# Tent Space Approach of Morrey Spaces and Their Application to Duality and Complex Interpolation

Authors: Takahiro Ono
Journal: Journal of Function Spaces
 (2023)

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## Abstract

The aim in this paper is to establish a new duality property of Morrey spaces and to discover the complex interpolation space between Morrey spaces and Lebesgue spaces. For that purpose, a new spaceNp,qsRn is introduced by using methods of tent spaces. The space generalizes Morrey spaces and Lebesgue spaces. Furthermore, this scale of spaces is amenable to the complex interpolation space.


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## Body

## 1. Introduction

The Morrey spaceMqsRn was introduced by Morrey [1] to investigate the existence and differentiability properties of solutions to the elliptic partial differential equations of second order. Tent spaces were introduced by Coifman et al. [2] to analyze Hardy spaces, and tent spaces have been applied for the theory of parabolic partial differential equations in the previous research. The aim in this paper is to generalize Morrey spaces by applying some properties of tent spaces. To our best knowledge, it seems that tent spaces are not used for the study of function spaces related to Morrey spaces.As the motivation of this study, let us recall concrete examples of equivalence of homogeneous Triebel-Lizorkin spaces. The homogeneous Triebel-Lizorkin spaceḞp,qsRn was introduced by Triebel (see [3]). The norm of Ḟp,qsRn is characterized in terms of tent spaces. The space Ḟp,qsRn generalizes Lebesgue spaces LpRn1<p<∞ and the bounded mean oscillation space BMORn. Furthermore, the following properties is known as a particular case:
(1)L2RnL2Rn,BMORn1/2BMORnḞ2,20RnḞ4,20RnḞ∞,20RnḞ2,20Rn∗Ḟ4/3,20Rn∗Ḟ1,20Rn∗.Here,X,Yθ is the complex interpolation space between X and Y. Furthermore, X∗ is the dual space of X.In this paper, the new function spaceNp,qsRn is introduced (see Section 2.2 below). The norm of the space Np,qsRn is also defined via tent spaces like homogeneous Triebel-Lizorkin spaces. The motivation of the study of the space Np,qsRn is to construct the corresponding chart for Lebesgue spaces and Morrey spaces. The following chart summarizes arguments shown in this paper:
(2)LpRnLpRn,MqsRnθMqsRnNp,p0RnNpθ,qθθsRnN∞,qsRnNp′,p′0Rn∗Npθ′,qθ′−θsRn∗N1,q′−sRn∗.The explicit expressions ofpθ and qθ are not given here. We content ourselves with mentioning that there are natural interpolation indices. One of important points in the above chart is that the description of the complex interpolation space between Lebesgue spaces and Morrey spaces is given. A description of the complex interpolation space between two Morrey spaces is known (see Lemma 11). As far as we know, other descriptions of the complex interpolation space between Morrey spaces and other spaces are not known in the recent research.


### 1.1. Notations

We use the following notations in this paper:(1)
We denote byL0Rn the set of all measurable functions on Rn(2)
For1≤p≤∞, the conjugate number p′ of p is defined by the number which realizes 1/p+1/p′=1(3)
Forx∈Rn and r>0, Bx,r is the ball with radius r centered at x(4)
Forf∈L0Rn and x∈Rn, we define the Hardy-Littlewood maximal operator M by
(3)Mfx≔supr>01Bx,r∫Bx,rfydy.More generally, for0<q<∞, we define its powered version by
(4)Mqf≔Mfq1/q.(5)
The spaceL10,∞;dt/t consists of measurable functions ϕ:0,∞⟶0,∞ satisfying
(5)ϕL10,∞;dt/t≔∫0∞ϕtdtt<∞.


### 1.2. Lebesgue Spaces and Morrey Spaces

In this subsection, we shall introduce the definitions of Lebesgue spaces and Morrey spaces. The most basic Banach spaces are Lebesgue spacesLpRn defined as follows:Let1≤p≤∞. Then, the space LpRn is the set of all functions f∈L0Rn satisfying
(6)fp≔∫Rnfxpdx1/p<∞for 1≤p<∞ and
(7)f∞≔esssupx∈Rnfx<∞for p=∞. Next, we recall the definition of Morrey spaces which is the main function spaces in this paper.Definition 1.
Let1≤q<∞ and 0≤s≤n/q. The Morrey space MqsRn is the set of all functions f∈L0Rn satisfying
(8)fMqs≔supx,r∈Rn×0,∞rs1Bx,r∫Bx,rfyqdy1/q<∞.We note some properties of Morrey spaces.Remark 2.
(1)
Letωn be the volume of the unit ball on Rn. Then, the equation
(9)fMpn/p=ωn−1/psupx,r∈Rn×0,∞∫Bx,rfypdy1/p=ωn−1/pfpimplies that
(10)Mpn/pRn=LpRn,with equivalence of norms for 1≤p<∞.(2)
For1≤q2≤q1<∞, the embedding
(11)Mq1sRn⊂Mq2sRnfollows from Hölder’s inequality.(3)
Combining (10) and (11), for any 1≤q≤n/s, we have
(12)Ln/sRn⊂MqsRn.


### 1.3. Dual Spaces

In the study of quasi-Banach spaces, the duality argument plays an important role. We recall how we identify the dual space of function spaces.Definition 3.
For a quasi-Banach spaceX, we write the dual space of X as X∗. The Banach space X∗ is defined as the set of all linear continuous functionals ℓ:X⟶C. We define
(13)ℓX∗≔supfX=1ℓf.It is known that a dual space of quasi-Banach space is also a quasi-Banach space with the above norm. We define the following duality relation between quasi-Banach spaces and their duals contained inL0Rn:Definition 4.
For quasi-Banach spacesX,Y⊂L0Rn, we say that X∗=Y if it satisfies the following:
(0) Iff∈X and g∈Y, then f·g∈L1Rn(i)
Forg∈Y, we set
(14)ℓgf=∫Rnfxgxdxf∈X.Then, we haveℓg∈X∗ and
(15)ℓgX∗≤CgY.(ii)
For anyℓ∈X∗, there uniquely exists g∈Y satisfying
(16)ℓf=∫Rnfxgxdxf∈X,and we have
(17)gY≤CℓX∗.Usually, the notion of duality relation is considered for Banach spaces of measurable functions. However, as we will see, the ones for quasi-Banach spaces will be needed since we would like to handle the quasi-Banach spaceNp,qsRn which will be defined in Section 2.2 below. For quasi-Banach spaces X and Y, we say that X is a predual space of Y if it satisfies X∗=Y. It is known that a predual space is not always unique. For Lebesgue spaces, we have the following duality property:Lemma 5.
Let1≤p<∞. Then,
(18)LpRn∗=Lp′Rn.For the proof, see, e.g., Grafakos [4].In the previous research, some predual spaces of Morrey spaces are known. The following block spaces were found by Long [5].Definition 6.
Lets∈R and 1<q<∞. We say that a measurable function a is a q,s-block in a ball B if a satisfies
(i)
suppa⊂B,(ii)
aq≤B−1/q′−s/n
The block spaceBqsRn is the set of all functions f∈L0Rn such that
(19)fBqs≔inf∑j∈Nλjf=∑j∈Nλjaj,λjj∈N⊂C,ajj∈N:q,s−blocks<∞,where the infimum is taken over all decompositions of f. Here, each aj satisfies (i) and (ii) for B replaced by the ball Bj. Note that Bj can vary according to j.The dual space ofBq′−sRn is the Morrey space MqsRn.Lemma 7 (see [5]).
Let1<q<∞ and 0<s≤n/q. Then,
(20)Bq′−sRn∗=MqsRn.We refer to Adams and Xiao [6] and Gogatishvili and Mustafayev [7] for other representations of predual spaces of Morrey spaces.


### 1.4. Complex Interpolation Spaces

The theory of complex interpolation spaces plays an important role in operator theory. Our second aim is to obtain the complex interpolation between Morrey spaces and Lebesgue spaces. For this purpose, we prepare the following.For quasi-Banach function spacesX,Y, let X+Y be the sum space such that
(21)X+Y≔f=f1+f2:f1∈X,f2∈Y.We defineFX,Y be the set of all mappings f:C⟶X+Y which are analytic in S≔z∈C:0<Rez<1, continuous on S¯≔z∈C:0≤Rez≤1 and satisfy
(i)
fz:z∈S⊂X+Y(ii)
fit:t∈R⊂X, limt′⟶tfit′−fitX=0, for each t∈R and supt∈RfitX<∞(iii)
f1+it:t∈R⊂Y, limt′⟶tf1+it′−f1+itY=0, for each t∈R and supt∈Rf1+itY<∞Furthermore, introducing the norm ofFX,Y by
(22)fFX,Y≔maxsupt∈RfitX,supt∈Rf1+itY,we define the complex interpolation space X,Yθ between X and Y with respect to 0≤θ≤1, by
(23)X,Yθ≔x∈X+Y:x=fθ,f∈FX,Yequipped with norm
(24)xX,Yθ≔inffFX,Yf∈FX,Y,x=fθ.We can interpolate the operator norm of bounded linear operator, which is the main thrust of investigating interpolation spaces.Lemma 8 (Bergh and Löfström [8]).
For pairs of Banach spacesX1,X2 and Y1,Y2, assume that a linear operator T is bounded from XjtoYjj=1,2. Then, T is also bounded from X1,X2θ to Y1,Y2θ and satisfies
(25)TX1,X2θ⟶Y1,Y2θ≤TX1⟶Y11−θTX2⟶Y2θ,for any 0<θ<1. Here, we define the operator norm by
(26)TX⟶Y≔supfX=1TfY,for an operator from a Banach space X to a Banach space Y.We shall give another characterization of complex interpolation spaces by using Carderón product spaces defined as follows:Definition 9.
For quasi-Banach spacesX,Y⊂L0Rn, and 0≤θ≤1, we define
(27)X1−θYθ≔f∈X+Y:fX1−θYθ<∞,where
(28)fX1−θYθ=inff1X1−θf2Yθ:fx≤f1x1−θf2xθa.e..The spaceX1−θYθ is called the Carderón product space between X and Y with respect to θ.Below, we recall the relationship between complex interpolation spaces and Calderón product spaces. We say that a quasi-Banach spaceX is a quasi-Banach function lattice if it satisfies
(29)gx≤fxa.e.forg∈L0Rn⟹g∈XwithgX≤fX,for any f∈X. We say that a quasi-Banach space X satisfies the Fatou property if
(30)0≤fk∈Xandsupk∈NfkX<∞,withfk⟶f∈L0Rna.e.⟹f∈XandfX=limk⟶∞fkX,for fkk∈N⊂X. Furthermore, we say that a quasi-Banach space X is r-convex for 0<r<∞ if it satisfies
(31)∑j=1kfjr1/rX≤∑j=1kfjXr1/r,for fjj=1k⊂X. In Kalton and Mitrea [9], it was shown that, if two quasi-Banach function lattices X1 and X2 satisfy the Fatou property, the ri-convexity (i=1,2), and if either of X1 or X2 is separable, then
(32)X1,X2θ=X11−θX2θ,for any 0<θ<1.We recall two classical formulas on complex interpolation for Lebesgue spaces and for Morrey spaces.Lemma 10 (see [4]).
Let1≤p1,p2≤∞ and 1<θ<1. Then,
(33)Lp1Rn,Lp2Rnθ=LpθRn,for 1/pθ=1−θ/p1+θ/p2.The complex interpolation of Morrey spaces is more complicated than that of Lebesgue spaces.Lemma 11 (Hakim and Sawano [10]).
Let0<θ<1, 1≤qi<∞, and 0<si≤n/pii=1,2. If s1q1=s2q2, then
(34)Mq1s1Rn,Mq2s2Rnθ=f∈Mqθsθ¯Rn:lima⟶0+χf<afMqs=0,for 1/qθ=1−θ/q1+θ/q2 and sθ=1−θs1+θs2. Here, Mqθsθ¯Rn denotes the closure with respect to MqθsθRn of the set of all essentially bounded functions in MqθsθRn.The organization of the remain part is as follows: In Section2, we introduce tent spaces and our new space Np,qsRn. In Section 3, we shall state the main theorems. Further, in Section 4, some properties and the proofs of the main theorems are given and in Section 5, we present their application.


## 1.1. Notations

We use the following notations in this paper:(1)
We denote byL0Rn the set of all measurable functions on Rn(2)
For1≤p≤∞, the conjugate number p′ of p is defined by the number which realizes 1/p+1/p′=1(3)
Forx∈Rn and r>0, Bx,r is the ball with radius r centered at x(4)
Forf∈L0Rn and x∈Rn, we define the Hardy-Littlewood maximal operator M by
(3)Mfx≔supr>01Bx,r∫Bx,rfydy.More generally, for0<q<∞, we define its powered version by
(4)Mqf≔Mfq1/q.(5)
The spaceL10,∞;dt/t consists of measurable functions ϕ:0,∞⟶0,∞ satisfying
(5)ϕL10,∞;dt/t≔∫0∞ϕtdtt<∞.


## 1.2. Lebesgue Spaces and Morrey Spaces

In this subsection, we shall introduce the definitions of Lebesgue spaces and Morrey spaces. The most basic Banach spaces are Lebesgue spacesLpRn defined as follows:Let1≤p≤∞. Then, the space LpRn is the set of all functions f∈L0Rn satisfying
(6)fp≔∫Rnfxpdx1/p<∞for 1≤p<∞ and
(7)f∞≔esssupx∈Rnfx<∞for p=∞. Next, we recall the definition of Morrey spaces which is the main function spaces in this paper.Definition 1.
Let1≤q<∞ and 0≤s≤n/q. The Morrey space MqsRn is the set of all functions f∈L0Rn satisfying
(8)fMqs≔supx,r∈Rn×0,∞rs1Bx,r∫Bx,rfyqdy1/q<∞.We note some properties of Morrey spaces.Remark 2.
(1)
Letωn be the volume of the unit ball on Rn. Then, the equation
(9)fMpn/p=ωn−1/psupx,r∈Rn×0,∞∫Bx,rfypdy1/p=ωn−1/pfpimplies that
(10)Mpn/pRn=LpRn,with equivalence of norms for 1≤p<∞.(2)
For1≤q2≤q1<∞, the embedding
(11)Mq1sRn⊂Mq2sRnfollows from Hölder’s inequality.(3)
Combining (10) and (11), for any 1≤q≤n/s, we have
(12)Ln/sRn⊂MqsRn.


## 1.3. Dual Spaces

In the study of quasi-Banach spaces, the duality argument plays an important role. We recall how we identify the dual space of function spaces.Definition 3.
For a quasi-Banach spaceX, we write the dual space of X as X∗. The Banach space X∗ is defined as the set of all linear continuous functionals ℓ:X⟶C. We define
(13)ℓX∗≔supfX=1ℓf.It is known that a dual space of quasi-Banach space is also a quasi-Banach space with the above norm. We define the following duality relation between quasi-Banach spaces and their duals contained inL0Rn:Definition 4.
For quasi-Banach spacesX,Y⊂L0Rn, we say that X∗=Y if it satisfies the following:
(0) Iff∈X and g∈Y, then f·g∈L1Rn(i)
Forg∈Y, we set
(14)ℓgf=∫Rnfxgxdxf∈X.Then, we haveℓg∈X∗ and
(15)ℓgX∗≤CgY.(ii)
For anyℓ∈X∗, there uniquely exists g∈Y satisfying
(16)ℓf=∫Rnfxgxdxf∈X,and we have
(17)gY≤CℓX∗.Usually, the notion of duality relation is considered for Banach spaces of measurable functions. However, as we will see, the ones for quasi-Banach spaces will be needed since we would like to handle the quasi-Banach spaceNp,qsRn which will be defined in Section 2.2 below. For quasi-Banach spaces X and Y, we say that X is a predual space of Y if it satisfies X∗=Y. It is known that a predual space is not always unique. For Lebesgue spaces, we have the following duality property:Lemma 5.
Let1≤p<∞. Then,
(18)LpRn∗=Lp′Rn.For the proof, see, e.g., Grafakos [4].In the previous research, some predual spaces of Morrey spaces are known. The following block spaces were found by Long [5].Definition 6.
Lets∈R and 1<q<∞. We say that a measurable function a is a q,s-block in a ball B if a satisfies
(i)
suppa⊂B,(ii)
aq≤B−1/q′−s/n
The block spaceBqsRn is the set of all functions f∈L0Rn such that
(19)fBqs≔inf∑j∈Nλjf=∑j∈Nλjaj,λjj∈N⊂C,ajj∈N:q,s−blocks<∞,where the infimum is taken over all decompositions of f. Here, each aj satisfies (i) and (ii) for B replaced by the ball Bj. Note that Bj can vary according to j.The dual space ofBq′−sRn is the Morrey space MqsRn.Lemma 7 (see [5]).
Let1<q<∞ and 0<s≤n/q. Then,
(20)Bq′−sRn∗=MqsRn.We refer to Adams and Xiao [6] and Gogatishvili and Mustafayev [7] for other representations of predual spaces of Morrey spaces.


## 1.4. Complex Interpolation Spaces

The theory of complex interpolation spaces plays an important role in operator theory. Our second aim is to obtain the complex interpolation between Morrey spaces and Lebesgue spaces. For this purpose, we prepare the following.For quasi-Banach function spacesX,Y, let X+Y be the sum space such that
(21)X+Y≔f=f1+f2:f1∈X,f2∈Y.We defineFX,Y be the set of all mappings f:C⟶X+Y which are analytic in S≔z∈C:0<Rez<1, continuous on S¯≔z∈C:0≤Rez≤1 and satisfy
(i)
fz:z∈S⊂X+Y(ii)
fit:t∈R⊂X, limt′⟶tfit′−fitX=0, for each t∈R and supt∈RfitX<∞(iii)
f1+it:t∈R⊂Y, limt′⟶tf1+it′−f1+itY=0, for each t∈R and supt∈Rf1+itY<∞Furthermore, introducing the norm ofFX,Y by
(22)fFX,Y≔maxsupt∈RfitX,supt∈Rf1+itY,we define the complex interpolation space X,Yθ between X and Y with respect to 0≤θ≤1, by
(23)X,Yθ≔x∈X+Y:x=fθ,f∈FX,Yequipped with norm
(24)xX,Yθ≔inffFX,Yf∈FX,Y,x=fθ.We can interpolate the operator norm of bounded linear operator, which is the main thrust of investigating interpolation spaces.Lemma 8 (Bergh and Löfström [8]).
For pairs of Banach spacesX1,X2 and Y1,Y2, assume that a linear operator T is bounded from XjtoYjj=1,2. Then, T is also bounded from X1,X2θ to Y1,Y2θ and satisfies
(25)TX1,X2θ⟶Y1,Y2θ≤TX1⟶Y11−θTX2⟶Y2θ,for any 0<θ<1. Here, we define the operator norm by
(26)TX⟶Y≔supfX=1TfY,for an operator from a Banach space X to a Banach space Y.We shall give another characterization of complex interpolation spaces by using Carderón product spaces defined as follows:Definition 9.
For quasi-Banach spacesX,Y⊂L0Rn, and 0≤θ≤1, we define
(27)X1−θYθ≔f∈X+Y:fX1−θYθ<∞,where
(28)fX1−θYθ=inff1X1−θf2Yθ:fx≤f1x1−θf2xθa.e..The spaceX1−θYθ is called the Carderón product space between X and Y with respect to θ.Below, we recall the relationship between complex interpolation spaces and Calderón product spaces. We say that a quasi-Banach spaceX is a quasi-Banach function lattice if it satisfies
(29)gx≤fxa.e.forg∈L0Rn⟹g∈XwithgX≤fX,for any f∈X. We say that a quasi-Banach space X satisfies the Fatou property if
(30)0≤fk∈Xandsupk∈NfkX<∞,withfk⟶f∈L0Rna.e.⟹f∈XandfX=limk⟶∞fkX,for fkk∈N⊂X. Furthermore, we say that a quasi-Banach space X is r-convex for 0<r<∞ if it satisfies
(31)∑j=1kfjr1/rX≤∑j=1kfjXr1/r,for fjj=1k⊂X. In Kalton and Mitrea [9], it was shown that, if two quasi-Banach function lattices X1 and X2 satisfy the Fatou property, the ri-convexity (i=1,2), and if either of X1 or X2 is separable, then
(32)X1,X2θ=X11−θX2θ,for any 0<θ<1.We recall two classical formulas on complex interpolation for Lebesgue spaces and for Morrey spaces.Lemma 10 (see [4]).
Let1≤p1,p2≤∞ and 1<θ<1. Then,
(33)Lp1Rn,Lp2Rnθ=LpθRn,for 1/pθ=1−θ/p1+θ/p2.The complex interpolation of Morrey spaces is more complicated than that of Lebesgue spaces.Lemma 11 (Hakim and Sawano [10]).
Let0<θ<1, 1≤qi<∞, and 0<si≤n/pii=1,2. If s1q1=s2q2, then
(34)Mq1s1Rn,Mq2s2Rnθ=f∈Mqθsθ¯Rn:lima⟶0+χf<afMqs=0,for 1/qθ=1−θ/q1+θ/q2 and sθ=1−θs1+θs2. Here, Mqθsθ¯Rn denotes the closure with respect to MqθsθRn of the set of all essentially bounded functions in MqθsθRn.The organization of the remain part is as follows: In Section2, we introduce tent spaces and our new space Np,qsRn. In Section 3, we shall state the main theorems. Further, in Section 4, some properties and the proofs of the main theorems are given and in Section 5, we present their application.


## 2. Tent Spaces and New Spaces

In this section, we recall the definition and some properties of tent spaces, and after that, we introduce the new spaceNp,qsRn. We organize this section as follows: In Section 2.1, we present the definition and investigate some properties of tent spaces. In Section 2.2, we give the definition of the space Np,qsRn.


### 2.1. Tent Spaces

In this section, we recall the definition and some properties of tent spaces. Tent spaces were initially introduced by Coifman et al. [2]. In the previous research, tent spaces have been applied for study of boundedness of Calderón-Zygmund operators (see Section 5 for the definition) and the theory of parabolic differential equations (see David and Journé [11] and Koch and Tataru [12]). We write R+n+1≔Rn×0,∞.Definition 12.
Let1≤p≤∞, 1≤q<∞, and s∈R. The tent space Tp,qsR+n+1 is the set of all measurable functions F:R+n+1⟶C satisfying
(35)FTp,qs≔∫0∞∫Bx,ttsqFy,tqdydttn+11/qLxpRn<∞,for p<∞, and
(36)FT∞,qs≔supx,r∈Rn×0,∞1B∫0r∫x−y<r−ttsqFy,tqdydtt1/q<∞,for p=∞.It is easy to show that the triangle inequality for∗Tp,qs holds. Moreover, we note that by applying Fubini’s lemma, we have
(37)FTp,p0~∫R+n+1Fx,tpdxdtt1/p,for 1≤p<∞ and F∈L0R+n+1.The following is the known duality theorem for tent spaces based on Definition4:Lemma 13 (Huang [13]).
Let1≤p<∞, 1<q<∞, and s∈R. If F∈Tp,qsR+n+1 and G∈Tp′,q′−sR+n+1, then F·G is integrable with respect to the measure dxdt/t and
(38)Tp,qsR+n+1∗=Tp′,q′−sR+n+1via the coupling
(39)F,G=∫R+n+1Fx,tGx,tdxdtt.
Furthermore, the following inequalities hold:(40)∫R+n+1Fx,tGx,tdxdtt≤CFTp,qsGTp′,q′−s,for 1<p<∞, and
(41)∫R+n+1Fx,tGx,tdxdtt≤CFT1,q−sGT∞,q′s,for p=1. Moreover, the inequality
(42)F·GTp0,q0s0≤CFTp1,q1s1GTp2,q2s2holds for s0=s1+s2∈R, 0≤1/p0=1/p1+1/p2≤1, and 0<1/q0+1/q1+1/q2<1 with 1≤p1,p2≤∞, and 1<q1,q2<∞.We note that Lemma13 is a particular case of Theorem 4.3 in [13] where β0=0, β=s, and r=q.We have also the Sobolev embedding theorem for tent spaces as follows:Lemma 14 (Amenta [14]).
Let1<q<∞. For 1<p1,p2≤∞ and s1≤s2, if s1+n/p1=s2+n/p2, one has
(43)Tp1,qs1R+n+1⊂Tp2,qs2R+n+1,with a norm inequality
(44)FTp2,q2s2≤CFTp1,q2s1,for any F∈Tp1,p2s1R+n+1.The following lemma states that the Calderón product for tent spaces can be calculated.Lemma 15 (see [13]).
Let1≤q1,q2<∞, 1≤p1,p2≤∞notboth∞, and s∈R. Then,
(45)Tp1,q1s1R+n+11−θTp2,q2s2R+n+1θ=Tp0,q0s0R+n+1,where
(46)s0=1−θs1+θs2,1p0=1−θp1+θp2,and1q0=1−θq1+θq2.


### 2.2. The Definition ofNp,qs-Spaces

In this section, we introduce new function spaces calledNp,qs-spaces by using tent spaces defined in Section 2.1. First, we define the Np,qs-norms with respect to ϕ∈L10,∞;dt/t.Definition 16.
Letϕ∈L10,∞;dt/t. For 1≤p≤∞, 1≤q<∞, and s≤n/q, we define
(47)fNp,qs,ϕ≔ϕt1/qfxTp,qs=∫0∞∫Bx,ttsqϕtfyqdydttn+11/qLxpRn,for 1≤p<∞, and
(48)fN∞,qs,ϕ≔ϕt1/qfxT∞,qs=supx,r∈Rn×0,∞1B∫0r∫x−y<r−ttsqϕtfyqdydtt1/q,for p=∞.
Using the above norm, we define theNp,qs-norm for s≥0.Definition 17.
For1≤q<∞, q≤p≤∞, and 0≤s≤n/q−n/p, we define
(49)Np,qsRn≔f∈L0Rn:fNp,qs,ϕ<∞foranyϕ∈L10,∞;dtt,where
(50)fNp,qs=supϕL10,∞;dt/t=1fNp,qs,ϕ.Remark that fors≥0, the definition of the Np,qs-norm demanded q≤p. The way of defining the Np,qs-norm for p≤q is different from the one for q≤p.Definition 18.
Let1≤p≤q≤∞, n/q−n/p≤−s≤0, and ϕ∈L10,∞,dt/t. We say that the function ftx is an Np,q−s,ϕ-atom if ftx≥0 for all x∈ℝn and t∈0,∞ and
(51)ϕt1/qftxTp,q−s≤1.
The spaceNp,q−sℝn is the set of all f∈L0ℝn such that there exists λ>0 and an Np,q−s,ϕ-atom ft for which
(52)fx≤λ∫0∞ϕtftxdtt,for almost all x∈ℝn. Let λϕ,ft be the infimum of such λ, and let
(53)fNp,q−s≔infλϕ,ft,where the infimum is taken over all functions ϕ which satisfy ϕL10,∞,dt/t=1 and all Np,q−s,ϕ-atoms ft.Remark that fors≤0, the definition of the Np,qs-norm demanded p≤q. For the case p=q, we defined the Np,p0-norm in two ways. However, those are essentially equivalent (see Lemma 24 in Section 4.1 below). The definition of the norm fNp,q−s is complicated; therefore, we introduce N~p,q−s-“norms” for convenience.Definition 19.
For1≤p≤q<∞ and n/q−n/p<−s≤0, we define
(54)fN~p,q−s≔infϕL10,∞;dt/t=1fNp,q−s,ϕ,for measurable function f. The space N~p,q−sRn is the set given by
(55)N~p,q−sRn≔f∈L0Rn:fN~p,q−s<∞.Then, we include the following:Lemma 20.
Assume that the parametersp,q,s satisfy the conditions in Definition 18. Then, one has N~p,q−sRn⊂Np,q−sRn, with the inequality
(56)fNp,q−s≤fN~p,q−s,for f∈N~p,q−sRn.Proof.
Iff=0, then the conclusion is clear. So, assume otherwise. Let f∈N~p,q−sRn. Then, for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 such that
(57)fNp,q−s,ϕ≤fN~p,q−s+ε.
Subsequently, fromϕL10,∞;dt/t=1, we have
(58)fx=∫0∞ϕtfxdtt=fNp,q−s,ϕ∫0∞ϕtfxfNp,q−s,ϕdtt.
Then, trivially,(59)ffNp,q−s,ϕNp,q−s,ϕ=1.
Thus, we obtain(60)fNp,q−s≤fNp,q−s,ϕ≤fN~p,q−s+ε,for any ε>0, which concludes the proof.It is unclear that theN~p,q−s-“norms” are quasinorms because they do not satisfy the quasitriangle inequality f+gN~p,q−s≤CfN~p,q−s+gN~p,q−s. Meanwhile, we claim that the Np,qs-norms do satisfy the quasitriangle inequality. For s≥0, it is evident from Definition 17. Thus, we consider the case where s≤0. Let f,g∈Np,q−sRn. Then, for any ε1,ε2>0, there exists ϕ1,ϕ2∈L10,∞;dt/t satisfying ϕ1L10,∞;dt/t=ϕ2L10,∞;dt/t=1, and we have
(61)fx≤fNp,q−s+ε1∫0∞ϕ1tftxdtt,gx≤gNp,q−s+ε2∫0∞ϕ2tgtxdtt,for some Np,q−s,ϕ1-atom ft and Np,q−s,ϕ2-atom gt. Note that
(62)ϕ1t1/qftTp,q−s,ϕ2t1/qgtNp,q−s,ϕ2≤1.If we putϕ0t=ϕ1t+ϕ2t/2, then ϕ0L10,∞;dt/t=1, and we obtain
(63)fx+gx≤21/q′fNp,q−s+gNp,q−s+ε3∫0∞ϕ0t121/q′ϕ1tϕ0tftx+ϕ2tϕ0tgtxdtt,where we put ε3=ε1+ε2. The triangle inequality in tent spaces implies that
(64)ϕ0t1/q121/q′ϕ1tϕ0tft+ϕ2tϕ0tgtTp,q−s≤ϕ0t1/q121/q′ϕ1tϕ0tftTp,q−s+ϕ0t1/q121/q′ϕ2tϕ0tgtTp,q−s.Then, from (62), we obtain
(65)ϕ0t1/q121/q′ϕ1tϕ0tftTp,q−s=12ϕ1tϕ1t+ϕ2t1/q′ftTp,q−s≤12.Similarly, we have(66)ϕ0t1/q121/q′ϕ2tϕ0tgtTp,q−s≤12.Thus, the functionϕ1/ϕ0tft+ϕ2/ϕ0tgt/21/q′ is an Np,q−s,ϕ0-atom so that we get
(67)f+gNp,q−s≤21/q′fNp,q−s+gNp,q−s+ε3,for any ε3>0. This shows that the norm ∗Np,q−s satisfies the quasitriangle inequality.


## 2.1. Tent Spaces

In this section, we recall the definition and some properties of tent spaces. Tent spaces were initially introduced by Coifman et al. [2]. In the previous research, tent spaces have been applied for study of boundedness of Calderón-Zygmund operators (see Section 5 for the definition) and the theory of parabolic differential equations (see David and Journé [11] and Koch and Tataru [12]). We write R+n+1≔Rn×0,∞.Definition 12.
Let1≤p≤∞, 1≤q<∞, and s∈R. The tent space Tp,qsR+n+1 is the set of all measurable functions F:R+n+1⟶C satisfying
(35)FTp,qs≔∫0∞∫Bx,ttsqFy,tqdydttn+11/qLxpRn<∞,for p<∞, and
(36)FT∞,qs≔supx,r∈Rn×0,∞1B∫0r∫x−y<r−ttsqFy,tqdydtt1/q<∞,for p=∞.It is easy to show that the triangle inequality for∗Tp,qs holds. Moreover, we note that by applying Fubini’s lemma, we have
(37)FTp,p0~∫R+n+1Fx,tpdxdtt1/p,for 1≤p<∞ and F∈L0R+n+1.The following is the known duality theorem for tent spaces based on Definition4:Lemma 13 (Huang [13]).
Let1≤p<∞, 1<q<∞, and s∈R. If F∈Tp,qsR+n+1 and G∈Tp′,q′−sR+n+1, then F·G is integrable with respect to the measure dxdt/t and
(38)Tp,qsR+n+1∗=Tp′,q′−sR+n+1via the coupling
(39)F,G=∫R+n+1Fx,tGx,tdxdtt.
Furthermore, the following inequalities hold:(40)∫R+n+1Fx,tGx,tdxdtt≤CFTp,qsGTp′,q′−s,for 1<p<∞, and
(41)∫R+n+1Fx,tGx,tdxdtt≤CFT1,q−sGT∞,q′s,for p=1. Moreover, the inequality
(42)F·GTp0,q0s0≤CFTp1,q1s1GTp2,q2s2holds for s0=s1+s2∈R, 0≤1/p0=1/p1+1/p2≤1, and 0<1/q0+1/q1+1/q2<1 with 1≤p1,p2≤∞, and 1<q1,q2<∞.We note that Lemma13 is a particular case of Theorem 4.3 in [13] where β0=0, β=s, and r=q.We have also the Sobolev embedding theorem for tent spaces as follows:Lemma 14 (Amenta [14]).
Let1<q<∞. For 1<p1,p2≤∞ and s1≤s2, if s1+n/p1=s2+n/p2, one has
(43)Tp1,qs1R+n+1⊂Tp2,qs2R+n+1,with a norm inequality
(44)FTp2,q2s2≤CFTp1,q2s1,for any F∈Tp1,p2s1R+n+1.The following lemma states that the Calderón product for tent spaces can be calculated.Lemma 15 (see [13]).
Let1≤q1,q2<∞, 1≤p1,p2≤∞notboth∞, and s∈R. Then,
(45)Tp1,q1s1R+n+11−θTp2,q2s2R+n+1θ=Tp0,q0s0R+n+1,where
(46)s0=1−θs1+θs2,1p0=1−θp1+θp2,and1q0=1−θq1+θq2.


## 2.2. The Definition ofNp,qs-Spaces

In this section, we introduce new function spaces calledNp,qs-spaces by using tent spaces defined in Section 2.1. First, we define the Np,qs-norms with respect to ϕ∈L10,∞;dt/t.Definition 16.
Letϕ∈L10,∞;dt/t. For 1≤p≤∞, 1≤q<∞, and s≤n/q, we define
(47)fNp,qs,ϕ≔ϕt1/qfxTp,qs=∫0∞∫Bx,ttsqϕtfyqdydttn+11/qLxpRn,for 1≤p<∞, and
(48)fN∞,qs,ϕ≔ϕt1/qfxT∞,qs=supx,r∈Rn×0,∞1B∫0r∫x−y<r−ttsqϕtfyqdydtt1/q,for p=∞.
Using the above norm, we define theNp,qs-norm for s≥0.Definition 17.
For1≤q<∞, q≤p≤∞, and 0≤s≤n/q−n/p, we define
(49)Np,qsRn≔f∈L0Rn:fNp,qs,ϕ<∞foranyϕ∈L10,∞;dtt,where
(50)fNp,qs=supϕL10,∞;dt/t=1fNp,qs,ϕ.Remark that fors≥0, the definition of the Np,qs-norm demanded q≤p. The way of defining the Np,qs-norm for p≤q is different from the one for q≤p.Definition 18.
Let1≤p≤q≤∞, n/q−n/p≤−s≤0, and ϕ∈L10,∞,dt/t. We say that the function ftx is an Np,q−s,ϕ-atom if ftx≥0 for all x∈ℝn and t∈0,∞ and
(51)ϕt1/qftxTp,q−s≤1.
The spaceNp,q−sℝn is the set of all f∈L0ℝn such that there exists λ>0 and an Np,q−s,ϕ-atom ft for which
(52)fx≤λ∫0∞ϕtftxdtt,for almost all x∈ℝn. Let λϕ,ft be the infimum of such λ, and let
(53)fNp,q−s≔infλϕ,ft,where the infimum is taken over all functions ϕ which satisfy ϕL10,∞,dt/t=1 and all Np,q−s,ϕ-atoms ft.Remark that fors≤0, the definition of the Np,qs-norm demanded p≤q. For the case p=q, we defined the Np,p0-norm in two ways. However, those are essentially equivalent (see Lemma 24 in Section 4.1 below). The definition of the norm fNp,q−s is complicated; therefore, we introduce N~p,q−s-“norms” for convenience.Definition 19.
For1≤p≤q<∞ and n/q−n/p<−s≤0, we define
(54)fN~p,q−s≔infϕL10,∞;dt/t=1fNp,q−s,ϕ,for measurable function f. The space N~p,q−sRn is the set given by
(55)N~p,q−sRn≔f∈L0Rn:fN~p,q−s<∞.Then, we include the following:Lemma 20.
Assume that the parametersp,q,s satisfy the conditions in Definition 18. Then, one has N~p,q−sRn⊂Np,q−sRn, with the inequality
(56)fNp,q−s≤fN~p,q−s,for f∈N~p,q−sRn.Proof.
Iff=0, then the conclusion is clear. So, assume otherwise. Let f∈N~p,q−sRn. Then, for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 such that
(57)fNp,q−s,ϕ≤fN~p,q−s+ε.
Subsequently, fromϕL10,∞;dt/t=1, we have
(58)fx=∫0∞ϕtfxdtt=fNp,q−s,ϕ∫0∞ϕtfxfNp,q−s,ϕdtt.
Then, trivially,(59)ffNp,q−s,ϕNp,q−s,ϕ=1.
Thus, we obtain(60)fNp,q−s≤fNp,q−s,ϕ≤fN~p,q−s+ε,for any ε>0, which concludes the proof.It is unclear that theN~p,q−s-“norms” are quasinorms because they do not satisfy the quasitriangle inequality f+gN~p,q−s≤CfN~p,q−s+gN~p,q−s. Meanwhile, we claim that the Np,qs-norms do satisfy the quasitriangle inequality. For s≥0, it is evident from Definition 17. Thus, we consider the case where s≤0. Let f,g∈Np,q−sRn. Then, for any ε1,ε2>0, there exists ϕ1,ϕ2∈L10,∞;dt/t satisfying ϕ1L10,∞;dt/t=ϕ2L10,∞;dt/t=1, and we have
(61)fx≤fNp,q−s+ε1∫0∞ϕ1tftxdtt,gx≤gNp,q−s+ε2∫0∞ϕ2tgtxdtt,for some Np,q−s,ϕ1-atom ft and Np,q−s,ϕ2-atom gt. Note that
(62)ϕ1t1/qftTp,q−s,ϕ2t1/qgtNp,q−s,ϕ2≤1.If we putϕ0t=ϕ1t+ϕ2t/2, then ϕ0L10,∞;dt/t=1, and we obtain
(63)fx+gx≤21/q′fNp,q−s+gNp,q−s+ε3∫0∞ϕ0t121/q′ϕ1tϕ0tftx+ϕ2tϕ0tgtxdtt,where we put ε3=ε1+ε2. The triangle inequality in tent spaces implies that
(64)ϕ0t1/q121/q′ϕ1tϕ0tft+ϕ2tϕ0tgtTp,q−s≤ϕ0t1/q121/q′ϕ1tϕ0tftTp,q−s+ϕ0t1/q121/q′ϕ2tϕ0tgtTp,q−s.Then, from (62), we obtain
(65)ϕ0t1/q121/q′ϕ1tϕ0tftTp,q−s=12ϕ1tϕ1t+ϕ2t1/q′ftTp,q−s≤12.Similarly, we have(66)ϕ0t1/q121/q′ϕ2tϕ0tgtTp,q−s≤12.Thus, the functionϕ1/ϕ0tft+ϕ2/ϕ0tgt/21/q′ is an Np,q−s,ϕ0-atom so that we get
(67)f+gNp,q−s≤21/q′fNp,q−s+gNp,q−s+ε3,for any ε3>0. This shows that the norm ∗Np,q−s satisfies the quasitriangle inequality.


## 3. Main Theorems

In this section, we introduce our main theorems. The first result is the duality theorem ofNp,qs-spaces. This derives the duality theorem on Morrey spaces by combining with Proposition 31 in Section 4.1 below.Theorem 21.
Let1<p≤∞, 1≤q<∞, and 0≤s<n/q−n/p. Then, one has Np′,q′−sRn∗=Np,qsRn in the sense of Definition 4.
(0) Letf∈Np′,q′−sRn and g∈Np,qsRn. Then,·g∈L1Rn(i)
Forg∈Np,qsRn, set
(68)ℓgf=∫Rnfxgxdx.Then,ℓg∈Np′,q′−sRn∗ and ℓgNp′,q′−s∗≤CgNp,qs.(ii)
For anyℓ∈Np′,q′−sRn∗, there uniquely exists g∈Np,qsRn satisfying ℓ=ℓg, and one has gNp,qs≤CℓNp′,q′−s∗.Lettingp=∞ and combining Theorem 21 with Proposition 31 in Section 4.1 below, we obtain the following corollary:Corollary 22.
Let1<q<∞ and 0≤s<n/q. Then,
(69)N1,q′−sRn∗=N∞,qsRn=MqsRn.For homogeneous Triebel-Lizorkin spacesḞp,qsRn and the bounded mean oscillation space BMORn, it is known that the duality Ḟ1,20Rn∗=BMORn holds (see Sawano [15] for their definitions and the precise statement of the duality). Furthermore, it is generalized to the duality Ḟp,qsRn∗=Ḟp′,q′−sRn (see Frazier and Jawerth [16]). For Morrey spaces, Theorem 21 realizes the above generalization of Ḟp,qsRn. There are several predual of Morrey spaces. The block spaces described in Section 1.3 were first introduced in [5]. Subsequently, other predual spaces were found (see [6, 7]). The space N1,q′−sRn is the new one we found.The second theorem concerns a complex interpolation between Lebesgue spaces and Morrey spaces.Theorem 23.
Let1≤p,q<∞, 0≤s≤n/q, and 0≤θ≤1. Then,
(70)LpRn,MqsRnθ=Npθ,qθθsRn,where pθ=p/1−θ and 1/qθ=1−θ/p+θ/q.We must assumes≤n/q−n/p in the definition of Np,qs-norms when s≥0. The above θs, pθ, and qθ satisfy the condition. In fact, from s≤n/q, we have
(71)θs≤θnq=1−θnp+θnq−1−θnp=nqθ−npθ.Theorem23 implies that Np,qs-spaces with s≥0 can be characterized by complex interpolation spaces between some Lebesgue space and some Morrey space.


## 4. Proofs

In this section, we prove some lemmas ofNp,qs-spaces, and after that, we prove the main theorems introduced in Section 3. We organize this section as follows: In Section 4.1, we investigate some properties of Np,qs-spaces. In Section 4.2, we prove Theorem 21. In Section 4.3, we prove Theorem 23.


### 4.1. Properties

In this section, we prove some properties ofNp,qs-spaces. At first, we prove that the space Np,qsRn generalizes Lebesgue spaces and Morrey spaces. This relation is similar to the fact that homogeneous Triebel-Lizorkin space Ḟp,qsRn generalizes LpRn and BMORn. The coincidence with Lebesgue spaces is the following. Here, we must keep in mind that we defined ∗Np,p0 in two ways, which are Np,ps-norms for s≥0 and for s≤0. We will show that Lemma 24 below holds in both senses.Lemma 24.
For1≤p<∞, we have LpRn=Np,p0Rn, and the following equivalence holds:
(72)fNp,p0~fp,for any f∈L0Rn.Proof.
Letf∈L0Rn. First, for Np,qs-norms for s≥0, (37) implies that
(73)fNp,p0=supϕL10,∞;dt/t=1ϕ1/pfTp,p0~supϕL10,∞;dt/t=1∫Rn∫0∞ϕtfxpdtdxt1/p=fp.
Conversely, forNp,ps-norms for s≤0, Lemma 20 and (37) imply that
(74)fNp,p0≤fN~p,p0=infϕL10,∞;dt/t=1ϕ1/pfTp,p0~fp.
Meanwhile, if we assumef∈Np,p0Rn in the sense of Definition 18, then for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 and Np,p0,ϕ-atom ft such that
(75)fx≤fNp,p0+ε∫0∞ϕtftxdtt.
Thus, by Hölder’s inequality fort, we have
(76)fp≤fNp,p0+ε∫0∞ϕtftdttp≤CfNp,p0+ε∫0∞ϕtdtt1/p′ϕ1/pftTp,p0≤fNp,p0+ε,for any ε>0. It concludes the proof in both senses.The scaleNp,qsRn is monotone in 1≤q<∞ as Lemma 25 shows, which is similar to homogeneous Triebel-Lizorkin spaces Ḟp,qsRn for 1≤q<∞.Lemma 25.
For any1≤p≤∞, 1≤q1≤q2<∞, and s∈R, one has Np,q2sRn⊂Np,q1sRn, and
(77)fNp,q1s≤CfNp,q2s,for any f∈Np,q2sRn.Proof.
If suffices to show thatfNp,q1s,ϕ≤CfNp,q2s,ϕ for any ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, it is easy to show for p=∞ using Hölder’s inequality. So, we assume that 1≤p<∞. Setting 1/q1=1/q2+1/q3 and using Hölder’s inequality twice, we learn
(78)fNp,q1s,ϕ=∫0∞∫Bx,ttsq1ϕtfyq1dydttn+11/q1Lxp≤C∫0∞ϕt∫Bx,ttsq2fyq2dytnq1/q2dtt1/q1Lxp≤C∫0∞ϕtdtt1/q3∫0∞∫Bx,ttsq2ϕtfyq2dydttn+11/q2Lxp=CfNp,q2s,ϕ,from ϕL10,∞;dt/t=1.The following is the Sobolev embedding inNp,qs-spaces:Lemma 26.
Let1<q<∞. For 1<p1,p2≤∞, and 0≤s1≤s2 (or s1≤s2≤0), if s1+n/p1=s2+n/p2, then one has Np1,qs1Rn⊂Np2,qs2Rn, and
(79)fNp2,qs2≤CfNp1,qs1,for any f∈Np1,qs1Rn.Lemma26 is a direct consequence of Lemma 14. We omit its proof. Lemmas 24 and 26 imply that Np,q−sRn⊂LqRn for −s+n/p=n/q. The following lemma claims that the converse embedding locally holds.Lemma 27.
Let1≤p≤q<∞, n/q−n/p<−s≤0, and B=Bx0,r⊂Rn be a ball. For f∈LqRn with suppf⊂B, one has
(80)fN~p,q−s≤Cr−s+n/p−n/qfq.In particular, the following inequality holds:(81)fN~1,q−s≤Cr−s+n/q′fq,for 0≤s<n/q′.Before proving Lemma27, we prepare the definition and some properties of Lorentz spaces.Definition 28.
Let0<p<∞ and 0<q≤∞. For f∈L0Rn, we define the decreasing rearrangement function f∗:0,∞⟶0,∞ of f as
(82)f∗t≔infα∈0,∞:dfα≤t,where df is the distribution function of f, given by
(83)dfα=x∈Rn:fx>αα>0.Here,inf∅ stands for ∞. Let f∈L0Rn. If 0<q<∞, then define
(84)fLp,q≔∫0∞t1/pf∗tqdtt1/q.Ifq=∞, then define
(85)fLp,q≔supt>0t1/pf∗t.We define the Lorentz spaceLp,qRn as the set of all functions f∈L0Rn for which fLp,q<∞.We note some properties of Lorentz spaces.Remark 29.
(i)
Let1<p1,p2,p3<∞ satisfy 1/p3=1/p1+1/p2 and 1≤q≤∞. Then, for f∈Lp1,qRn and g∈Lp2,q′Rn, we have
(86)fgLp3≤fLp1,qgLp2,q′.(ii)
For0<q<∞ and f∈LqRn, we have
(87)MqfLq,∞≤Cfq,where Mq is the q-powered Hardy–Littlewood maximal operator (see Section 1.1).(iii)
For any0<p<∞ and ball B=Bx0,r⊂Rn, we have
(88)χBLp,1~Bn/p.See, e.g., [4] for the above properties.All preparations to prove Lemma27 were given.Proof of Lemma 30.
For a fixedα>sq, let ϕt=cχ0,rttα/rα, where c>0 is a normalization parameter satisfying ϕL10,∞;dt/t=1. Since suppf⊂Bx0,r, for y∈suppf, we get y−x0<r. Fix x∈Rn and 0<t<r with suppf∩Bx,t≠∅. Then, for y∈suppf∩Bx,t, we have
(89)x−x0≤x−y+y−x0<t+r<2r.
Keeping in mind that in Remark29, we have
(90)fN~p,q−s≤∫0∞∫Bx,tt−sqϕtfyqdydttn+11/qLxpRn≤C∫0r∫Bx,ttα−sqrαfyqdydttn+11/qLxpBx0,2r≤Cr−sMqfLpBx0,2r≤Cr−sMqfLq,∞χBx0,2rLp0,1≤Cr−s+n/p−n/qfq,where 1/p=1/q+1/p0.The following proposition shows thatNp,qs-spaces generalize Morrey spaces.Proposition 31.
For1≤q<∞ and 0<s<n/q, one has N∞,qsRn=MqsRn with the norm equivalence
(91)fN∞,qs~fMqs,for any f∈MqsRn.Proof.
We can easily show thatfN∞,qs≤fMqs for f∈MqsRn:
(92)fN∞,qs=supϕL10,∞;dt/t=1supx,r∈Rn×0,∞1B∫0r∫x−y<r−ttsqϕtfyqdydtt1/q≤supϕL10,∞;dt/t=1supx,r∈Rn×0,∞rsqB∫0∞∫x−y<rϕtfyqdydtt1/q=fMqs.
Conversely, we assumef∈N∞,qsRn. For B=Bx0,r⊂Rn, we have
(93)fLqB~supgLq′B=1∫Bfxgxdx.
Then, Lemma27 implies gχB∈Lq′Rn⊂N1,q′−sRn for any g∈Lq′B. Moreover, for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 such that ϕ1/q′gχBT1,q′−s≤gχBN~1,q′−s+ε. Thus, Hölder’s inequality and Lemmas 13 and 27 imply that
(94)supgLq′B=1∫Bfxgxdx=supgLq′B=1∫0∞∫RnϕtfxgχBxdxdtt≤CsupgLq′B=1ϕ1/qfT∞,qsϕ1/q′gχBT1,q′−s≤CsupgLq′B=1fN∞,qsgχBN~1,q′−s+ε≤CsupgLq′B=1fN∞,qsr−s+n/qgχBq′+ε=Cr−s+n/qfN∞,qs+fN∞,qsε,for any ε>0. Consequently, the arbitrariness of ε>0 implies that
(95)rs−n/qfLqB≤CfN∞,qs,for any ball B=Bx0,r, and it concludes the proof.Lemma25 shows that the Np,qs-spaces are vested. We combine Lemmas 25 and 26 and Proposition 31.Remark 32.
Let1≤q<p≤∞ and 0≤s≤n/q−n/p. Then,
(96)Np,qsRn⊂N∞,qs+n/pRn=Mqs+n/pRn.Here, we used Lemma26 and Proposition 31.Furthermore, we obtain the following:Remark 33.
Let1≤p≤q<∞ and n/q−n/p<−s≤0. Then,
(97)Np,q−sRn⊂Np0,q0Rn⊂Lp0Rn,where p0=1/p−s/n−1<q. Here, we used Lemmas 26 and 25.


### 4.2. Proof of Theorem21

In this section, we prove Theorem21.Proof of Theorem 34 (0), (i).
Letf∈Np′,q′−sRn and g∈Np,qsRn. Then, for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 and Np′,q′−s,ϕ-atom ft such that
(98)fx≤fNp′,q′−s+ε∫0∞ϕtftxdtt.Thus, (40) implies that
(99)∫Rnfxgxdx≤fNp′,q′−s+ε∫0∞∫Rnϕtftxgxdxdtt≤CfNp′,q′−s+εϕ1/q′ftTp′,q′−sϕ1/qgTp,qs≤fNp′,q′−s+εgNp,qs,for any ε>0. Thus, we have
(100)∫Rnfxgxdx≤CfNp′,q′−sgNp,qs,and it concludes the proof of Theorem 34 (0). From (100), we obtain
(101)ℓgNp′,q′−s∗=supfNp′,q′−s=1∫Rnfxgxdx≤CgNp,qs.It shows Theorem21 (i).Proof of Theorem34 (ii). We assume that ℓ is a bounded linear functional of Np′,q′−sRn. Fix a ball B=Bx0,r. If f is supported on Bx0,r, then
(102)fNp′,q′−s≤Cr−s+n/q−n/pfχBx0,rq′from Lemma 27. Hence, ℓ induces a bounded linear functional on Lq′Bx0,r and acts with gB∈LqBx0,r. By taking Bj=B0,jj∈N, we have gBj=gBj+1 on Bj, and thus we obtain a unique function g on Rn that is locally in LqRn, such that ℓf=∫Rnfxgxdx when f∈Np′,q′−sRn is supported on some ball. We can extend ℓ as a global functional over Rn using the following lemma:Lemma 35.
Let1≤p≤q<∞, and n/q−n/p<−s≤0. For f∈Np,q−sRn, let fj=fχBj, where Bj=B0,jj∈N. Then,
(103)limj⟶∞f−fjNp,q−s=0.Proof.
Becausef∈Np,q−sRn, for any ε1>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 and an Np,q−s,ϕ-atom ft such that
(104)f−fj=fχBjc≤fNp,q−s+ε1∫0∞ϕtftχBjcdtt.
From the dominated convergence theorem, for anyε2>0, there exists N∈N such that
(105)ϕ1/qftχBjcTp,q−s<ε2,for j≥N. Thus, the function ftχBjc/ε2 is an Np,q−s,ϕ-atom, and we have
(106)f−fjNp,q−s≤fNp,q−s+ε1·ε2,for j≥N. This concludes the proof.To prove Theorem21 (ii), it suffices to prove gNp,qs≤CℓNp′,q′−s∗. At first, we prove gN∞,qs≤CℓN1,q′−s∗, which is the case when p=∞. Let B=Bx0,r. From Lemma 27, we have
(107)∫Bx0,rgxqdx1/q~supfLq′B=1∫Bfxgxdx≤supfLq′B=1ℓN1,q′−s∗fχBN1,q′−s≤Cr−s+n/qℓN1,q′−s∗,for any x0∈Rn and r>0. Thus, we have gN∞,qs≤CℓN1,q′−s∗ by Proposition 31. Next, we prove Theorem 21 (ii) for the case when 1<p<∞. To prove gNp,qs≤CℓNp′,q′−s∗ for 1<p<∞, we present the following lemma.Lemma 36.
Let1<q≤p<∞ and n/q′−n/p′≤−s≤0. For ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, define
(108)πqs,ϕFx=∫0∞ϕt1/qtsFx,tdtt.
Then,(109)πqs,ϕFNp′,q′−s≤FTp′,q′0,for F∈Tp′,q′0R+n+1.Proof.
From the definition ofπqs,ϕFx, we get
(110)πqs,ϕFx≤FTp′,q′0∫0∞ϕt1FTp′,q′0ϕt−1/q′tsFx,tdtt.
Note that(111)ϕt1/q′1FTp′,q′ϕt−1/q′tsFx,tTp′,q′−s=1.
The function1/FTp′,q′ϕt−1/q′tsFx,t is an Np′,q′−s,ϕ-atom, and we obtain
(112)πqs,ϕFNp′,q′−s≤FTp′,q′0.We now prove thatgNp,qs≤CℓNp′,q′−s∗ for 1<p<∞. For any ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, by applying Lemma 13, (40), and Lemma 36, we have
(113)gNp,qs,ϕ=tsϕ1/qgTp,q0~supFTp′,q′0=1∫0∞∫Rntsϕt1/qgxFx,tdxdtt=supFTp′,q′0=1∫Rngxπqs,ϕFxdx≤supFTp′,q′0=1ℓN1,q′−s∗πqs,ϕFNp′,q′−s≤CsupFTp′,q′0=1ℓN1,q′−s∗FTp′,q′0=CℓN1,q′−s∗.Taking the supremum overϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, we obtain the conclusion.


### 4.3. Proof of Theorem23

In this section, we prove Theorem23. Notice that LpRn1≤p<∞ and MqsRn0≤s≤n/q are Banach function lattices, satisfy the Fatou property, and are 1-convex. Furthermore, LpRn1≤p<∞ is separable. Thus, thanks to (32), it suffices to demonstrate the following:Theorem 37.
Let1≤p,q<∞, 0≤s≤n/q, and 0≤θ≤1. Then,
(114)LpRn1−θMqsRnθ=Np/1−θ,qθθsRnwhere 1/qθ=1−θ/p+θ/q.Before proving Theorem37, we check the consistency of Theorem 37 for the critical cases s=0 and s=n/q.Lemma 38.
Let1≤p<∞. Then,
(115)fNp,q0~fp,for any 1≤q≤p and f∈L0Rn.Proof.
From Lemmas24 and 25, we have
(116)fNp,q0≤fp,for any q≤p. Conversely, we set
(117)ϕεt=εn+δtn+δχε,∞t,for some fixed δ>0. We also set
(118)c1≔∫1∞1tn+δdttandc2≔∫1∞1tδdtt.
Then, by Lebesgue differentiation theorem andc1=ϕεL10,∞;/dt/t, we have
(119)fp=liminfε⟶01εn∫Bx,εfyqdy1/qLxp=1c1liminfε⟶0∫0∞∫Bx,εϕεt1εnfyqdydtt1/qLxp≤1c1liminfε⟶0∫0∞∫Bx,tεδtδχε,∞tfyqdydttn+11/qLxp≤c2c1fNp,q0.
So, we are done.Lettings=0 in Theorem 37, we see that the property Mq0Rn=L∞Rn for any 1≤q<∞ yields
(120)LpRn1−θL∞Rnθ=Np/1−θ,qθ0Rn.From Lemma38, we have Np/1−θ,qθ0Rn=Lp/1−θRn, and it does not contradict Lemma 10. Furthermore, we have the following.Proposition 39.
Ifs=n/q−n/p, one has
(121)fNp,qs~fq,for any f∈LqRn.Proof.
From Proposition31 and Lemma 26, we have
(122)fq=fMqn/q~fN∞,qn/q≤CfNp,qn/q−n/p.
Next, Lemmas24 and 26 imply that
(123)fNp,qn/q−n/p≤CfNq,q0=fq.Settings=n/q in Theorem 37, we learn that Mqn/qRn=LqRn implies that
(124)LpRn1−θLqRnθ=Np/1−θ,qθθn/qRn.Fromθn/q=n/qθ−1−θn/p and Proposition 39, we obtain Np/1−θ,qθθn/qRn=LqθRn, and it does not contradict Lemma 10.We now prove Theorem37. First, we prove that LpRn1−θMqsRnθ⊂Np/1−θ,qθθsRn. Let f∈LpRn1−θMqsRnθ. Then, for any ε>0, there exists f1∈LpRn and f2∈MqsRn such that fx≤f1x1−θf2xθ and
(125)f1Lp1−θf2Mqsθ≤fLp1−θMqsθ+ε.It should be noted thatfθTp,qs=fTθp,θqs/θθ for any θ>0. From (42), for any ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, we have
(126)ϕt1/qθfxTp/1−θ,qθθs≤ϕt1/qθf1x1−θf2xθTp/1−θ,q0θs≤Cϕt1−θ/pf1x1−θTp/1−θ,p/1−θ0ϕtθ/qf2xθT∞,q/θθs≤Cf1p1−θf2Mqsθ≤CfLp1−θMqsθ+ε,for any ε>0. Thus, we obtain fNp/1−θ,qθθs≤CfLp1−θMqsθ, which concludes the proof of LpRn1−θMqsRnθ⊂Np/1−θ,qθθsRn.Next, we must proveNp/1−θ,qθθsRn⊂LpRn1−θMqsRnθ. Let ϕ0t=ctαχ0,1t for a sufficiently large α>0, where c>0 denotes the number that satisfies ϕ0L10,∞;dt/t=1. We assume that f∈Np/1−θ,qθθsRn. Then, we have ϕ0t1/qθfx∈Tp/1−θ,qθθsR+n+1. Thus, Lemma 15 implies that for any ε>0, there exists functions Fx,t∈Tp,p0R+n+1 and Gx,t∈T∞,qsR+n+1, such that
(127)ϕ0t1/qθfx≤Fx,t1−θGx,tθ,(128)FTp,p01−θGT∞,qsθ≤ϕ0t1/qθfxTp/1−θ,qθθs+ε.From (127), we obtain
(129)ϕ0tfx≤ϕ0t1/qθ′Fx,t1−θGx,tθ.Sinceϕ0L10,∞;dt/t=1 and 1/qθ=1−θ/p′+θ/q′, integrating both sides of the above inequality against the measure dt/t and using Hölder’s inequality, we obtain
(130)fx≤∫0∞ϕ0t1/qθ′Fx,t1−θGx,tθdtt≤∫0∞ϕ0t1/p′Fx,tdtt1−θ∫0∞ϕ0t1/q′Gx,tdttθ≕f1x1−θf2xθ.Usingϕ0L10,∞;dt/t=1 and Hölder’s inequality again, we obtain
(131)f1p≤∫0∞ϕ0tdtt1/p′∫0∞Fx,tdtt1/pp~FTp,p0,where we used (37) in the last equality. Next, we estimate f2. Using (41), we have
(132)f2LqBx0,1~supgLq′Bx0,1=1∫0∞∫Rnϕ0t1/q′Gx,tgxχBx0,1dxdtt≤CsupgLq′Bx0,1=1GT∞,qsϕ0t1/q′gχBx0,1xT1,q′−s,for any x0∈Rn. Using the argument in the proof of Lemma 27, we obtain
(133)ϕ0t1/q′gχBx0,1xT1,q′−s≤CgLq′Bx0,1,which implies that
(134)f2LqBx0,1≤CGT∞,qs,for any x0∈Rn. By combining (131), (134), and (128), we obtain that for any ε>0, there exists functions f1∈LpRn and f2∈LlocqRn, such that fx≤f1x1−θf2xθ and
(135)f1p1−θsupx0∈Rnf2LqBx0,1θ≤CFTp,p01−θGT∞,qsθ≤ϕ01/qθfTp/1−θ,qθθs+ε,for any f∈Np/1−θ,qθθsRn. Thus, we obtain
(136)f1p1−θsupx0∈Rnf2LqBx0,1θ≤CfNp/1−θ,qθθs.To conclude, we investigate the dilation property.Lemma 40.
Let1≤p≤∞, 1≤q<∞, and 0≤s≤n/q−n/p. Define frx=fx/r for r>0 and f∈L0Rn. Then,
(137)frNp,qs=rs+n/pfNp,qs,for any f∈Np,qsRn.Proof.
Letf∈Np,qsRn and ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1. Considering that ϕr·∈L10,∞;dt/t with ϕr·L10,∞;dt/t=1 for any r>0, we have
(138)frNp,qs,ϕ=∫0∞∫Bx,ttsqϕtfyrqdydttn+11/qLxp=∫0∞∫Bx,trsqtrsqϕrtrfyrqdydttn+11/qLxp=rs+n/p∫0∞∫Bx,ttsqϕrtfyqdydttn+11/qLxp=rs+n/pfNp,qs,ϕr·.
Thus, we obtainfrNp,qs=rs+n/pfNp,qs.Now, all preparations to proveNp/1−θ,qθθsRn⊂LpRn1−θMqsRnθ have been provided. From Lemma 40, we have f∈Np/1−θ,qθθsRn if and only if fr∈Np/1−θ,qθθsRn for any r>0. We fix an arbitrary r>0. Then, from (136) and Lemma 40, we obtain
(139)f1rp1−θsupx0∈Rnrs1rn∫Bx0,rf2ryqdy1/qθ=rθs+n1−θ/pf1psupx0∈Rnf2LqBx0,1θ≤Crθs+n1−θ/pfNp/1−θ,qθθs=frNp/1−θ,qθθs.Note that(140)fx≤f1x1−θf2xθ⇔frx≤f1rx1−θf2rxθ,for any r>0. By replacing fr=g for any g∈Np/1−θ,qθθsRn, there exist meaurable functions g1,g2 such that
(141)gx≤g1x1−θg2xθ,g1p1−θsupx0∈Rnrs1rn∫Bx0,rg2yqdy1/qθ≤CgNp/1−θ,qθθs.Becauser>0 is arbitrary, we have the conclusion.


## 4.1. Properties

In this section, we prove some properties ofNp,qs-spaces. At first, we prove that the space Np,qsRn generalizes Lebesgue spaces and Morrey spaces. This relation is similar to the fact that homogeneous Triebel-Lizorkin space Ḟp,qsRn generalizes LpRn and BMORn. The coincidence with Lebesgue spaces is the following. Here, we must keep in mind that we defined ∗Np,p0 in two ways, which are Np,ps-norms for s≥0 and for s≤0. We will show that Lemma 24 below holds in both senses.Lemma 24.
For1≤p<∞, we have LpRn=Np,p0Rn, and the following equivalence holds:
(72)fNp,p0~fp,for any f∈L0Rn.Proof.
Letf∈L0Rn. First, for Np,qs-norms for s≥0, (37) implies that
(73)fNp,p0=supϕL10,∞;dt/t=1ϕ1/pfTp,p0~supϕL10,∞;dt/t=1∫Rn∫0∞ϕtfxpdtdxt1/p=fp.
Conversely, forNp,ps-norms for s≤0, Lemma 20 and (37) imply that
(74)fNp,p0≤fN~p,p0=infϕL10,∞;dt/t=1ϕ1/pfTp,p0~fp.
Meanwhile, if we assumef∈Np,p0Rn in the sense of Definition 18, then for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 and Np,p0,ϕ-atom ft such that
(75)fx≤fNp,p0+ε∫0∞ϕtftxdtt.
Thus, by Hölder’s inequality fort, we have
(76)fp≤fNp,p0+ε∫0∞ϕtftdttp≤CfNp,p0+ε∫0∞ϕtdtt1/p′ϕ1/pftTp,p0≤fNp,p0+ε,for any ε>0. It concludes the proof in both senses.The scaleNp,qsRn is monotone in 1≤q<∞ as Lemma 25 shows, which is similar to homogeneous Triebel-Lizorkin spaces Ḟp,qsRn for 1≤q<∞.Lemma 25.
For any1≤p≤∞, 1≤q1≤q2<∞, and s∈R, one has Np,q2sRn⊂Np,q1sRn, and
(77)fNp,q1s≤CfNp,q2s,for any f∈Np,q2sRn.Proof.
If suffices to show thatfNp,q1s,ϕ≤CfNp,q2s,ϕ for any ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, it is easy to show for p=∞ using Hölder’s inequality. So, we assume that 1≤p<∞. Setting 1/q1=1/q2+1/q3 and using Hölder’s inequality twice, we learn
(78)fNp,q1s,ϕ=∫0∞∫Bx,ttsq1ϕtfyq1dydttn+11/q1Lxp≤C∫0∞ϕt∫Bx,ttsq2fyq2dytnq1/q2dtt1/q1Lxp≤C∫0∞ϕtdtt1/q3∫0∞∫Bx,ttsq2ϕtfyq2dydttn+11/q2Lxp=CfNp,q2s,ϕ,from ϕL10,∞;dt/t=1.The following is the Sobolev embedding inNp,qs-spaces:Lemma 26.
Let1<q<∞. For 1<p1,p2≤∞, and 0≤s1≤s2 (or s1≤s2≤0), if s1+n/p1=s2+n/p2, then one has Np1,qs1Rn⊂Np2,qs2Rn, and
(79)fNp2,qs2≤CfNp1,qs1,for any f∈Np1,qs1Rn.Lemma26 is a direct consequence of Lemma 14. We omit its proof. Lemmas 24 and 26 imply that Np,q−sRn⊂LqRn for −s+n/p=n/q. The following lemma claims that the converse embedding locally holds.Lemma 27.
Let1≤p≤q<∞, n/q−n/p<−s≤0, and B=Bx0,r⊂Rn be a ball. For f∈LqRn with suppf⊂B, one has
(80)fN~p,q−s≤Cr−s+n/p−n/qfq.In particular, the following inequality holds:(81)fN~1,q−s≤Cr−s+n/q′fq,for 0≤s<n/q′.Before proving Lemma27, we prepare the definition and some properties of Lorentz spaces.Definition 28.
Let0<p<∞ and 0<q≤∞. For f∈L0Rn, we define the decreasing rearrangement function f∗:0,∞⟶0,∞ of f as
(82)f∗t≔infα∈0,∞:dfα≤t,where df is the distribution function of f, given by
(83)dfα=x∈Rn:fx>αα>0.Here,inf∅ stands for ∞. Let f∈L0Rn. If 0<q<∞, then define
(84)fLp,q≔∫0∞t1/pf∗tqdtt1/q.Ifq=∞, then define
(85)fLp,q≔supt>0t1/pf∗t.We define the Lorentz spaceLp,qRn as the set of all functions f∈L0Rn for which fLp,q<∞.We note some properties of Lorentz spaces.Remark 29.
(i)
Let1<p1,p2,p3<∞ satisfy 1/p3=1/p1+1/p2 and 1≤q≤∞. Then, for f∈Lp1,qRn and g∈Lp2,q′Rn, we have
(86)fgLp3≤fLp1,qgLp2,q′.(ii)
For0<q<∞ and f∈LqRn, we have
(87)MqfLq,∞≤Cfq,where Mq is the q-powered Hardy–Littlewood maximal operator (see Section 1.1).(iii)
For any0<p<∞ and ball B=Bx0,r⊂Rn, we have
(88)χBLp,1~Bn/p.See, e.g., [4] for the above properties.All preparations to prove Lemma27 were given.Proof of Lemma 30.
For a fixedα>sq, let ϕt=cχ0,rttα/rα, where c>0 is a normalization parameter satisfying ϕL10,∞;dt/t=1. Since suppf⊂Bx0,r, for y∈suppf, we get y−x0<r. Fix x∈Rn and 0<t<r with suppf∩Bx,t≠∅. Then, for y∈suppf∩Bx,t, we have
(89)x−x0≤x−y+y−x0<t+r<2r.
Keeping in mind that in Remark29, we have
(90)fN~p,q−s≤∫0∞∫Bx,tt−sqϕtfyqdydttn+11/qLxpRn≤C∫0r∫Bx,ttα−sqrαfyqdydttn+11/qLxpBx0,2r≤Cr−sMqfLpBx0,2r≤Cr−sMqfLq,∞χBx0,2rLp0,1≤Cr−s+n/p−n/qfq,where 1/p=1/q+1/p0.The following proposition shows thatNp,qs-spaces generalize Morrey spaces.Proposition 31.
For1≤q<∞ and 0<s<n/q, one has N∞,qsRn=MqsRn with the norm equivalence
(91)fN∞,qs~fMqs,for any f∈MqsRn.Proof.
We can easily show thatfN∞,qs≤fMqs for f∈MqsRn:
(92)fN∞,qs=supϕL10,∞;dt/t=1supx,r∈Rn×0,∞1B∫0r∫x−y<r−ttsqϕtfyqdydtt1/q≤supϕL10,∞;dt/t=1supx,r∈Rn×0,∞rsqB∫0∞∫x−y<rϕtfyqdydtt1/q=fMqs.
Conversely, we assumef∈N∞,qsRn. For B=Bx0,r⊂Rn, we have
(93)fLqB~supgLq′B=1∫Bfxgxdx.
Then, Lemma27 implies gχB∈Lq′Rn⊂N1,q′−sRn for any g∈Lq′B. Moreover, for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 such that ϕ1/q′gχBT1,q′−s≤gχBN~1,q′−s+ε. Thus, Hölder’s inequality and Lemmas 13 and 27 imply that
(94)supgLq′B=1∫Bfxgxdx=supgLq′B=1∫0∞∫RnϕtfxgχBxdxdtt≤CsupgLq′B=1ϕ1/qfT∞,qsϕ1/q′gχBT1,q′−s≤CsupgLq′B=1fN∞,qsgχBN~1,q′−s+ε≤CsupgLq′B=1fN∞,qsr−s+n/qgχBq′+ε=Cr−s+n/qfN∞,qs+fN∞,qsε,for any ε>0. Consequently, the arbitrariness of ε>0 implies that
(95)rs−n/qfLqB≤CfN∞,qs,for any ball B=Bx0,r, and it concludes the proof.Lemma25 shows that the Np,qs-spaces are vested. We combine Lemmas 25 and 26 and Proposition 31.Remark 32.
Let1≤q<p≤∞ and 0≤s≤n/q−n/p. Then,
(96)Np,qsRn⊂N∞,qs+n/pRn=Mqs+n/pRn.Here, we used Lemma26 and Proposition 31.Furthermore, we obtain the following:Remark 33.
Let1≤p≤q<∞ and n/q−n/p<−s≤0. Then,
(97)Np,q−sRn⊂Np0,q0Rn⊂Lp0Rn,where p0=1/p−s/n−1<q. Here, we used Lemmas 26 and 25.


## 4.2. Proof of Theorem21

In this section, we prove Theorem21.Proof of Theorem 34 (0), (i).
Letf∈Np′,q′−sRn and g∈Np,qsRn. Then, for any ε>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 and Np′,q′−s,ϕ-atom ft such that
(98)fx≤fNp′,q′−s+ε∫0∞ϕtftxdtt.Thus, (40) implies that
(99)∫Rnfxgxdx≤fNp′,q′−s+ε∫0∞∫Rnϕtftxgxdxdtt≤CfNp′,q′−s+εϕ1/q′ftTp′,q′−sϕ1/qgTp,qs≤fNp′,q′−s+εgNp,qs,for any ε>0. Thus, we have
(100)∫Rnfxgxdx≤CfNp′,q′−sgNp,qs,and it concludes the proof of Theorem 34 (0). From (100), we obtain
(101)ℓgNp′,q′−s∗=supfNp′,q′−s=1∫Rnfxgxdx≤CgNp,qs.It shows Theorem21 (i).Proof of Theorem34 (ii). We assume that ℓ is a bounded linear functional of Np′,q′−sRn. Fix a ball B=Bx0,r. If f is supported on Bx0,r, then
(102)fNp′,q′−s≤Cr−s+n/q−n/pfχBx0,rq′from Lemma 27. Hence, ℓ induces a bounded linear functional on Lq′Bx0,r and acts with gB∈LqBx0,r. By taking Bj=B0,jj∈N, we have gBj=gBj+1 on Bj, and thus we obtain a unique function g on Rn that is locally in LqRn, such that ℓf=∫Rnfxgxdx when f∈Np′,q′−sRn is supported on some ball. We can extend ℓ as a global functional over Rn using the following lemma:Lemma 35.
Let1≤p≤q<∞, and n/q−n/p<−s≤0. For f∈Np,q−sRn, let fj=fχBj, where Bj=B0,jj∈N. Then,
(103)limj⟶∞f−fjNp,q−s=0.Proof.
Becausef∈Np,q−sRn, for any ε1>0, there exists ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1 and an Np,q−s,ϕ-atom ft such that
(104)f−fj=fχBjc≤fNp,q−s+ε1∫0∞ϕtftχBjcdtt.
From the dominated convergence theorem, for anyε2>0, there exists N∈N such that
(105)ϕ1/qftχBjcTp,q−s<ε2,for j≥N. Thus, the function ftχBjc/ε2 is an Np,q−s,ϕ-atom, and we have
(106)f−fjNp,q−s≤fNp,q−s+ε1·ε2,for j≥N. This concludes the proof.To prove Theorem21 (ii), it suffices to prove gNp,qs≤CℓNp′,q′−s∗. At first, we prove gN∞,qs≤CℓN1,q′−s∗, which is the case when p=∞. Let B=Bx0,r. From Lemma 27, we have
(107)∫Bx0,rgxqdx1/q~supfLq′B=1∫Bfxgxdx≤supfLq′B=1ℓN1,q′−s∗fχBN1,q′−s≤Cr−s+n/qℓN1,q′−s∗,for any x0∈Rn and r>0. Thus, we have gN∞,qs≤CℓN1,q′−s∗ by Proposition 31. Next, we prove Theorem 21 (ii) for the case when 1<p<∞. To prove gNp,qs≤CℓNp′,q′−s∗ for 1<p<∞, we present the following lemma.Lemma 36.
Let1<q≤p<∞ and n/q′−n/p′≤−s≤0. For ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, define
(108)πqs,ϕFx=∫0∞ϕt1/qtsFx,tdtt.
Then,(109)πqs,ϕFNp′,q′−s≤FTp′,q′0,for F∈Tp′,q′0R+n+1.Proof.
From the definition ofπqs,ϕFx, we get
(110)πqs,ϕFx≤FTp′,q′0∫0∞ϕt1FTp′,q′0ϕt−1/q′tsFx,tdtt.
Note that(111)ϕt1/q′1FTp′,q′ϕt−1/q′tsFx,tTp′,q′−s=1.
The function1/FTp′,q′ϕt−1/q′tsFx,t is an Np′,q′−s,ϕ-atom, and we obtain
(112)πqs,ϕFNp′,q′−s≤FTp′,q′0.We now prove thatgNp,qs≤CℓNp′,q′−s∗ for 1<p<∞. For any ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, by applying Lemma 13, (40), and Lemma 36, we have
(113)gNp,qs,ϕ=tsϕ1/qgTp,q0~supFTp′,q′0=1∫0∞∫Rntsϕt1/qgxFx,tdxdtt=supFTp′,q′0=1∫Rngxπqs,ϕFxdx≤supFTp′,q′0=1ℓN1,q′−s∗πqs,ϕFNp′,q′−s≤CsupFTp′,q′0=1ℓN1,q′−s∗FTp′,q′0=CℓN1,q′−s∗.Taking the supremum overϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, we obtain the conclusion.


## 4.3. Proof of Theorem23

In this section, we prove Theorem23. Notice that LpRn1≤p<∞ and MqsRn0≤s≤n/q are Banach function lattices, satisfy the Fatou property, and are 1-convex. Furthermore, LpRn1≤p<∞ is separable. Thus, thanks to (32), it suffices to demonstrate the following:Theorem 37.
Let1≤p,q<∞, 0≤s≤n/q, and 0≤θ≤1. Then,
(114)LpRn1−θMqsRnθ=Np/1−θ,qθθsRnwhere 1/qθ=1−θ/p+θ/q.Before proving Theorem37, we check the consistency of Theorem 37 for the critical cases s=0 and s=n/q.Lemma 38.
Let1≤p<∞. Then,
(115)fNp,q0~fp,for any 1≤q≤p and f∈L0Rn.Proof.
From Lemmas24 and 25, we have
(116)fNp,q0≤fp,for any q≤p. Conversely, we set
(117)ϕεt=εn+δtn+δχε,∞t,for some fixed δ>0. We also set
(118)c1≔∫1∞1tn+δdttandc2≔∫1∞1tδdtt.
Then, by Lebesgue differentiation theorem andc1=ϕεL10,∞;/dt/t, we have
(119)fp=liminfε⟶01εn∫Bx,εfyqdy1/qLxp=1c1liminfε⟶0∫0∞∫Bx,εϕεt1εnfyqdydtt1/qLxp≤1c1liminfε⟶0∫0∞∫Bx,tεδtδχε,∞tfyqdydttn+11/qLxp≤c2c1fNp,q0.
So, we are done.Lettings=0 in Theorem 37, we see that the property Mq0Rn=L∞Rn for any 1≤q<∞ yields
(120)LpRn1−θL∞Rnθ=Np/1−θ,qθ0Rn.From Lemma38, we have Np/1−θ,qθ0Rn=Lp/1−θRn, and it does not contradict Lemma 10. Furthermore, we have the following.Proposition 39.
Ifs=n/q−n/p, one has
(121)fNp,qs~fq,for any f∈LqRn.Proof.
From Proposition31 and Lemma 26, we have
(122)fq=fMqn/q~fN∞,qn/q≤CfNp,qn/q−n/p.
Next, Lemmas24 and 26 imply that
(123)fNp,qn/q−n/p≤CfNq,q0=fq.Settings=n/q in Theorem 37, we learn that Mqn/qRn=LqRn implies that
(124)LpRn1−θLqRnθ=Np/1−θ,qθθn/qRn.Fromθn/q=n/qθ−1−θn/p and Proposition 39, we obtain Np/1−θ,qθθn/qRn=LqθRn, and it does not contradict Lemma 10.We now prove Theorem37. First, we prove that LpRn1−θMqsRnθ⊂Np/1−θ,qθθsRn. Let f∈LpRn1−θMqsRnθ. Then, for any ε>0, there exists f1∈LpRn and f2∈MqsRn such that fx≤f1x1−θf2xθ and
(125)f1Lp1−θf2Mqsθ≤fLp1−θMqsθ+ε.It should be noted thatfθTp,qs=fTθp,θqs/θθ for any θ>0. From (42), for any ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1, we have
(126)ϕt1/qθfxTp/1−θ,qθθs≤ϕt1/qθf1x1−θf2xθTp/1−θ,q0θs≤Cϕt1−θ/pf1x1−θTp/1−θ,p/1−θ0ϕtθ/qf2xθT∞,q/θθs≤Cf1p1−θf2Mqsθ≤CfLp1−θMqsθ+ε,for any ε>0. Thus, we obtain fNp/1−θ,qθθs≤CfLp1−θMqsθ, which concludes the proof of LpRn1−θMqsRnθ⊂Np/1−θ,qθθsRn.Next, we must proveNp/1−θ,qθθsRn⊂LpRn1−θMqsRnθ. Let ϕ0t=ctαχ0,1t for a sufficiently large α>0, where c>0 denotes the number that satisfies ϕ0L10,∞;dt/t=1. We assume that f∈Np/1−θ,qθθsRn. Then, we have ϕ0t1/qθfx∈Tp/1−θ,qθθsR+n+1. Thus, Lemma 15 implies that for any ε>0, there exists functions Fx,t∈Tp,p0R+n+1 and Gx,t∈T∞,qsR+n+1, such that
(127)ϕ0t1/qθfx≤Fx,t1−θGx,tθ,(128)FTp,p01−θGT∞,qsθ≤ϕ0t1/qθfxTp/1−θ,qθθs+ε.From (127), we obtain
(129)ϕ0tfx≤ϕ0t1/qθ′Fx,t1−θGx,tθ.Sinceϕ0L10,∞;dt/t=1 and 1/qθ=1−θ/p′+θ/q′, integrating both sides of the above inequality against the measure dt/t and using Hölder’s inequality, we obtain
(130)fx≤∫0∞ϕ0t1/qθ′Fx,t1−θGx,tθdtt≤∫0∞ϕ0t1/p′Fx,tdtt1−θ∫0∞ϕ0t1/q′Gx,tdttθ≕f1x1−θf2xθ.Usingϕ0L10,∞;dt/t=1 and Hölder’s inequality again, we obtain
(131)f1p≤∫0∞ϕ0tdtt1/p′∫0∞Fx,tdtt1/pp~FTp,p0,where we used (37) in the last equality. Next, we estimate f2. Using (41), we have
(132)f2LqBx0,1~supgLq′Bx0,1=1∫0∞∫Rnϕ0t1/q′Gx,tgxχBx0,1dxdtt≤CsupgLq′Bx0,1=1GT∞,qsϕ0t1/q′gχBx0,1xT1,q′−s,for any x0∈Rn. Using the argument in the proof of Lemma 27, we obtain
(133)ϕ0t1/q′gχBx0,1xT1,q′−s≤CgLq′Bx0,1,which implies that
(134)f2LqBx0,1≤CGT∞,qs,for any x0∈Rn. By combining (131), (134), and (128), we obtain that for any ε>0, there exists functions f1∈LpRn and f2∈LlocqRn, such that fx≤f1x1−θf2xθ and
(135)f1p1−θsupx0∈Rnf2LqBx0,1θ≤CFTp,p01−θGT∞,qsθ≤ϕ01/qθfTp/1−θ,qθθs+ε,for any f∈Np/1−θ,qθθsRn. Thus, we obtain
(136)f1p1−θsupx0∈Rnf2LqBx0,1θ≤CfNp/1−θ,qθθs.To conclude, we investigate the dilation property.Lemma 40.
Let1≤p≤∞, 1≤q<∞, and 0≤s≤n/q−n/p. Define frx=fx/r for r>0 and f∈L0Rn. Then,
(137)frNp,qs=rs+n/pfNp,qs,for any f∈Np,qsRn.Proof.
Letf∈Np,qsRn and ϕ∈L10,∞;dt/t satisfying ϕL10,∞;dt/t=1. Considering that ϕr·∈L10,∞;dt/t with ϕr·L10,∞;dt/t=1 for any r>0, we have
(138)frNp,qs,ϕ=∫0∞∫Bx,ttsqϕtfyrqdydttn+11/qLxp=∫0∞∫Bx,trsqtrsqϕrtrfyrqdydttn+11/qLxp=rs+n/p∫0∞∫Bx,ttsqϕrtfyqdydttn+11/qLxp=rs+n/pfNp,qs,ϕr·.
Thus, we obtainfrNp,qs=rs+n/pfNp,qs.Now, all preparations to proveNp/1−θ,qθθsRn⊂LpRn1−θMqsRnθ have been provided. From Lemma 40, we have f∈Np/1−θ,qθθsRn if and only if fr∈Np/1−θ,qθθsRn for any r>0. We fix an arbitrary r>0. Then, from (136) and Lemma 40, we obtain
(139)f1rp1−θsupx0∈Rnrs1rn∫Bx0,rf2ryqdy1/qθ=rθs+n1−θ/pf1psupx0∈Rnf2LqBx0,1θ≤Crθs+n1−θ/pfNp/1−θ,qθθs=frNp/1−θ,qθθs.Note that(140)fx≤f1x1−θf2xθ⇔frx≤f1rx1−θf2rxθ,for any r>0. By replacing fr=g for any g∈Np/1−θ,qθθsRn, there exist meaurable functions g1,g2 such that
(141)gx≤g1x1−θg2xθ,g1p1−θsupx0∈Rnrs1rn∫Bx0,rg2yqdy1/qθ≤CgNp/1−θ,qθθs.Becauser>0 is arbitrary, we have the conclusion.


## 5. An Application of Complex Interpolation Spaces

As basic properties of function spaces, we will prove the boundedness of Calderón–Zygmund operators. Calderón–Zygmund operators are defined by the following:For a bounded linear operatorT:L2Rn⟶L2Rn, we say that T is a Calderón–Zygmund operator if there exists a kernel kx,y, and it satisfies the following conditions:
(i)
Tfx=∫Rnkx,yfydy for x∉suppf(ii)
kx,y≤C1/x−yn for any x,y∈Rn(iii)
kx,y−kz,y+ky,x−ky,z≤Cz−y/x−yn+1 whenever z−y<1/2x−yIn fact, these operators are bounded on Lebesgue spacesLpRn (1<p<∞) (see Stein [17]) and Morrey spaces MqsRn (1<q<∞, 0<s<n/q) (see F. Chiarenza and M. Frasca [18]). Using Theorem 37, we obtain the boundedness in Np,qs-spaces.Theorem 41.
For1<p≤∞, 1<q<∞, and 0≤s≤n/q−n/p, any Calderón–Zygmund operator T are bounded on Np,qsRn.Proof.
For the case wheres=0, it immediately follows by the boundedness of T in Lebesgue spaces and Lemma 38. Furthermore, for the case where p=∞, we use the boundedness in Morrey spaces and Proposition 31. We prove the other cases. Combining Lemma 8 with Theorem 23 and the boundedness of T in LpRn and MqsRn, we see
(142)TNp/1−θ,qθθs⟶Np/1−θ,qθθs≤CTLp⟶Lp1−θTMqs⟶Mqsθ,for any 0<θ<1, 1<p,q<∞, and 0<s<n/q. Thus, we obtain TNp,qs⟶Np,qs<∞ for any 1<p≤∞, 1<q<∞, and 0≤s≤n/q−n/p.



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*Source: 5822846-2023-01-16.xml*