CALDERÓN COMPLEX INTERPOLATION OF MORREY SPACES

In this note we will discuss some results related to complex interpolation of Morrey spaces. We first recall the Riesz-Thorin interpolation theorem in Section 1. After that, we discuss a partial generalization of this theorem in Morrey spaces proved in [19]. We also discuss non-interpolation property of Morrey spaces given in [3, 17]. In Section 3, we recall the definition of Calderón’s complex interpolation method and the description of complex interpolation of Lebesgue spaces. In Section 4, we discuss the description of complex interpolation of Morrey spaces given in [6, 10, 14, 15]. Finally, we discuss the description of complex interpolation of subspaces of Morrey spaces in the last section. This note is a summary of the current research about interpolation of Morrey spaces, generalized Morrey spaces, and their subspaces in [6, 9, 10, 11, 12, 14, 15].


The Riesz-Thorin interpolation theorem
We first recall the definition of Lebesgue spaces. Let 1 ≤ p ≤ ∞. The Lebesgue space L p = L p (R n ) is defined to be the set of all measurable function f on R n such that the norm where a j ∈ C and {A j } k j=1 is a collection of disjoint subsets of R n with finite measure. In this case, Note that the L p space is a Banach space with the norm defined in (1) Moreover, we also have the log-convexity property of L p -norm as follows.
Note that, Lemma 1.1 can be viewed as the inclusion L p0 ∩ L p1 ⊆ L p . A complement to this result is the following lemma. Here, L p0 + L p1 is defined to be the set of all functions f for which f = f 0 + f 1 for some f 0 ∈ L p0 and f 1 ∈ L p1 .
As a preparation for proving the Riesz-Thorin interpolation theorem, we prove Hadamard's three lines lemma. Lemma 1.3. Let S := {z ∈ C : 0 < Re(z) < 1} and S be its closure. Let F be any continuous function on S such that F is holomorphic in S and F is bounded on S. Then, for every θ ∈ (0, 1) and s ∈ R, we have Proof. Let M 0 := sup t∈R |F (it)|, M 1 := sup t∈R |F (1+it)|, and M := sup z∈S |F (z)|. Define Theorem 1.4. Let 0 ≤ θ ≤ 1 and 1 ≤ p 0 , p 1 , q 0 , q 1 ≤ ∞. Suppose that T is a bounded linear operator from L p0 to L q0 and L p1 to L q1 . Then T is bounded from L p to L q , where p and q are defined by Moreover, Proof. The proof follows the idea in [7]. We only handle the case where p 0 and p 1 are finite. Let M 0 := T L p 0 →L q 0 and M 1 := T L p 1 →L q 1 . Let f be a simple function and write where a j > 0, α j ∈ R, and {A j } k j=1 is a collection of pairwise disjoint subsets of R n with |A j | < ∞. Note that Now, let g be fixed and write where b > 0, β ∈ R, and {B } m =1 is a collection of pairwise disjoint subsets of R n with finite measure. Then, by linearity of T , we have Let S := {z ∈ C : 0 < Re(z) < 1} and S be its closure. For every z ∈ S, define where q 0 := q0 q0−1 , q 1 := q1 q1−1 , and q := q q−1 . Since a j and b are positive, we see that F is continuous on S and F is holomorphic in S. Morever, by (7) and (9), we have By Hölder's inequality and the boundedness of T from L p0 to L q0 , for every z ∈ S, we have Therefore, sup z∈S |F (z)| < ∞. Note that F (z) can be rewritten as By Hölder's inequality and the boundedness of T from L p0 to L q0 , for every t ∈ R, we have Since A j 's are pairwise disjoint, we have Combining these calculations with (11), we obtain Similarly, By the three-lines lemma and (12)-(13), we have (8) and (10), we have

Combining this inequality with
for every simple function f . Finally, (14) can be extended for all f ∈ L p by using the density of simple functions in L p .

Interpolation of linear operators in Morrey spaces
Remark: Stampacchia proved the following extension of the Riesz-Thorin interpolation theorem.
Define p, r, and s by If T is a bounded linear operator from L p0 to M r0 s0 and from L p1 to M r1 s1 , then T is bounded from L p to M r s .
Unfortunately, if the domain of the operator T is Morrey spaces, there are some counterexamples given by A. Ruiz and L. Vega [17] for the case n > 1 and by O. Blasco et al. in [3] for the case n = 1. Let us recall the result in [3].
Proof. According to the definition of q, we know that q 1 < q < q 0 . Hence, we may choose Let N 0 ∈ N be such that where j = 0, 1, . . . , N − 1 and set E N := ∪ N −1 j=0 I N j . Observe that the choice of β allows {E N } ∞ N =1 to be disjoint. Note that r 0 < r 1 , so r 0 < r < r 1 . Therefore, we may choose With this choice of γ, we construct an operator T by the formula for every measurable function f . By the Hölder inequality, for every f ∈ L q0 we have It follows from (17) that Consequently, for some C 1 > 0 and for every f ∈ M q0 q1 . Since {E N } ∞ N =N0 is a collection of disjoint sets and q 1 = r 1 , we get Combining (19) and for each j = 0, 1, . . . , N − 1, we get According to (17), we have This implies (18). The proof of the unboundedness of T from M q0 q to L r goes as follows. Define Note that, for every N ∈ N, we have It follows from (15) that q(β+1) q0 − β < 0. This implies On the other hand, we claim Indeed, (20) follows from In view of Theorem 2.3, the Riesz-Thorin theorem can not be naturally generalized to Morrey spaces. However, by adding some mild assumptions, there are recent researches about complex interpolation interpolation of Morrey spaces and their generalization (see [6,9,10,11,12,13,14,15]).

Calderón's complex interpolation method
In this section we recall the complex interpolation method introduced by Calderón in [4]. We follow the terminology and presentation in [1,4,12]. In Subsections 3.1 and 3.2, we recall the definition of Calderón's first and second complex interpolation method. For the proof of our results in the next section, we shall discuss the Calderón product of Banach spaces in Section 3.3.
3.1. The first complex interpolation method. A pair (X 0 , X 1 ) is said to be a compatible couple of Banach spaces if there exists a Hausdorff topological vector space Z such that X 0 and X 1 are subspaces of Z and that the embedding of X 0 and X 1 into Z is continous. From now on, let S := {z ∈ C : 0 ≤ Re(z) ≤ 1} and S be its interior.
The norm on F(X 0 , X 1 ) is defined by Definition 3.2 (Calderón's first complex interpolation spaces). Let θ ∈ (0, 1) and (X 0 , X 1 ) be a compatible couple of Banach spaces. The complex interpolation space The fact that [X 0 , X 1 ] θ is a Banach space can be seen in [4] and [1, Theorem 4.1.2]. When X 0 and X 1 are Lebesgue spaces, Calderón gave the following description of [X 0 , X 1 ] θ .
Note that the Riesz-Thorin complex interpolation theorem can be seen as a corollary of Theorem 3.3 and the following Calderón's result.
We also invoke the following useful lemma.
3.2. The second complex interpolation method. First let us recall the definition of Banach space-valued Lipschitz continuous functions. Let X be a Banach space. Denote by Lip(R, X) the set of all functions f : R → X such that be a compatible couple of Banach spaces. Denote by G(X 0 , X 1 ) the set of all continuous functions G :S → X 0 + X 1 such that: are Lipschitz continuous on R for j = 0, 1.
The space G(X 0 , X 1 ) is equipped with the norm The relation between the second complex interpolation and the interpolation of linear operators is given as follows.
Proof. Let f ∈ [X 0 , X 1 ] θ . Then f = G (θ) for some G ∈ G(X 0 , X 1 ). By using the following inequalities The relation between the first and second complex interpolation functors is given in the following lemma: Proof. We give a simplified proof of [10, Lemma 2.4]. The proof is adapted from [11]. The continuity and holomorphicity of H k is a consequence of the corresponding property of G. Let j ∈ {0, 1} be fixed. Since t ∈ R → G(j + it) ∈ X j is Lipschitzcontinuous, we see that t ∈ R → H k (j + it) ∈ X j is bounded and continuous on R.
We shall also use the following useful connection between the first and second complex interpolation, obtained by Bergh [2].
3.3. Calderón product. In order to describe the first complex interpolation spaces, sometimes it is easier to calculate the Calderón product of Banach lattices and applying the result of Sestakov in [18]. The definition of the Calderón product and Sestakov's lemma are given as follows.
Definition 3.13. Let θ ∈ (0, 1) and (X 0 , X 1 ) be a compatible couple of Banach spaces of measurable functions in R n . The Calderón product X 0 1−θ X 1 θ of X 0 and X 1 is defined by x ∈ R n }. Theorem 3.14. Let θ ∈ (0, 1) and (X 0 , X 1 ) be a compatible couple of Banach spaces of measurable functions in R n . Then Proof. This result was due to Calderón [4]. For the convenience of the reader, we give the detailed proof. We first prove the triangle inequality in for any measurable function ϕ : R n → (0, ∞). Since we conclude that This Then as before, Since X 0 and X 1 are Banach spaces, we see that converge in X 0 and X 1 , respectively. Consequently, ∞ j=1 f j (x) converges absolutely for almost all x ∈ R n and belongs to X 0 which also yields that By virtue of the Hölder inequality and factorization, for 1 ≤ p 0 , p 1 ≤ ∞ where p is defined by 1 p := 1−θ p0 + θ p1 . We now recall the following result by Sestakov.
Lemma 3.15. [18] Let (X 0 , X 1 ) be a compatible couple of Banach spaces of measurable functions in R n . Then for every θ ∈ (0, 1), we have

The description of complex interpolation of Morrey spaces
In this section, we will discuss the first and second complex interpolation of Morrey spaces. The interpolation by using the first method can be found in [6,9,10,15]. Meanwhile, the result on the second complex interpolation is given by Lemarié-Rieusset [14]. The presentation in this section and Section 5 follows [12].
4.1. The first complex interpolation of Morrey spaces. The first result about the description of the first complex interpolation of Morrey spaces was due to Cobos et al. [6].
where p and q are defined by (26).

Theorem 4.4. [10]
Keep the same assumption as in Theorem 4.2 and assume also that q 0 = q 1 . Then we have Note that Theorem 4.4 is an improvement of Theorems 4.2, in the sense that, [M p0 q0 , M p1 q1 ] θ is now written in terms of the parameters p and q only and this description is more explicit than the right-hand side of (28). In order to prove Theorem 4.4, we need two lemmas. The first one is the fact that the set in the right-hand side of (36) is closed. The second lemma tells us that this set contains M p0 q0 ∩ M p1 q1 . Lemma 4.5. Let 1 ≤ p ≤ q < ∞. Then the set
Consequently, for every f ∈ M p0 q0 ∩ M p1 q1 , we have We are now ready to prove Theorem 4.4.
Proof of Theorem 4.4. By virtue of Theorem 4.2 and Lemmas 4.5 and 4.6, we have Conversely, let f ∈ A. For every N ∈ N, define f N := χ { 1 N ≤|f |≤N } f . As in the proof of Lemma 4.6, we may assume that q 0 < q 1 . Then q 0 < q < q 1 . This implies Observe that the function f (x) := |x| −n/p does not belong to the set in the right-hand side of (36), but this function is in M p q . From this observation, one may inquire whether we can interpolate Morrey spaces and that the output is also Morrey spaces. The affirmative answer was given by Lemarié-Rieusset [14]. He proved the following result about the second complex interpolation of Morrey spaces.
It is written in the book [1, p. 90] that the first complex interpolation space is the main interest in this book and the second complex interpolation method is considered as a technical tool. Therefore, Theorem 4.7 can be seen as an example of the importance of the second complex interpolation method. In order to prove Theorem 4.7, we prove the following lemmas about the construction of the second complex interpolation functor. and respectively. Define F 0 , F 1 , G 0 , G 1 : S → L 0 by: and Then, for any z ∈ S, we have For any z ∈ C with ε < Re(z) < 1 − ε and w ∈ C with |w| 1, we have where the constant C ε depending only on ε ∈ (0, 1/2).
By the triangle inequality, we have Writing out the definitions in full, we obtain Since q 0 > q 1 , we have By a similar argument, we also have as desired. Proof. It follows from (45) that G(z) ∈ M p0 q0 + M p1 q1 and Now let z 1 , z 2 ∈ S. Then, by inequality (48), we get This shows the continuity of G : S → M p0 q0 + M p1 q1 . The proof of holomorphicity of G : S → M p0 q0 + M p1 q1 goes as follows. Let ε ∈ (0, 1 2 ) and define S ε := {z ∈ S : ε < Re(z) < 1 − ε}.
Note that we can not use the function F defined by (41) as the first complex interpolation functor because F does not belong to F(M p0 q0 , M p1 q1 ) when f (x) := |x| −n/p . This fact is a consequence of the following proposition. Proof. Assume that p 0 > p 1 and define Q : Using (51) and letting R := exp((Qt) −1 ), we get where we use sin Qt log |x| 2 > sin 1 2 for every R < |x| < 2R. Thus, (52) implies Now we arrive at our main result in this section.
Observe that M p q coincides with C ∞ c M p q . These subspaces can be unified by introducing the following definition.
Definition 5.2. Assume that a linear subspace of measurable functions U satisfies the condition: g ∈ U whenever f ∈ U and |g| ≤ |f |. Then In order to prove Theorem 5.4, we need to prove the following lemmas: Then, for each k = 0, 1, we have This shows that f ∈ U M p q . The proof of Theorem 5.4 is given as follows: Proof of Theorem 5.4. We assume that q 1 > q 0 . By using Lemma 5.6, the inclu- Note that, for any 0 < b < c < ∞, we have a pointwise estimate: Observe that for every w ∈ S, we have Then we have for all z, z ∈ S. Thus, F N : S → U M p0 q0 + U M p1 q1 is a continuous function. Likewise we can check that F N | S : S → U M p0 q0 + U M p1 q1 is a holomorphic function. Note that, for all t ∈ R and j = 0, 1, we have pj qj . Furthermore, by using (56), we get for all t, t ∈ R. This shows that t ∈ R → F N (j + it) ∈ U M pj qj are continuous functions. In total, we have showed that F N ∈ F(U M p0 q0 , U M p1 q1 ). Note that, for M, N ∈ N with N < M , we have Then we have From now on, we shall always use the assumption of Theorem 5.7. To prove Theorem 5.7, we shall invoke and prove several lemmas.
Lemma 5.8. [10] Keep the assumption in Theorem 5.7. Then Proof. Without loss of generality, assume that q 0 > q 1 . Let f ∈ U M p q . Since Decompose Lemma 5.9.
Theorem 5.7. In view of Lemma 5.8, we only need to show that q0 , U M p1 q1 ] θ . Then there exists G ∈ G(U M p0 q0 , U M p1 q1 ) such that G (θ) = f . For z ∈ S and k ∈ N, define H k (z) by (22). By virtue of Lemmas 5.6 and 5.9, we have H k (θ) ∈ U M p q . Since H k (θ) converges to G (θ) = f in M p0 q0 +M p1 q1 , we see that f ∈ U M p q .
By substituting U := L ∞ c , L 0 c , L ∞ , we have the following result. Corollary 5.11. Keep the same assumption as in the previous theorems. Then