Applications of Littlewood-Paley Theory for ̇B σ-Morrey Spaces to the Boundedness of Integral Operators

1 School of High Technology for Human Welfare, Tokai University, 317 Nishino Numazu, Shizuoka 410-0395, Japan 2 College of Economics, Nihon University, 1-3-2 Misaki-cho, Chiyoda-ku, Tokyo 101-8360, Japan 3Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan 4Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

The space  ∞ comp (R  \ {0}) denotes the set of all compactly supported  ∞ (R  )-functions supported on R  \ {0}.Now we formulate our main result in the simplest form.
Theorem 1.Let 1 <  < ∞,  > 0 and  ∈ [−/, −).Let  ∈  ∞  (R  \ {0}) be a function such that      ()      ) then  ∈ S  (R  ) and where F −1 denotes the inverse Fourier transform and the implicit constant in ∼ does not depend on .     ()      ) measures local regularities of functions, and the parameter  plays the role of global regularity.Moreover, the proof of Theorem 13 reveals this aspect of our studying both local and global regularities of functions simultaneously.Theorem 1 concerns the Littlewood-Paley theory.The Littlewood-Paley theory is one of the most powerful tools in harmonic analysis.Roughly speaking, this is a technique of transforming functions into good ones in order to measure the norms.Another side of Littlewood-Paley theory is that the functions are broken into good pieces of functions.We shall illustrate that the Littlewood-Paley theory is very useful by applying this to the fractional integral operator   of order .The fractional integral operator   (0 <  < ), which is given by plays an important role not only in harmonic analysis but also in partial differential equations.It is well known that   can be seen as the inverse operator of (−Δ) /2 modulo a multiplicative constant, and hence   has the smoothing effect.However, due to this smoothing effect,   still seems to have a lot to be investigated.
To describe the related function spaces and formulate Theorem 1 in the full statement, we now fix some more notations.Since cubes play the central role, let us fix our notation of cubes.We denote by Q the set of all cubes of the form (, ) with  ∈ R  and  > 0. Also, we denote by D the set of dyadic cubes: where to define the right-hand side we used the Minkowski sum.Given  ∈ D∪Q, we denote by () the center of a cube  and by || its volume.The notation ℓ() stands for || 1/ .Given a set  ⊂ R  , we define D () := { ∈ D :  ⊂ } , With these notations in mind, let us recall the definition of the function space Ḃ  ( , )(R  ) and related function spaces defined in [10].Here we redefine Ḃ  ( , )(R  ) in terms of dyadic cubes.However, a geometric observation shows that the definition of Ḃ  ( , )(R  ) through dyadic cubes and that through cubes are equivalent.
As we remarked above, (3) and (10) for any measurable function  on R  .In view of (12), we identify the right-hand side and the left-hand side in these formulas.Note that from the definition of norms, we have          Ḃ  ( , ) ∼ sup ∈R  ,>0 for any measurable function  on R  .We need to pay attention to the word "local, " when we discuss Ḃ  ( , )(R  ).Burenkov and Guliyev, together with their successors, investigated local Morrey-type spaces in [11].By "local" in [11], they meant that they defined the "local Morrey-type" norm by sup which indicates that sup appearing in the definition is restricted to all balls centered at the origin.However, by "local" in the present paper we meant that we measure the regularity of functions by (6).Nowadays there are many different definitions related to classical Morrey spaces, so we need to carefully distinguish the different definitions and the names given to the definitions.See [12] for related usage of the word "local" such as central mean oscillation, central BMO, -central bounded mean oscillation, and -central Morrey spaces.
The goal of the present paper is to show that these function spaces fall under the scope of the Littlewood-Paley theory.As an application of this fact we show the boundedness property of   and singular integral operators.The Littlewood-Paley theory is a powerful tool to investigate the boundedness property of   .To consider the connection between Ḃ  ( , )(R  ) and the Littlewood-Paley theory, we present definitions.Here and below we use the definition of the Fourier transform below for definiteness: Following [13], we define It may be helpful to observe that for each  ∈ S(R  ).
In the present paper, the following function space of Littlewood-Paley type will play a key role.Here and below we denote   := −log 2 ℓ() for a cube  ∈ D. Observe that The function space Ḃ  (  , )(R  ) denotes the set of all tempered distributions  ∈ S  (R  ) for which the quantity ‖‖ Ḃ  (  , ) is finite.
In Theorem 1, we did not mention what happens if  ∈ S  (R  ), and the right-hand side of (4) is finite.Here including this problem, we reformulate and reinforce Theorem 1.
More precisely, we have the following.Instead of Theorem 1 itself, we shall prove Theorem 4 in the present paper.
To establish Theorem 4, we will need an auxiliary vectorvalued estimate of the Hardy-Littlewood maximal operator .Define the Hardy-Littlewood maximal operator  by The following proposition is proven in our earlier paper [12].This is an extension of [23] to Ḃ ( , )(R  ).
With Theorem 4 and Proposition 5 in mind, we investigate the boundedness property of   again as we announced in the beginning.More precisely, we shall provide an alternative proof of the following theorems, which were proven earlier in [10,12].Theorem 6 (see [10,12]).Suppose that the parameters , , , , , and  satisfy Assume in addition that Then   is a bounded operator from Ḃ  ( , )(R  ) to Ḃ  ( , )(R  ).
Theorem 7 (see [10,12]).Suppose that the parameters , , , , and  satisfy (26) and with  = 1.Then   is a bounded linear operator from We can also consider Campanato spaces and Lipschitz spaces in this framework.First, let P  (R  ) be the set of all polynomials having a degree at most  ∈ {0, 1, 2, . ..}.For a cube , a locally integrable function  over  and a nonnegative integer , there exists a unique polynomial  ∈ P  (R  ) such that, for all  ∈ P  (R  ), Denote this unique polynomial  by    .It follows immediately from the definition that     =  if  ∈ P  (R  ).We can characterize Λ spaces and Λ spaces (cf.[28]).Note that, in particular, with norm coincidence.For the definition of CMO , (R  ) and CBMO , (R  ), we refer to [12].
where we use the obvious modification when  = ∞.
Now we define a function space by way of difference.For ℎ ∈ R  , an integer  ∈ N and a function  : R  → C, we define inductively.
Definition 11.For  ≥ 0,  > 0 and  ∈ {0, Remark 14.It is absolutely necessary to assume that  is a continuous function in Theorem 13, when  ≥ 1.We remark that there exists a discontinuous function  such that Δ +1 ℎ () = 0 for all , ℎ ∈ R  .See [29, Proposition A1] for such an example constructed algebraically.Now we explain notations and we describe its organization of the present paper.We use the following notations for the inequalities.First, we use standard notation for inequalities.For example, in the present paper a chain of inequalities of the form appears in (110) below.The inequality (38) means that there exist  1 ,  2 ,  3 > 0 such that If the implicit constants in ≲ or ∼ do depend on some important parameters , , . .., then we write ≲ ,,... or ∼ ,,... .We shall prove Theorem 4 in Section 3. We prove Theorem 6 in Section 4. Theorem 7 is covered in Section 5. We prove Theorem 13 in Section 6. Finally in Section 7 we present another application of Theorem 4 by showing that the Fourier multiplier is bounded on Ḃ  ( , )(R  ).

Preliminaries
In the present paper, we frequently use the following fundamental inequalities.
Lemma 15 (see [30, page 466]).Let ],  ∈ Z, ,  > 0, and for some and that for some   ∈ R  .Then we have To formulate the next lemma, we recall the definition of   with 1 ≤  < ∞.A measurable function , which takes values in (0, ∞) almost everywhere, is said to be an   weight or belongs to the class   , if For all  > 1, it is easy to see that  1 ⊂   and that   () ≤  1 ().
It is just a matter of handling the left-hand side carefully.But, for the sake of convenience, we supply the proof.
In the course of the proof of Theorem 6, we need another piece of information on the space   ( , )(R  ).Let The next lemma concerns the norm of the translation operator.
We need the following sequence of functions.

Lemma 21. There exists a sequence of functions {𝑟
In the present paper the sequence {  } ∞ =−∞ above is called a Rademacher sequence.

Littlewood-Paley Characterization of
Ḃ  ( , )(R  ) for any constant function .Note that this implies that Then, since  > 1, we can use the Hölder inequality and we have for  ≤  ≤ 2.Assuming that  +  < 0, the sequence { 0   (0)} >0 is convergent by the Cauchy test.Thus, we can consider the mapping Let us check that the range of  is in Ḃ  ( , )(R  ).
As for the first term, directly from the definition of ‖‖ L (0) , , we obtain that Also, a geometric observation shows that the second term can be estimated similarly.Since  < 0, we obtain that By using (64) and  < 0, we can handle the third term: In view of the way in which we chose , we obtain that In summary, It remains to estimate the fourth term.We employ the following estimate: Since  < 0, (72) is summable and we obtain that where the implicit constant in (73) is independent of .

Proof of Theorem 4 Part (a)
. For 1 <  < ∞ and  ∈ R, let us define the function space - where, for  ∈ Z  , we write ⟨⟩ := √ 1 + || 2 , and the norm is given by Then from the definition of the norms ( 10) and (78), the following chain of continuous inclusion holds.For −/ ≤ , Thus, (a) follows.

Proof of Theorem 4 Part (b).
Let  ∈ S(R  ) be a fixed function satisfying (17).It follows from (17) that there exist  − and  + such that Let  ∈ Ḃ  (  , )(R  ) and fix a dyadic cube  * such that 0 ∈  * .We are going to show that converges in the topology of  ∞ ( * ) ∩ S  (R  ), and that converges in the topology of   ( * ) ∩ S  (R  ).
The presice meaning of (82) and ( 83 as  → ∞.Once (82) and ( 83) are proven, we will have converges in the topology of   ( * ) ∩ S  (R  ) and that  −  is a polynomial.Hence it follows that  ∈   loc (R  ).Remark also that the convergence in S  (R  ) of the sum defining  2 is a generality.So let us prove (82) and (83).
Let us begin with proving (82).To this end we take a dyadic cube  containing  * .Since we are assuming (81), we deduce that showing (83).With (82) and ( 83), the proof of (b) is now complete.
Remark 23.We did not use the structure of   () in the proof.Therefore, we can deduce the following variant.
The heart of the matter is how to construct the inverse mapping from exists in the topology of   loc (R  )∩S  (R  ) in view of ( 92) and (99).We claim that Here and below in proving (112), we assume that f =  by replacing  with f.Let us prove (112).Suppose that we are given a dyadic cube  and  > 0 such that  ∈ D(  ).We estimate First, we expand  by using (111).If we denote φ () := for  ∈ Z, then we have where the convergence, as we have established in Part (b), takes places in   loc (R  ) ∩ S  (R  ).We set for  ∈ Z.
We decompose (83) by using  1 and  2, . 1 by using (115).We estimate  1 .Let  be an auxiliary parameter again, which is taken so that By using the maximal operator , we have By Lemma 16, we have ()  ) We write We decompose again the right-hand side of (120) dyadically.

Proof of Theorem 4 Part (d).
An important corollary of Proposition 5 is that we can relax the condition (17) on , which is stronger than (d) in Theorem 4.
Then the norm equivalence Proof.Choose integers  − and  + so that Define Then  * * ∈  ∞ comp (R  \ {0}) by virtue of (138).Observe also that To prove Theorem 6, we need the lemmas.Note that Lemma 26 can be seen as the Plancherel-Polya-Nikolskij inequality for Ḃ  (  , ).
Lemma 26.Suppose that the parameters , , and  satisfy Let  be chosen so that it satisfies (17).Then we have for all  > 0 and  ∈ D(  ).
Proof.The right-hand inequality is a consequence of Lemma 20.
Let  ∈  be fixed.We choose  ∈ N sufficiently large.Also, take a compactly supported function  so that  equals 1 on supp() as we did in (138).Then we have We decompose R  dyadically.Then we have Thus, the proof is now complete.
In the course of the proof, we obtained the following chain of inequalities.
Corollary 27.Suppose that the parameters , , and  satisfy Let  be chosen so that it satisfies (17).
The following estimate is somehow well known.Here we remark that the following form was recorded in [34].Here we change variables   → − to transform the result into the one needed in the present paper.
Now we prove Theorem 6.Given  ∈ Ḃ  ( , ), we shall define  The definition of    can be justified in the same was as Theorem 6.So we assume that    is already defined.Let  > 0 and a cube  be fixed so that  ⊂   .Set  0 := /( +  − ) and  1 := /( +  + ) for some small  > 0.More precisely, we choose  > 0 so that By interpolation described in [35] Let us define From ( 172) and (173), we deduce that If we insert this estimate and (174), then we have The proof is therefore complete.
To prove Theorem 13, we need the following lemma.
(3) Suppose that { P } ∞ =1 is a sequence of polynomials of degree  such that ℎ = lim This concludes the proof of Theorem 13.