# Boundedness for the Modified Fractional Integral Operator from Mixed Morrey Spaces to the Bounded Mean Oscillation Space and Lipschitz Spaces

Authors: Mingquan Wei; Lanyin Sun
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we establish the boundedness of the modified fractional integral operator from mixed Morrey spaces to the bounded mean oscillation space and Lipschitz spaces, respectively.


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## Body

## 1. Introduction

Let0<α<n, and the fractional integral operator Iα and the modified fractional integral operator I^α are defined by(1)Iαfx=∫ℝnfyx−yn−αdy,I^αfx=∫ℝn1x−yn−α−χy≥1yyn−αfydy,where f is a locally integrable function on ℝn.The well-known Hardy–Littlewood–Sobolev inequality yields the boundedness ofIα from Lpℝn to Lqℝn, where 1<p<q<∞ and 1/p−1/q=α/n. For the endpoint situation, we know Iα is bounded from H1ℝn to Ln/n−αℝn (see [1]). Moreover, Peetre [2] proved that I^α is bounded from Ln/αℝn to BMOℝn. Here, BMOℝn is the bounded mean oscillation space, which consists of all locally integrable functions f on ℝn such that(2)fBMO=supx∈ℝn,r>01Bx,r∫Bx,rfy−fBx,rdy<∞,where Bx,r is the ball centered at x with radius r, Bx,r denotes its Lebesgue measure, and fBx,r=1/Bx,r∫Bx,rfydy. If one regards two functions whose difference is a constant as one, the space BMOℝn is a Banach space with respect to the norm ⋅BMO.Other than Lebesgue spaces, Morrey spaces are also important function spaces to study the boundedness of integral operators in harmonic analysis. The classical Morrey space was introduced by Morrey [3] to study the regularity of elliptic partial differential equations. Now, we recall the definition of the Morrey space Mqpℝn.For1≤q≤p<∞, the classical Morrey space Mqpℝn consists of all functions f∈Llocqℝn with(3)fMqp=supx∈ℝn,r>0Bx,r1/p−1/qfLqBx,r<∞.One can see that Morrey spaces are natural generalizations of Lebesgue spaces. The mapping properties ofI^α on Morrey spaces were first studied by Peetre [2] and further generalized by Adams [4]. We refer readers to [5–19] and the references therein for more studies about boundedness of the fractional integral operator on Morrey-type and anisotropic spaces. Recently, the mapping properties of I^α from Morrey spaces to BMOℝn and Lipschitz spaces were also obtained in [20–22]. Here, we review the definition of Lipschitz spaces briefly. Let 0<β<1; we say a locally integrable function f belongs to the Lipschitz space Lipβℝn if(4)fLipβ=supx∈ℝn,r>01Bx,r1+β/n∫Bx,rfy−fBx,rdy<∞.If one regards two functions whose difference is a constant as one, the spaceLipβℝn is a Banach space with respect to the norm ⋅Lipβ. As we know, a locally integrable function f which belongs to Lipβℝn also means that there exists a constant C such that fx−fy≤Cx−yβ for all x,y∈ℝn. If C0 is the smallest constant satisfying the inequality, then fLipβ∼C0.With the development of the theory of function spaces, Morrey spaces have been extended to different settings. One of the extensions is the mixed Morrey space, which was recently defined by Nogayama et al. [23–25] to uniform mixed Lebesgue spaces and Morrey spaces. Note that there exists another mixed Morrey space by using the iteration of Morrey norm introduced by Ragusa and Scapellato [26]. We refer the readers to [21, 27, 28] for the boundedness of various operators on these mixed Morrey spaces of iteration type.To give the definition of mixed Morrey spacesMq⟶pℝn, we first recall the definition of mixed Lebesgue spaces introduced in [29]. Let p⟶=p1,…,pn∈0,∞n. Then, the mixed Lebesgue norm ⋅p⟶ is defined by(5)fp⟶=∫ℝ⋯∫ℝ∫ℝfx1,x2,…,xnp1dx1p2/p1dx2p3/p2⋯dxn1/pn,where f is a measurable function. If pj=∞ for some j=1,…,n, then we have to make appropriate modifications. We define the mixed Lebesgue space Lp⟶ℝn to be the set of all measurable functions f with fp⟶<∞.Now, we can give the definition of mixed Morrey spaces introduced by Nogayama [24].Definition 1.
Let1≤q⟶<∞,1≤p<∞, and n/p≤∑i=1n1/qi. A measurable function f belongs to the mixed Morrey space Mq⟶pℝn if and only if(6)fMq⟶p=supx∈ℝn,r>0Bx,r1/p−1/n∑i=1n1/qifχBx,rq⟶<∞.
In [24], the author proved the boundedness of Iα from one mixed Morrey space to another, which inspires us to consider the mapping of I^α at or beyond the endpoint situation, i.e., the boundedness of I^α from Mq⟶pℝn to BMOℝn or Lipβℝn.
Throughout the paper, we use the following notations.
LetLlocℝn be the collection of all locally integrable functions on ℝn. We use χE and E to denote the characteristic function and the Lebesgue measure of a measurable set E.
The letterp⟶ denotes n-tuples of the numbers in 0,∞, i.e., p⟶=p1,…,pn, where n∈ℕ and n≥2. By definition, the inequality, for example, 0<p⟶<∞, means 0<pi<∞ for all i. For 1≤p⟶≤∞, we denote p⟶′=p1′,…,pn′, where pi′ satisfies 1/pi+1/pi′=1.
ByA≲B, we mean that A≤CB for some constant C>0, and A∼B means that A≲B and B≲A.


## 2. Main Results

We first recall the boundedness ofIα on mixed Lebesgue spaces, which will be used to prove our main results. Here and in the following, we denote q1⟶=q11,q12,…,q1n and q2⟶=q21,q22,…,q2n.Lemma 1.
Let0<α<n and 1<q1⟶<q2⟶<∞. Then,(7)IαfLq2⟶≲fLq1⟶if and only if(8)α=∑i=1n1q1i−∑i=1n1q2i.The proof of Lemma1 can be found in Lemma 3.1 of [30].In [24], Nogayama established the following result on the boundedness of Iα from one mixed Morrey space to another.Lemma 2.
Let0<α<n, 1<p1,p2<∞, and 1<q1⟶,q2⟶<∞. Assume that n/p1≤∑i=1n1/q1i and n/p2≤∑i=1n1/q2i. Also, assume that 1/p2=1/p1−α/n and q1⟶/p1=q2⟶/p2. Then, for f∈Mq1⟶p1ℝn,(9)IαfMq2⟶p2≲fMq1⟶p1.For more studies on the boundedness of operators on mixed Morrey spaces, we refer the readers to [31, 32]. One can see that p1 in Lemma 2 satisfies 1<p1<n/α. It is natural to ask what happens if p1=n/α or p1>n/α. In this section, we give an affirmative answer. More precisely, we will establish the boundedness of modified fractional integral operator I^α from Mq1⟶p1ℝn to BMOℝn or Lipβℝn.Whenp1=n/α in Lemma 2, we have the following theorem.Theorem 1.
Let0<α<n, 1<q1⟶<∞, and α<∑i=1n1/q1i. Then, for all f∈Mq1⟶n/αℝn, we have(10)I^αfBMO≲fMq1⟶n/α.Proof.
By the definition ofBMOℝn, we only need to show for any B=Bx0,r and f∈Mq1⟶n/αℝn, there holds(11)1B∫BI^αfx−I^αfBdx≲fMq1⟶n/α.
Write2jB=Bx0,2jr, and let(12)c1=∫ℝn\2Bfyx0−yn−αdy.
Then, we have(13)1B∫BI^αfx−I^αfBdx≤2B∫B∫2Bfyx−yn−αdydx+2B∫B∫ℝn\2Bfy|x−y|n−αdy−c1dx≔I+II.
Defineq2⟶ such that 1<q1⟶<q2⟶<∞ and α=∑i=1n1/q1i−∑i=1n1/q2i. For the term I, by using Lemma 1 and Hölder’s inequality on mixed Lebesgue spaces, we obtain(14)I≲1B∫BIαfχ2Bxdx≲1BIαfχ2Bq2⟶χBq2⟶′≲1Bfχ2Bq1⟶χBq2⟶′≲r∑i=1n1/q1i−αBfMq1⟶n/αχBq2⟶′≲r∑i=1n1/q1i−∑i=1n1/q2i−αfMq1⟶n/α≲fMq1⟶n/α.
Now, we turn to estimateII. By a direct computation, we have(15)II=1B∫B∫ℝn\2Bydydx≲rB∫B∫ℝn\2B1x0−yn−α+1fydydx≲∑j=1∞r2jrn−α+1∫2j+1B\2jBfydy≲∑j=1∞r2jrn−α+1fχ2j+1Bq1⟶χ2j+1Bq1⟶′≲∑j=1∞r2jr∑i=1n1/q1i−α+1fχ2j+1Bq1⟶=∑j=1∞r2jr12jr∑i=1n1/q1i−αfχ2j+1Bq1⟶≲∑j=1∞12jfMq1⟶n/α=fMq1⟶n/α.
From the estimates ofI and II, we get (11), which finishes the proof.
Whenp1>n/α in Lemma 2, we have the following mapping property of I^α from Mq1⟶p1ℝn to Lipα−n/p1ℝn.Theorem 2.
Let0<α<n, 0<α−n/p1<1, 1<q1⟶<∞, and α<∑i=1n1/q1i. Then, for f∈Mq1⟶p1ℝn, we have(16)I^αfLipα−n/p1≲fMq1⟶p1.Proof.
The proof is similar to that of Theorem1.
By the definition of Lipschitz spaces, we only need to show for anyB=Bx0,r and f∈Mq1⟶p1ℝn, there holds(17)1B1+α/n−1/p1∫BI^αfx−I^αfBdx≲fMq1⟶p1.
We also write2jB=Bx0,2jr and let(18)c1=∫ℝn\2Bfyx0−yn−αdy.
Similar to (13), we have(19)1B1+α/n−1/p1∫BI^αfx−I^αfBdx≤2B1+α/n−1/p1∫B∫2Bfyx−yn−αdydx+2B1+α/n−1/p1∫B∫ℝn\2Bfyx−yn−αdy−c1dx=I1+I2.
Chooseq2⟶ such that 1<q1⟶<q2⟶<∞ and α=∑i=1n1/q1i−∑i=1n1/q2i. For the term I1, by using Lemma 1 and Hölder’s inequality on mixed Lebesgue spaces, we get(20)I1=1B1+α/n−1/p1∫BIαfχ2Bxdx≲1B1+α/n−1/p1Iαfχ2Bq2⟶χBq2⟶′≲1B1+α/n−1/p1fχ2Bq1⟶χBq2⟶′≲r∑i=1n1/q1i−n/p1r∑i=1n1/q2i+α−n/p1fMq1⟶p1=fMq1⟶p1since α=∑i=1n1/q1i−∑i=1n1/q2i.
ForI2, by a similar method as in the proof of Theorem 1, we have(21)I2=1B1+α/n−1/p1∫B∫ℝn\2Bfydydx≲rB1+α/n−1/p1∫B∫ℝn\2B1x0−yn−α+1fydydx≲∑j=1∞r1−α+n/p12jrn−α+1∫2j+1B\2jBfydy≲∑j=1∞r1−α+n/p12jrn−α+1fχ2j+1Bq1⟶χ2j+1Bq1⟶′≲∑j=1∞r1−α+n/p12jr∑i=1n1/q1i−n/p12jr∑i=1n1/q1i−α+1fχ2j+1Bq1⟶=∑j=1∞r1−α+n/p12jr1−α+n/p112jr∑i=1n1/q1i−n/p1fχ2j+1Bq1⟶≲∑j=1∞12j1−α+n/p1fMq1⟶p1≲fMq1⟶p1since 1−α+n/p1>0.
From the estimates ofI1 and I2, we get (17). The proof is complete.
It is worth mentioning that our results in Theorems1 and 2 extend the corresponding results of classical Morrey spaces to mixed Morrey spaces.



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