Theorem 3.8. Let 0< D <2ⁿ, q⁰ < p < n/α, 1/r= 1/p−α/n, b∈BM O(Rⁿ) and the sequence {jC_j} ∈l¹. If F_t¹(g) = 0 for any g ∈L^u(Rⁿ)(1 < u < ∞) and T^k,4(f) = ||F_t^k,4(f)|| are the bounded operators on L^p,ϕ(Rⁿ) for 1< p < ∞(1≤ k ≤m), then T^b is bounded from L^p,ϕ(Rⁿ) to L^r,ϕ(Rⁿ).

Proof. Choose q⁰ < s < p in Theorem 3.3, we have, by Lemmas 2.5 and 2.6,

||T^b(f)||_L^r,ϕ ≤ kM(T^b(f))k_L^r,ϕ ≤CkM^#(T^b(f))k_L^r,ϕ

≤ C||b||BM O m

X

k=1

(kMs(IαT^k,2(f))kL^r,ϕ+kMα,s(T^k,4(f))kL^r,ϕ)

≤ C||b||BM O m

X

k=1

(kIαT^k,2(f)kL^r,ϕ+kT^k,4(f)kL^p,ϕ)

≤ C||b||_{BM O}kfk_L^p,ϕ.

This completes the proof of the theorem.

4. Applications

In this section we shall apply Theorems 3.1-3.8 to some particular operators such as the Littlewood-Paley operator, Marcinkiewicz operator and Bochner- Riesz operator.

Application 1.Littlewood-Paley operator.

Fixed ε >0. Let ψ be a fixed function which satisfies:

(1) R

Rⁿψ(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|;

Let ψ_t(x) = t⁻ⁿψ(x/t) for t > 0 and F_t(f)(x) = R

Rⁿf(y)ψ_t(x−y)dy. The Littlewood-Paley operator is defined(see [23])

g_ψ(f)(x) = Z ∞

0

|F_t(f)(x)|²dt t

1/2

.

Set H be the space H =

(

h:||h||= Z ∞

0

|h(t)|²dt/t 1/2

<∞ )

.

Letb be a locally integrable function on Rⁿ. The Toeplitz type operator related to the Littlewood-Paley operator is defined by

g^b_ψ(f)(x) = Z ∞

0

|F_t^b(f)(x)|²dt t

1/2

, where

F_t^b =

m

X

k=1

(F_t^k,1M_bI_αF_t^k,2 +F_t^k,3I_αM_bF_t^k,4),

moreover, F_t^k,1(f) are F_t(f) or ±I(the identity operator), T^k,2(f) = ||F_t^k,2(f)||

and T^k,4(f) = ||F_t^k,4(f)|| are the bounded linear operators on L^p(Rⁿ) for 1 <

p < ∞, T^k,3 = ±I, k = 1,· · · , m, M_b(f) = bf and I_α is the fractional integral operator(0 < α < n). It is easily to see that g_ψ^b satisfies the conditions of Theorems 3.1-3.8 (see [12],[13],[14]), thus these theorems hold for g_ψ^b.

Application 2.Marcinkiewicz operator.

Fixed 0< γ ≤1. Let Ω be homogeneous of degree zero onRⁿ withR

Sⁿ⁻¹Ω(x⁰) dσ(x⁰) = 0. Assume that Ω∈Lip_γ(Sⁿ⁻¹). Set F_t(f)(x) =R

|x−y|≤t

Ω(x−y)

|x−y|ⁿ⁻¹f(y)dy.

The Marcinkiewicz operator is defined by (see [24]) µ_Ω(f)(x) =

Z ∞

0

|F_t(f)(x)|²dt t³

1/2

.

Set H be the space H =

(

h:||h||= Z ∞

0

|h(t)|²dt/t³ 1/2

<∞ )

.

Letb be a locally integrable function on Rⁿ. The Toeplitz type operator related to the Marcinkiewicz operator is defined by

µ^b_Ω(f)(x) = Z ∞

0

|F_t^b(f)(x)|²dt t³

1/2

, where

F_t^b =

m

X

k=1

(F_t^k,1M_bI_αF_t^k,2 +F_t^k,3I_αM_bF_t^k,4),

moreover, F_t^k,1(f) are F_t(f) or ±I(the identity operator), T^k,2(f) = ||F_t^k,2(f)||

and T^k,4(f) = ||F_t^k,4(f)|| are the bounded linear operators on L^p(Rⁿ) for 1 <

p < ∞, T^k,3 = ±I, k = 1,· · · , m, M_b(f) = bf and I_α is the fractional integral operator(0 < α < n). It is easily to see that µ^b_Ω satisfies the conditions of Theorems 3.1-3.8 (see [12],[13],[14],[24]), thus these theorems hold for µ^b_Ω.

Application 3.Bochner-Riesz operator.

Let δ > (n−1)/2, F_t^δ(f)(ξ) = (1ˆ −t²|ξ|²)^δ₊fˆ(ξ) and B_t^δ(z) = t⁻ⁿB^δ(z/t) for t >0. The maximal Bochner-Riesz operator is defined by(see [15])

Bδ,∗(f)(x) = sup

t>0

|F_t^δ(f)(x)|.

SetH be the spaceH ={h:||h||= sup

t>0

|h(t)|<∞}. Letb be a locally integrable function onRⁿ. The Toeplitz type operator related to the maximal Bochner-Riesz operator is defined by

B_δ,∗^b (f)(x) = sup

t>0

|B_δ,t^b (f)(x)|, where

B_δ,t^b =

m

X

k=1

(F_t^k,1M_bI_αF_t^k,2+F_t^k,3I_αM_bF_t^k,4),

moreover, F_t^k,1(f) are F_t(f) or ±I(the identity operator), T^k,2(f) = ||F_t^k,2(f)||

and T^k,4(f) = ||F_t^k,4(f)|| are the bounded linear operators on L^p(Rⁿ) for 1 <

p < ∞, T^k,3 = ±I, k = 1,· · · , m, M_b(f) = bf and I_α is the fractional integral