The Boundedness of Marcinkiewicz Integral Associated with Schrödinger Operator and Its Commutator Dongxiang

The authors prove that Marcinkiewicz integral operator is not only are bounded on , for , but also a bounded map from to weak . Meanwhile, the -boundedness and -boundedness are also obtained. Finally, the -boundedness and -boundedness for the commutator of Marcinkiewicz integral of schrodinger type are established.


Introduction and Notation
Let us consider the Schrödinger operator in   ,  ≥ 3. The function  is nonnegative,  ̸ = 0, and belongs to a reverse Hölder class   , for some exponent  > /2; that is, there exists a constant  such that for every ball  ⊂   .We introduce the definition of the reverse Hölder index of  as  0 = sup{ :  ∈   }.It is known that  ∈   implies  ∈  + , for some  > 0. Therefore, under the assumption  ∈  /2 , we may conclude  0 > /2.
The classical Marcinkiewicz integral operator  is defined by The above operator was introduced by Stein in [1] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimension.Meanwhile, Stein [1] showed that if Ω ∈ Lip  ( −1 ), for some 0 <  ≤ 1, then  is a bounded operator on   (  ), for 1 <  ≤ 2, and is a bounded map from  1 (  ) to weak  1 (  ).Benedek et al. [2] showed that if Ω is continuously differentiable in  ̸ = 0, then  is a bounded operator on   (  ), for 1 <  ≤ ∞.Ding et al. [3] proved that the Marcinkiewicz function  is bounded from  1 (  ) to  1 (  ) with Ω satisfying cancelation condition on  −1 and  1 -Dini condition.
Similar to the classical Marcinkiewicz function , we define the Marcinkiewicz integral    associated with the Schrödinger operator  by  3 ) For a given potential  ∈   , with  > /2, we introduce the auxiliary function ≤ 1} ,  ∈   .(7) The above assumptions () are finite, for all  ∈   .Proposition 1 (see [4]).There exist  and  0 ≥ 1 such that for all ,  ∈   .
A norm for  ∈ BMO  (), denoted by []  , is given by the infimum of the constants in the inequalities above.Notice that if we let  = 0, we obtain the John-Nirenberg space BMO.
In the note, we devote ourselves to establish the following boundedness of the commutators of Marcinkiewicz integral of schrodinger operator type.Theorem 6.Let  ∈   ,  ∈ BMO ∞ (), and  0 such that 1/ 0 = (1/ 0 − 1/) + , where  0 is the reverse Hölder index of ; then, for all  ∈   , we have Throughout this paper,  denotes the constants that are independent of the main parameters involved but whose value may differ from line to line.By  ⋍ , we mean that there exists a constant  > 1 such that 1/ ≤ / ≤ .We use the symbol  ≲  to denote that there exists a positive constant  such that  ≤ .
Proof.Let  ∈   (  ) and  = ( 0 , ( 0 )).We first observe and so we have to deal with the average on  of each term.Thanks to Hölder's inequality with 1 <  <  0 and Lemma 10, one has For the second term, we split again  =  1 +  2 .By choosing 1 < s <  <  0 and denoting ] = s/( − s), by the boundedness of    on  s(  ) and Hölder's inequality, we obtain where, in the last inequality, we have used Lemma 9.
For the remaining term, let 1 = 1/ + 1/s, () ⋍ ( 0 ) using Lemma 9, Lemma 10, and Minkowski's inequality, we arrive to Since  can be chosen large enough, the last series converges.Thus, we finished the proof.
Remark 12.It is easy to check that if the critical ball  is replaced by 2, the above lemma also holds.
Proof.We write Splitting into annuli, we have where  0 is the least integer such that 2  0 ≥ ( 0 )/.Then, To deal with  12 , by using Lemma 10 and choosing  >   , we have Thanks to Hölder's inequality and Lemma 10, we have To deal with  22 , similarly as  12 , we have We have completed the proof of the lemma.

The Boundedness of Marcikiewicz Integral and Its Commutator
In this section, we first employ the same technique in [6] to prove Theorem 5.
Proof.Similarly as [6], it suffices to prove the following pointwise estimate: Obviously, For  3 , using Lemma 4 again, we get With the help of the   -boundedness and weak (1, 1)boundedness of   and , we can get the same boundedness of    .
For the BMO  -boundedness and ( 1  ,  1 )-boundedness of    , we only make similar modification in the procedure of the same estimate in [6].Here, we omit it.
Next we will establish some boundedness for the commutator of Marcinkiewicz integral of Schrödinger operator type.
We start with the proof of Theorem 6.