Boundedness of Toeplitz type operators associated to Riesz potential operator with general kernel on Orlicz space

Abstract In this paper, the boundedness properties for some Toeplitz type operators associated to the Riesz potential and general integral operators from Lebesgue spaces to Orlicz spaces are proved. The general integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

and there is a sequence of positive constant numbers fC k g such that for any k 1, 0 B @ Z 2 k jz yjÄjx yj<2 kC1 jz yj .jK.x; y/ K.x; z/j C jK.y; x/ K.z; x/j/ q dy where 1 < q 0 < 2 and 1=q C 1=q 0 D 1. Let b be a locally integrable function on R n . The Toeplitz type operator related to T is defined by where T k;1 are T or˙I (the identity operator), T k;2 and T k;4 are the bounded linear operators on L p .R n / for 1 < p < 1, T k;3 D˙I for k D 1; :::; m, M b .f / D bf and I˛is the Riesz potential operator (0 <˛< n) (see [2]).
Definition 1.2. Let F .x; y; t / be defined on R n R n OE0; C1/ and b be a locally integrable function on R n , set x; y/f .y/dy and for every bounded and compactly supported function f , where F k;1 t are F t .f / or˙I (the identity operator), F k;2 t and F k;4 t are linear operators, F k;3 t D˙I for k D 1; :::; m. Let H be the Banach space H D fh W jjhjj < 1g. For each fixed x 2 R n , we view F t .f /.x/ and F b t .f /.x/ as the mappings from OE0; C1/ to H , and F satisfies: Z 2jy zj<jx yj .jjF .x; y; t / F .x; z; t /jj C jjF .y; x; t / F .z; x; t /jj/dx Ä C; and there is a sequence of positive constant numbers fC k g such that for any k 1, 0 B @ Z 2 k jz yjÄjx yj<2 kC1 jz yj .jjF .x; y; t / F .x; z; t /jj C jjF .y; x; t / F .z; x; t /jj/ q dy where 1 < q 0 < 2 and 1=q C 1=q 0 D 1. Set S.f /.x/ D jjF t .f /.x/jj: The Toeplitz type operator related to F t is defined by .f /jj and S k;4 .f / D jjF k;4 t .f /jj are the bounded operators on L p .R n / for 1 < p < 1 and k D 1; :::; m.
Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 1.1 with C j D 2 j ı (see [4,19]). Note that the commutator OEb; T .f / D bT .f / T .bf / is a particular operator of the Toeplitz type operators T b and S b . It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [2, 3, 5-14, 16-18, 21, 22]). The main purpose of this paper is to prove the boundedness properties for the Toeplitz type operators T b and S b from Lebesgue spaces to Orlicz spaces.
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 D jQj 1 R Q f .x/dx. It is well-known that (see [4,19]) Let M be the Hardy-Littlewood maximal operator defined by jf .y/jdy: : We say that f belongs to BMO.R n / if f # belongs to L 1 .R n / and jjf jj BMO D jjf # jj L 1 . More generally, let be a non-decreasing positive function on OE0; C1/ and define BMO .R n / as the space of all functions f such that jf .y/ f Q jdy Ä C .r/: Forˇ> 0, the Lipschitz space Lipˇ.R n / is the space of functions f such that jjf jj LipˇD sup x¤y jf .x/ f .y/j=jx yjˇ< 1: For f , m f denotes the distribution function of f , that is m f .t / D jfx 2 R n W jf .x/j > tgj.
Let be a non-decreasing convex function on OE0; C1/ with .0/ D 0. 1 denotes the inverse function of . The Orlicz space L .R n / is defined by the set of functions f such that R R n . jf .x/j/dx < 1 for some > 0. The norm is given by jf .x/j/dx/: We shall prove the following theorems in Section 2.
Theorem 1.3. Let 0 <ˇÄ 1, q 0 < p < n=.˛Cˇ/ and ', be the two non-decreasing positive functions on OE0; C1/ with Let T be the same as in Definition 1.1 and the sequence fjC j g 2 l 1 . If T 1 .g/ D 0 for any g 2 L u .R n /.
Theorem 1.4. Let 0 <ˇÄ 1, q 0 < p < n=.˛Cˇ/ and ', be the two non-decreasing positive functions on OE0; C1/ with Let S be the same as in Definition 1.2 and the sequence fjC j g 2 l 1 . If F 1 t .g/ D 0 for any g 2 L u .R n /.1 < u < 1/, then S b is bounded from L p .R n / to L .R n / if b 2 BMO.R n /.

Proofs of theorems
We begin with the following preliminary lemmas.
). Let T and S be the same as Definitions 1.1 and 1.2, the sequence fC j g 2 l 1 . Then T and S are bounded on L p .R n / for 1 < p < 1.
). Let 0 < p < 1. Then, for any smooth function f for which the left-hand side is finite, ). Suppose that 0 < ı < n, 1 Ä s < p < n=ı and 1=r D 1=p ı=n. Then Let be a non-decreasing positive function on OE0; C1/ and Á be an infinitely differentiable function on R n with compact support such that ). Let 0 <ˇ< 1 orˇD 1 and be a non-decreasing positive function on OE0; C1/. Then To prove the theorems of the paper, we need the following Key Lemma. Let T and S be the same as in Definitions 1.1 and 1.2. Suppose that Q D Q.x 0 ; d / is a cube with supp f .2Q/ c and x; Q x 2 Q. (I) If b 2 BMO.R n / and the sequence fjC j g 2 l 1 , then Lipˇ.R n / and the sequence fC j g 2 l 1 , then R n / and the sequence fjC j g 2 l 1 , then Lipˇ.R n / and the sequence fjC j g 2 l 1 , then Proof. For suppf .2Q/ c and x; Q x 2 Q, we have Note that jx yj jx 0 yj for x 2 Q and y 2 R n n 2Q. Recalling r > q 0 , taking 1 < u < 1; 1 < v < r with 1=q C 1=u C 1=v D 1, we obtain, by the conditions on K, similarly as in the proof of (I), we obtain The same argument as in the proof of (I) and (II) will give the proof of (III) and (VI), we omit the details.
Now we are in position to prove our theorems.
By using the same arguments as in the proof of Theorem 1.3 will give the proof of Theorem 1.4, we omit the details.

Applications
In this section we shall apply the Theorems 1.3 and 1.4 to some particular operators such as the Calderón-Zygmund singular integral operator and Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Application 1. Calderón-Zygmund singular integral operator
Let T be the Calderón-Zygmund operator (see [3,4,19,20]), the Toeplitz type operator related to T is defined by Then it is easy to verify that Key Lemma holds for T b (see [3,4,19,20]), thus T satisfies the conditions in Theorem 1.3 and Theorem 1.3 holds for T b .
The Toeplitz type operator related to the Littlewood-Paley operator is defined by where and t .x/ D t n .x=t / for t > 0. We write that F t .f / D t f . We also define that which is the Littlewood-Paley operator (see [20]).
x/ may be viewed as a mapping from OE0; C1/ to H , and it is clear that It is easy to see that g b satisfies the conditions of Theorem 1.4 (see [9][10][11][12][13]), thus Theorem 1.4 holds for g b .

Application 3. Marcinkiewicz operator
Let be homogeneous of degree zero on R n and R S n 1 .x 0 /d .x 0 / D 0. Assume that 2 Lip .S n 1 / for 0 < Ä 1, that is there exists a constant M > 0 such that for any x; y 2 S n 1 , j . We also define that which is the Marcinkiewicz operator (see [21] It is easy to see that b satisfies the conditions of Theorem 1.4 (see [10][11][12][13][14]21]), thus Theorem 1.4 holds for b .