Boundedness on Orlicz space of Toeplitz type operators related to multiplier operator and mean oscillation

In this paper, the boundedness for certain Toeplitz type operator related to the multiplier operator from Lebesgue space to Orlicz space is obtained.


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It is one of the core problems in analysis mathematics of studying the boundedness of integral operators on function spaces. The Toeplitz type operators are important ones, which are the non-trival and natural generalizations of the commututor operator. It is an advanced and hot research topics in harmonic analysis to study the boundedness of Toeplitz type operators on function spaces. In this paper, the boundedness for certain Toeplitz type operator related to the multiplier operator from Lebesgue space to Orlicz space is obtained.
First, let us introduce some notations. Throughout this paper, Q will denote a cube of R n with sides parallel to the axes. For any locally integrable function f , the sharp function of f is defined by where, and in what follows, f Q = |Q| −1 ∫ Q f (x)dx. It is well-known that (see Garcia-Cuerva & Rubio de Francia, 1985) Let M be the Hardy-Littlewood maximal operator defined by We say that f belongs to BMO(R n ) if f # belongs to L ∞ (R n ) and ||f || BMO = ||f # || L ∞. More generally, let be a non-decreasing positive function on [0, + ∞) and define BMO (R n ) as the space of all functions f such that Let be a non-decreasing convex function on [0, + ∞) with (0) = 0. −1 denotes the inverse function of . The Orlicz space L (R n ) is defined by the set of functions f such that ∫ R n ( |f (x)|)dx < ∞ for some > 0. The norm is given by

Results
In this paper, we will study the multilinear operator as following (see Kurtz & Wheeden, 1979).
A bounded measurable function k defined on R n ⧵ {0} is called a multiplier. The multiplier operator T associated with k is defined by where f denotes the Fourier transform of f and S(R n ) is the Schwartz test function class. Now, we recall the definition of the class M(s, l). Denote by |x| ∼ t the fact that the value of x lies in the annulus {x ∈ R n :ct < |x| < Ct}, where 0 < c < C < ∞ are values specified in each instance.
Definition 1 Let l ≥ 0 be a real number and 1 ≤ s ≤ 2. we say that the multiplier k satisfies the condition M(s, l), if for all R > 0 and multi-indices with | | ≤ l, when is a positive integer, and, in addition, if for all |z| < R∕2 and all multi-indices with | | = [l], the integer part of ,i.e.
[l] is the greatest integer less than or equal to , and l = [l]+ when is not an integer.
̂∈ D(R n ) and ̂ vanishes in a neighbourhood of the origin}. The following boundedness property of T on L p (R n ) is proved by Strömberg and Torkinsky (see Kurtz & Wheeden, 1979).
Definition 2 For a real number ̃l ≥ 0 and 1 ≤s < ∞, we say that K verifies the condition M (s,l), and write K ∈M(s,l), if for all multi-indices |̃| ≤l and, in addition, if for all |z| > R 2 , R < 0, and all multi-indices ̃ with |̃| = u, where u denotes the largest integer strictly less than ̃l with ̃l = u + v.
Lemma 1 (see Kurtz & Wheeden, 1979) Let k ∈ M(s, l),1 ≤ s ≤ 2, and l > n s . Then the associated Lemma 2 (see Kurtz & Wheeden, 1979) Lemma 3 (see Kurtz & Wheeden, 1979) Let 1 ≤ s < ∞, suppose that is a positive real number with l > n∕r, 1∕r = max{1∕s,1 − 1∕s}, and k ∈ M(s, l). Then there is a positive constant a, such that Now we can define the Toeplitz type operator associated to the multiplier operator as following.
Definition 3 Let b be a locally integrable function on R n and T be the multiplier operator. By Lemma 1, T(f )(x) = (K * f )(x) for K(x) =ǩ(x). The Toeplitz type operator associated to T is defined by where T k, 1 are T or ±I(the identity operator), T k, 2 are the bounded linear operators on L p (R n ) for 1 < p < ∞ and k = 1, … , m, is a particular operator of the Toeplitz type operator T b . It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see Janson, 1978;Janson & Peetre, 1988;Krantz & Li, 2001;Paluszynski, 1995;Pérez & Pradolini, 2001;Pérez & Trujillo-Gonzalez, 2002). The main purpose of this paper is to prove the boundedness properties for the Toeplitz type operator T b from Lebesgue spaces to Orlicz spaces.
We shall prove the following results in Section 4.
Theorem 1 Let T be the multiplier operator as Definition 3. Suppose that Q = Q(x 0 , d) is a cube with supp f ⊂ (2Q) c and x,x ∈ Q.
Corollary 2 Let 1 < p < s < ∞ and T be the multiplier operator as Definition 3. If T 1 (g) = 0 for any

Some lemma
We need the following preliminary lemmas.
Lemma 4 (see Kurtz & Wheeden, 1979) Let T be the multiplier operator as Definition 3. Then T is bounded on L p (R n ) for 1 < p < ∞.

Proofs of theorems
Now we are in position to prove our results.
Proof of Theorem 1 For suppf ⊂ (2Q) c and x,x ∈ Q, note that |x − y| ∼ |x 0 − y| for x ∈ Q and y ∈ R n ⧵ 2Q. We have (I) By the Hölder's inequality and Lemma 3, we obtain, for 1 < s, t < ∞ with 1∕r + 1∕s + 1∕t = 1, thus For I 2 , by using Theorem 1, We now put these estimates together and take the supremum over all Q such that x ∈ Q, we obtain Thus, taking r such that 1 < r < p, we obtain, by Lemma 5, Secondly, we prove that, if b ∈ Lip (R n ), for any r with 1 < r < n∕ . In fact, similar to the proof of (1) and by Theorem 1, we obtain (T b (f )) # ≤ C||b|| Lip m ∑ k=1 M , r (T k, 2 (f ))