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

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 space 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.

1. Introduction

The Morrey space 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 space was introduced by Triebel (see [3]). The norm of is characterized in terms of tent spaces. The space generalizes Lebesgue spaces and the bounded mean oscillation space . Furthermore, the following properties is known as a particular case:

Here, is the complex interpolation space between and . Furthermore, is the dual space of .

In this paper, the new function space is introduced (see Section 2.2 below). The norm of the space is also defined via tent spaces like homogeneous Triebel-Lizorkin spaces. The motivation of the study of the space is to construct the corresponding chart for Lebesgue spaces and Morrey spaces. The following chart summarizes arguments shown in this paper:

The explicit expressions of and 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 by the set of all measurable functions on (2)For , the conjugate number of is defined by the number which realizes (3)For and , is the ball with radius centered at (4)For and , we define the Hardy-Littlewood maximal operator by  More generally, for , we define its powered version by (5)The space consists of measurable functions satisfying

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 spaces defined as follows:

Let . Then, the space is the set of all functions satisfying for and for . Next, we recall the definition of Morrey spaces which is the main function spaces in this paper.

Definition 1. Let and . The Morrey space is the set of all functions satisfying

We note some properties of Morrey spaces.

Remark 2. (1)Let be the volume of the unit ball on . Then, the equation implies that with equivalence of norms for .(2)For , the embedding follows from Hölder’s inequality.(3)Combining (10) and (11), for any , we have

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 space , we write the dual space of as . The Banach space is defined as the set of all linear continuous functionals . We define

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 in :

Definition 4. For quasi-Banach spaces , we say that if it satisfies the following:
(0) If and , then (i)For , we set  Then, we have and (ii)For any , there uniquely exists satisfying and we have

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 space which will be defined in Section 2.2 below. For quasi-Banach spaces and , we say that is a predual space of if it satisfies . It is known that a predual space is not always unique. For Lebesgue spaces, we have the following duality property:

Lemma 5. Let . Then,

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. Let and . We say that a measurable function is a -block in a ball if satisfies (i)(ii)The block space is the set of all functions such that where the infimum is taken over all decompositions of . Here, each satisfies (i) and (ii) for replaced by the ball . Note that can vary according to .

The dual space of is the Morrey space .

Lemma 7 (see [5]). Let and . Then,

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 spaces , let be the sum space such that

We define be the set of all mappings which are analytic in , continuous on and satisfy (i)(ii), , for each and (iii), , for each and

Furthermore, introducing the norm of by we define the complex interpolation space between and with respect to , by equipped with norm

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 spaces and , assume that a linear operator is bounded from to . Then, is also bounded from to and satisfies for any . Here, we define the operator norm by for an operator from a Banach space to a Banach space .

We shall give another characterization of complex interpolation spaces by using Carderón product spaces defined as follows:

Definition 9. For quasi-Banach spaces , and , we define where

The space is called the Carderón product space between and with respect to .

Below, we recall the relationship between complex interpolation spaces and Calderón product spaces. We say that a quasi-Banach space is a quasi-Banach function lattice if it satisfies for any . We say that a quasi-Banach space satisfies the Fatou property if for . Furthermore, we say that a quasi-Banach space is -convex for if it satisfies for . In Kalton and Mitrea [9], it was shown that, if two quasi-Banach function lattices and satisfy the Fatou property, the -convexity (), and if either of or is separable, then for any .

We recall two classical formulas on complex interpolation for Lebesgue spaces and for Morrey spaces.

Lemma 10 (see [4]). Let and . Then, for .

The complex interpolation of Morrey spaces is more complicated than that of Lebesgue spaces.

Lemma 11 (Hakim and Sawano [10]). Let , , and . If , then for and . Here, denotes the closure with respect to of the set of all essentially bounded functions in .

The organization of the remain part is as follows: In Section 2, we introduce tent spaces and our new space . 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 space . 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 .

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 .

Definition 12. Let , , and . The tent space is the set of all measurable functions satisfying for , and for .

It is easy to show that the triangle inequality for holds. Moreover, we note that by applying Fubini’s lemma, we have for and .

The following is the known duality theorem for tent spaces based on Definition 4:

Lemma 13 (Huang [13]). Let , , and . If and , then is integrable with respect to the measure and via the coupling Furthermore, the following inequalities hold: for , and for . Moreover, the inequality holds for , , and with , and .

We note that Lemma 13 is a particular case of Theorem 4.3 in [13] where , , and .

We have also the Sobolev embedding theorem for tent spaces as follows:

Lemma 14 (Amenta [14]). Let . For and , if , one has with a norm inequality for any .

The following lemma states that the Calderón product for tent spaces can be calculated.

Lemma 15 (see [13]). Let , , and . Then, where

2.2. The Definition of -Spaces

In this section, we introduce new function spaces called -spaces by using tent spaces defined in Section 2.1. First, we define the -norms with respect to .

Definition 16. Let . For , , and , we define for , and for .
Using the above norm, we define the -norm for .

Definition 17. For , , and , we define where

Remark that for , the definition of the -norm demanded . The way of defining the -norm for is different from the one for .

Definition 18. Let , , and . We say that the function is an -atom if for all and and The space is the set of all such that there exists and an -atom for which for almost all . Let be the infimum of such , and let where the infimum is taken over all functions which satisfy and all -atoms .

Remark that for , the definition of the -norm demanded . For the case , we defined the -norm in two ways. However, those are essentially equivalent (see Lemma 24 in Section 4.1 below). The definition of the norm is complicated; therefore, we introduce -“norms” for convenience.

Definition 19. For and , we define for measurable function . The space is the set given by

Then, we include the following:

Lemma 20. Assume that the parameters satisfy the conditions in Definition 18. Then, one has , with the inequality for .

Proof. If , then the conclusion is clear. So, assume otherwise. Let . Then, for any , there exists satisfying such that Subsequently, from , we have Then, trivially, Thus, we obtain for any , which concludes the proof.

It is unclear that the -“norms” are quasinorms because they do not satisfy the quasitriangle inequality . Meanwhile, we claim that the -norms do satisfy the quasitriangle inequality. For , it is evident from Definition 17. Thus, we consider the case where . Let . Then, for any , there exists satisfying , and we have for some -atom and -atom . Note that

If we put , then , and we obtain where we put . The triangle inequality in tent spaces implies that

Then, from (62), we obtain

Similarly, we have

Thus, the function is an -atom so that we get for any . This shows that the norm satisfies the quasitriangle inequality.

3. Main Theorems

In this section, we introduce our main theorems. The first result is the duality theorem of -spaces. This derives the duality theorem on Morrey spaces by combining with Proposition 31 in Section 4.1 below.

Theorem 21. Let , , and . Then, one has in the sense of Definition 4.
(0) Let and . Then,(i)For , set  Then, and .(ii)For any , there uniquely exists satisfying , and one has .

Letting and combining Theorem 21 with Proposition 31 in Section 4.1 below, we obtain the following corollary:

Corollary 22. Let and . Then,

For homogeneous Triebel-Lizorkin spaces and the bounded mean oscillation space , it is known that the duality holds (see Sawano [15] for their definitions and the precise statement of the duality). Furthermore, it is generalized to the duality (see Frazier and Jawerth [16]). For Morrey spaces, Theorem 21 realizes the above generalization of . 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 is the new one we found.

The second theorem concerns a complex interpolation between Lebesgue spaces and Morrey spaces.

Theorem 23. Let , , and . Then, where and .

We must assume in the definition of -norms when . The above , , and satisfy the condition. In fact, from , we have

Theorem 23 implies that -spaces with can be characterized by complex interpolation spaces between some Lebesgue space and some Morrey space.

4. Proofs

In this section, we prove some lemmas of -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 -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 of -spaces. At first, we prove that the space generalizes Lebesgue spaces and Morrey spaces. This relation is similar to the fact that homogeneous Triebel-Lizorkin space generalizes and . The coincidence with Lebesgue spaces is the following. Here, we must keep in mind that we defined in two ways, which are -norms for and for . We will show that Lemma 24 below holds in both senses.

Lemma 24. For , we have , and the following equivalence holds: for any .

Proof. Let . First, for -norms for , (37) implies that Conversely, for -norms for , Lemma 20 and (37) imply that Meanwhile, if we assume in the sense of Definition 18, then for any , there exists satisfying and -atom such that Thus, by Hölder’s inequality for , we have for any . It concludes the proof in both senses.

The scale is monotone in as Lemma 25 shows, which is similar to homogeneous Triebel-Lizorkin spaces for .

Lemma 25. For any , , and , one has , and for any .

Proof. If suffices to show that for any satisfying , it is easy to show for using Hölder’s inequality. So, we assume that . Setting and using Hölder’s inequality twice, we learn from .

The following is the Sobolev embedding in -spaces:

Lemma 26. Let . For , and (or ), if , then one has , and for any .

Lemma 26 is a direct consequence of Lemma 14. We omit its proof. Lemmas 24 and 26 imply that for . The following lemma claims that the converse embedding locally holds.

Lemma 27. Let , , and be a ball. For with , one has

In particular, the following inequality holds: for .

Before proving Lemma 27, we prepare the definition and some properties of Lorentz spaces.

Definition 28. Let and . For , we define the decreasing rearrangement function of as where is the distribution function of , given by

Here, stands for . Let . If , then define

If , then define

We define the Lorentz space as the set of all functions for which .

We note some properties of Lorentz spaces.

Remark 29. (i)Let satisfy and . Then, for and , we have (ii)For and , we have where is the -powered Hardy–Littlewood maximal operator (see Section 1.1).(iii)For any and ball , we have

See, e.g., [4] for the above properties.

All preparations to prove Lemma 27 were given.

Proof of Lemma 30. For a fixed , let , where is a normalization parameter satisfying . Since , for , we get . Fix and with . Then, for , we have Keeping in mind that in Remark 29, we have where .

The following proposition shows that -spaces generalize Morrey spaces.

Proposition 31. For and , one has with the norm equivalence for any .

Proof. We can easily show that for : Conversely, we assume . For , we have Then, Lemma 27 implies for any . Moreover, for any , there exists satisfying such that . Thus, Hölder’s inequality and Lemmas 13 and 27 imply that for any . Consequently, the arbitrariness of implies that for any ball , and it concludes the proof.

Lemma 25 shows that the -spaces are vested. We combine Lemmas 25 and 26 and Proposition 31.

Remark 32. Let and . Then,

Here, we used Lemma 26 and Proposition 31.

Furthermore, we obtain the following:

Remark 33. Let and . Then, where . Here, we used Lemmas 26 and 25.

4.2. Proof of Theorem 21

In this section, we prove Theorem 21.

Proof of Theorem 34 (0), (i). Let and . Then, for any , there exists satisfying and -atom such that

Thus, (40) implies that for any . Thus, we have and it concludes the proof of Theorem 34 (0). From (100), we obtain

It shows Theorem 21 (i).

Proof of Theorem 34 (ii). We assume that is a bounded linear functional of . Fix a ball . If is supported on , then from Lemma 27. Hence, induces a bounded linear functional on and acts with . By taking , we have on , and thus we obtain a unique function on that is locally in , such that when is supported on some ball. We can extend as a global functional over using the following lemma:

Lemma 35. Let , and . For , let , where . Then,

Proof. Because , for any , there exists satisfying and an -atom such that From the dominated convergence theorem, for any , there exists such that for . Thus, the function is an -atom, and we have for . This concludes the proof.

To prove Theorem 21 (ii), it suffices to prove . At first, we prove , which is the case when . Let . From Lemma 27, we have for any and . Thus, we have by Proposition 31. Next, we prove Theorem 21 (ii) for the case when . To prove for , we present the following lemma.

Lemma 36. Let and . For satisfying , define Then, for .

Proof. From the definition of , we get Note that The function is an -atom, and we obtain

We now prove that for . For any satisfying , by applying Lemma 13, (40), and Lemma 36, we have

Taking the supremum over satisfying , we obtain the conclusion.

4.3. Proof of Theorem 23

In this section, we prove Theorem 23. Notice that and are Banach function lattices, satisfy the Fatou property, and are 1-convex. Furthermore, is separable. Thus, thanks to (32), it suffices to demonstrate the following:

Theorem 37. Let , , and . Then, where .

Before proving Theorem 37, we check the consistency of Theorem 37 for the critical cases and .

Lemma 38. Let . Then, for any and .

Proof. From Lemmas 24 and 25, we have for any . Conversely, we set for some fixed . We also set Then, by Lebesgue differentiation theorem and , we have So, we are done.

Letting in Theorem 37, we see that the property for any yields

From Lemma 38, we have , and it does not contradict Lemma 10. Furthermore, we have the following.

Proposition 39. If , one has for any .

Proof. From Proposition 31 and Lemma 26, we have Next, Lemmas 24 and 26 imply that

Setting in Theorem 37, we learn that implies that

From and Proposition 39, we obtain , and it does not contradict Lemma 10.

We now prove Theorem 37. First, we prove that . Let . Then, for any , there exists and such that and

It should be noted that for any . From (42), for any satisfying , we have for any . Thus, we obtain , which concludes the proof of .

Next, we must prove . Let for a sufficiently large , where denotes the number that satisfies . We assume that . Then, we have . Thus, Lemma 15 implies that for any , there exists functions and , such that

From (127), we obtain

Since and , integrating both sides of the above inequality against the measure and using Hölder’s inequality, we obtain

Using and Hölder’s inequality again, we obtain where we used (37) in the last equality. Next, we estimate . Using (41), we have for any . Using the argument in the proof of Lemma 27, we obtain which implies that for any . By combining (131), (134), and (128), we obtain that for any , there exists functions and , such that and for any . Thus, we obtain

To conclude, we investigate the dilation property.

Lemma 40. Let , , and . Define for and . Then, for any .

Proof. Let and satisfying . Considering that with for any , we have Thus, we obtain .

Now, all preparations to prove have been provided. From Lemma 40, we have if and only if for any . We fix an arbitrary . Then, from (136) and Lemma 40, we obtain

Note that for any . By replacing for any , there exist meaurable functions such that

Because 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 operator , we say that is a Calderón–Zygmund operator if there exists a kernel , and it satisfies the following conditions: (i) for (ii) for any (iii) whenever

In fact, these operators are bounded on Lebesgue spaces () (see Stein [17]) and Morrey spaces (, ) (see F. Chiarenza and M. Frasca [18]). Using Theorem 37, we obtain the boundedness in -spaces.

Theorem 41. For , , and , any Calderón–Zygmund operator are bounded on .

Proof. For the case where , it immediately follows by the boundedness of in Lebesgue spaces and Lemma 38. Furthermore, for the case where , 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 in and , we see for any , , and . Thus, we obtain for any , , and .

Data Availability

Data is not applicable in this manuscript. I introduce some previous researches I referred. A. Amenta, Tent spaces over metric measure spaces under doubling and related assumptions, Operator Theory in Harmonic and Non-commutative Analysis Vol. 240 (2014), 1-29. Y. Huang, Weighted tent spaces with Whitney averages: factorizations, interpolation, and duality, Math. Zeitschrift\textbf {24} (2016), 913-933. R. R. Coifman, Y. Meyer, E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. Vol. 62 No. 2 (1985), 304-335.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This study was supported by the Japan Society of Promotion of Science (202020606).