Calderón’s First and Second Complex Interpolations of Closed Subspaces of Morrey Spaces

Abstract

In this paper, we are going to describe the first and second complex interpolations of closed subspaces of Morrey spaces, based on our previous results in [11]. Our results will be general enough because we are going to deal with abstract linear subspaces satisfying the lattice condition only. We also consider the closure in Morrey spaces on bounded domains of the set of smooth functions with compact support. Here, we do not require the smoothness condition on domains.

1 Introduction

For $1 \le q\le p<\infty$, the Morrey space ${\mathcal M}^p_q={\mathcal M}^p_q({\mathbb R}^n)$ is defined to be the set of all q-locally integrable functions f on ${\mathbb R}^n$ such that

$$\begin{aligned} \Vert f\Vert _{{\mathcal M}^p_q}:=\sup _{x\in {\mathbb R}^n,r>0} |B(x,r)|^{\frac{1}{p}-\frac{1}{q}} \left( \int _{B(x,r)} |f(y)|^q \ dy\right) ^{\frac{1}{q}} <\infty . \end{aligned}$$

(1.1)

Here, B(x, r) denotes the ball centered at $x\in {\mathbb R}^n$ with radius r. The interpolations of Morrey spaces date back to the papers around 1960s. Campanato and Murthy [5], Spanne [31], and Peetre [24] obtained some results on the boundedness of operators on Morrey spaces and the interpolation spaces.

Based on the definition of the complex interpolation functors $(X_0,X_1) \mapsto [X_0,X_1]_\theta$ and $(X_0,X_1) \mapsto [X_0,X_1]^\theta$, introduced by Calderón in [4], whose definition we recall in Sect. 2, the following results are known:

Theorem 1.1

[14, 16] Suppose that $\theta \in (0,1)$, $1\le q_0 \le p_0<\infty$, $1\le q_1 \le p_1<\infty$, and $\frac{p_0}{q_0}=\frac{p_1}{q_1}$. Assume $q_0 \ne q_1$. Define

$$\begin{aligned} \frac{1}{p}:=\frac{1-\theta }{p_0}+\frac{\theta }{p_1} \quad \mathrm{and} \quad \frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}. \end{aligned}$$

Then

• (i) (Lu et al. [16]) $\left[ {\mathcal M}^{p_0}_{q_0}, {\mathcal M}^{p_1}_{q_1}\right] _\theta = \overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}$

• (ii) (Lemarié-Rieusset [14]) $\left[ {\mathcal M}^{p_0}_{q_0}, {\mathcal M}^{p_1}_{q_1}\right] ^\theta = {\mathcal M}^p_q.$

One of our main results in this note refines Theorem 1.1 (i):

Theorem 1.2

Keep the same assumption as in Theorem 1.1. Then we have

$$\begin{aligned} \left[ {\mathcal M}^{p_0}_{q_0}, {\mathcal M}^{p_1}_{q_1}\right] _\theta&= \left\{ f\in \overline{{\mathcal M}^p_q}:\lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^p_q}=0\right\} . \end{aligned}$$

(1.2)

Note that the right-hand side is independent of $p_0,p_1,q_0$, and $q_1$. We shall prove (1.2) in a more general framework.

The aim of this paper is to investigate the effect of the first and second complex interpolation functors through closed subspaces.

We use the following notation for closed subspaces of the Morrey space ${\mathcal M}^p_q$:

Definition 1.3

Assume that a linear subspace $U\subset L^0$ enjoys the lattice property: $g \in U$ whenever $f \in U$ and $|g|\le |f|$. For $1\le q<p<\infty$, define

$$\begin{aligned} U{\mathcal M}^p_q&:=\overline{U\cap {\mathcal M}^p_q}^{{\mathcal M}^p_q} \end{aligned}$$

(1.3)

$$\begin{aligned} U\bowtie {\mathcal M}^p_q&:=\left\{ f\in {\mathcal M}^p_q:\chi _{\{a\le |f|\le b\}}f\in U{\mathcal M}^p_q {\ \mathrm for \ all \ } 0<a<b<\infty \right\} . \end{aligned}$$

(1.4)

To investigate further the role of the closed subspace U in the second complex interpolation, we prove the following theorem:

Theorem 1.4

Suppose that $\theta \in (0,1)$, $1\le q_0 \le p_0<\infty$, $1\le q_1 \le p_1<\infty$, and $\frac{p_0}{q_0}=\frac{p_1}{q_1}$. Assume $q_0 \ne q_1$. Define

$$\begin{aligned} \frac{1}{p}:=\frac{1-\theta }{p_0}+\frac{\theta }{p_1} \quad \mathrm{and} \quad \frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}. \end{aligned}$$

Then we have

$$\begin{aligned} \left[ U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}\right] _\theta&=U {\mathcal M}^p_q \cap \left[ {\mathcal M}^{p_0}_{q_0},{\mathcal M}^{p_1}_{q_1}\right] _\theta \nonumber \\&= \left\{ f\in U{\mathcal M}^p_q\cap \overline{{\mathcal M}^{p}_q}: \lim _{a\rightarrow 0^+}\Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^p_q}=0\right\} . \end{aligned}$$

(1.5)

and

$$\begin{aligned} \left[ U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}\right] ^\theta =U \bowtie {\mathcal M}^p_q. \end{aligned}$$

(1.6)

We will prove (1.5) in a more general framework.

More and more attention has been paid for the closed subspaces of the Morrey space ${\mathcal M}^p_q$ with $1 \le q<p<\infty$ [29, 40]. Some of them are realized as $U{\mathcal M}^p_q$ for some U in Definition 1.3.

Definition 1.5

Let $1 \le q \le p<\infty$.

• 1. [11, p. 5] The space $\widetilde{{\mathcal M}^p_q}$ is defined to be the closure of $L^\infty _\mathrm{c}$ in ${\mathcal M}^p_q$.

• 2. [29, Definition 4.5] A function f in ${\mathcal M}^p_q$ is said to have “absolutely continuous norm” in ${\mathcal M}^p_q$ if $\Vert f\chi _{E_k}\Vert _{{\mathcal M}^p_q}\rightarrow 0$ for every sequence $\{E_k\}_{k=1}^{\infty }$ satisfying $\chi _{E_k}(x)\rightarrow 0$ a.e. The set of all functions in ${\mathcal M}^p_q$ of absolutely continuous norm is denoted by $\widehat{\mathcal M}^p_q$.

• 3. [40, Definition 2.23] $\overset{\diamond }{\mathcal M}{}^p_q$ denotes the closure with respect to ${\mathcal M}^p_q$ of the set of all smooth functions f such that $\partial ^\alpha f \in {\mathcal M}^p_q$ for all multi-indexes $\alpha$.

• 4. [40, Definition 2.23] $\overset{\circ }{{\mathcal M}}{}^p_q$ denotes the closure with respect to ${\mathcal M}^p_q$ of $C^\infty _\mathrm{c}$, or equivalently, the closure of $\mathcal {S}$ in ${\mathcal M}^p_q$.

• 5. [40, Sect. 2] $\overset{*}{\mathcal M}{}^p_q$ denotes the closure with respect to ${\mathcal M}^p_q$ of the set of all compactly supported functions $L^0$ in ${\mathcal M}^p_q$.

• 6. [6, p. 1] $\overline{{\mathcal M}^p_q}$ denotes the closure with respect to ${\mathcal M}^p_q$ of the set of all essentially bounded functions in ${\mathcal M}^p_q$.

From the definition, it is easy to see that $\widetilde{{\mathcal M}}{}^p_q=\overset{\circ }{{\mathcal M}}{}^p_q$ and that $\widetilde{{\mathcal M}}{}^p_q$, $\overset{*}{\mathcal M}{}^p_q$, and $\overline{{\mathcal M}^p_q}$ are realized as $U{\mathcal M}^p_q$ for some linear space U enjoying the lattice property; $U=L^\infty _\mathrm{c}$, $L^0$, and $L^\infty$ do the job, respectively.

The closed subspaces $\widehat{\mathcal M}^p_q$ and $\overset{\diamond }{\mathcal M}{}^p_q$ arise naturally. We refer to [40, Theorem 2.29] for $\overset{\diamond }{\mathcal M}{}^p_q$ and to [29, Theorem 4.3] for $\widehat{\mathcal M}^p_q$.

Let $\theta \in (0,1)$, $1< q \le p < \infty$, $1< q_0 \le p_0 < \infty$, and $1< q_1 \le p_1 < \infty$ satisfy $p_0< p < p_1$ and

$$\begin{aligned} \frac{1}{p} = \frac{1-\theta }{p_0} +\frac{\theta }{p_1}, \quad \frac{1}{q} = \frac{1-\theta }{q_0} +\frac{\theta }{q_1}, \quad \frac{q_0}{p_0} = \frac{q_1}{p_1}. \end{aligned}$$

Then we have the following relations:

$$\begin{aligned} { \overset{\circ }{{\mathcal M}}{}^p_q \subsetneq \widetilde{\mathcal M}^p_q = \widehat{\mathcal M}^p_q \subsetneq \overset{*}{\mathcal M}{}^p_q },\end{aligned}$$

(1.7)

$$\begin{aligned} \overset{\circ }{{\mathcal M}}{}^p_q \subsetneq \overset{\diamond }{\mathcal M}{}^p_q,\end{aligned}$$

(1.8)

$$\begin{aligned} \widetilde{\mathcal M}^p_q \subsetneq \overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q},\end{aligned}$$

(1.9)

$$\begin{aligned} \widetilde{\mathcal M}^p_q \subsetneq [\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta , \end{aligned}$$

(1.10)

$$\begin{aligned}{}[\overset{\circ }{{\mathcal M}}{}^{p_0}_{q_0}, \overset{\circ }{{\mathcal M}}{}^{p_1}_{q_1}]_\theta = \overset{\circ }{{\mathcal M}}{}^p_q. \end{aligned}$$

(1.11)

according to [40, Lemma 2.33], [40, Remark 2.36], [40, Lemma 2.37, Corollary 2.38], [40, Remark 2.36], [29, Theorem 4.3] and [37, Corollary 1.4], respectively. We have no further inclusion; see [11, Sect. 9].

In this paper we also consider the complex interpolations of generalized Morrey spaces, introduced by Nakai [19]. The thrust is that generalized Morrey spaces can be contained in the space $L^\infty$. In fact, in many results the indicator function of the level set of f comes into play as is the case with many results presented in this paper. Recall that for $1\le q<\infty$ and a function $\varphi :(0,\infty )\rightarrow (0,\infty )$, the generalized Morrey space ${\mathcal M}^\varphi _q$ is defined to be the set of all functions $f\in L^q_\mathrm{loc}$ such that

$$\begin{aligned} \Vert f\Vert _{{\mathcal M}^\varphi _q}:= \sup _{x\in {\mathbb R}^n, r>0}\varphi (r) \left( \frac{1}{|B(x,r)|} \int _{B(x,r)} |f(y)|^q \ dy\right) ^{\frac{1}{q}} \end{aligned}$$

is finite. We assume that $\varphi$ belongs to ${\mathcal G}_q$, that is, $\varphi$ is increasing but that $t\mapsto t^{-n/q}\varphi (t)$ is decreasing; see the work [20, p. 446] which justifies this assumption. Note that, for $\varphi (t):=t^{n/p}$, where $1\le q\le p<\infty$, we have ${\mathcal M}^\varphi _q={\mathcal M}^p_q$. See Sect. 2.3 for more examples of $\varphi$. Our previous results on the complex interpolation of generalized Morrey spaces are given as follows:

Theorem 1.6

[11, Theorem 2] Let $\theta \in (0,1)$, $1\le q_0<\infty$, $1\le q_1<\infty$, $\varphi _0\in {\mathcal G}_{q_0}$, $\varphi _1\in {\mathcal G}_{q_1}$, and $\varphi _0^{q_0}=\varphi _1^{q_1}$. Define $\varphi :=\varphi _0^{1-\theta }\varphi _1^{\theta }$ and $\frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}$. Then we have

$$\begin{aligned} \left[ {\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}\right] _\theta =\overline{{\mathcal M}^{\varphi _0}_{q_0} \cap {\mathcal M}^{\varphi _1}_{q_1}}^{{\mathcal M}^\varphi _q} \ \mathrm{and} \ \left[ {\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}\right] ^\theta ={{\mathcal M}^\varphi _q}. \end{aligned}$$

The following is our interpolation result which includes (1.2).

Theorem 1.7

Let $\theta \in (0,1)$, $1\le q_0<\infty$, $1\le q_1<\infty$, $\varphi _0\in {\mathcal G}_{q_0}$, $\varphi _1\in {\mathcal G}_{q_1}$, and $\varphi _0^{q_0}=\varphi _1^{q_1}$. Define $\varphi :=\varphi _0^{1-\theta }\varphi _1^{\theta }$ and $\frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}$. Then we have

$$\begin{aligned} \left[ {\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}\right] _\theta = \left\{ f\in \overline{{\mathcal M}^\varphi _q}:\lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0\right\} . \end{aligned}$$

(1.12)

Note again that the right-hand side is independent of $\varphi _0, \varphi _1, q_0$, and $q_1$.

We remark that (1.12) refines the general result asserting that $[{\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}]_\theta$ is the closure of ${\mathcal M}^{\varphi _0}_{q_0} \cap {\mathcal M}^{\varphi _1}_{q_1}$ in ${\mathcal M}^\varphi _q$.

We use the following notation for closed subspaces of generalized Morrey spaces:

Definition 1.8

Let U be the same as in Definition 1.3, $1\le q<\infty$, and $\varphi \in {\mathcal G}_q$. Define $U{\mathcal M}^\varphi _q:=\overline{U\cap {\mathcal M}^\varphi _q}^{{\mathcal M}^\varphi _q}$ and

$$\begin{aligned} U\bowtie {\mathcal M}^\varphi _q:=\left\{ f\in {\mathcal M}^\varphi _q:\chi _{\{a\le |f|\le b\}}f\in U{\mathcal M}^\varphi _q {\ \mathrm for \ all \ } 0<a<b<\infty \right\} . \end{aligned}$$

The complex interpolation result for $U{\mathcal M}^\varphi _q$ is given in the following theorem:

Theorem 1.9

Suppose that $\theta \in (0,1)$, $1\le q_0<\infty$, $1\le q_1<\infty$, and $\varphi _0^{q_0}=\varphi _1^{q_1}$. Define

$$\begin{aligned} \varphi :=\varphi _0^{1-\theta }\varphi _1^{\theta } \quad \mathrm{and} \quad \frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}. \end{aligned}$$

Then we have

$$\begin{aligned} \left[ U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}\right] _\theta&= U{\mathcal M}^\varphi _q \cap \left[ {\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}\right] _\theta \end{aligned}$$

(1.13)

$$\begin{aligned}&= \left\{ f\in U{\mathcal M}^\varphi _q\cap \overline{{\mathcal M}^{\varphi }_q}: \lim _{a\rightarrow 0^+}\Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0\right\} ,\end{aligned}$$

(1.14)

$$\begin{aligned} \left[ U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}\right] ^\theta&= U \bowtie {\mathcal M}^\varphi _q. \end{aligned}$$

(1.15)

As a special case for these examples, we have the following results:

Corollary 1.10

[11, Theorems 5.2 and 5.12] Suppose that $\theta \in (0,1)$, $1\le q_0<\infty$, $1\le q_1<\infty$, and $\varphi _0^{q_0}=\varphi _1^{q_1}$. Define $\varphi :=\varphi _0^{1-\theta }\varphi _1^{\theta }$ and $\frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}$.

• 1. The description of the first interpolation functor of these closed subspaces is as follows:

  $$\begin{aligned} \left[ \widetilde{{\mathcal M}^{\varphi _0}_{q_0}}, \widetilde{{\mathcal M}^{\varphi _1}_{q_1}} \right] _{\theta }= \widetilde{{\mathcal M}^\varphi _q}, \quad \left[ \overset{*}{{\mathcal M}^{\varphi _0}_{q_0}}, \overset{*}{{\mathcal M}^{\varphi _1}_{q_1}} \right] _{\theta }= \widetilde{{\mathcal M}^\varphi _q}, \quad \left[ \overline{{\mathcal M}^{\varphi _0}_{q_0}}, \overline{{\mathcal M}^{\varphi _1}_{q_1}} \right] _{\theta }= \overline{{\mathcal M}^\varphi _q}. \end{aligned}$$

• 2. The description of the second interpolation functor of these closed subspaces is as follows:

  $$\begin{aligned}&\left[ \widetilde{{\mathcal M}^{\varphi _0}_{q_0}}, \widetilde{{\mathcal M}^{\varphi _1}_{q_1}}\right] ^\theta = \left[ \overset{*}{{\mathcal M}^{\varphi _0}_{q_0}}, \overset{*}{{\mathcal M}^{\varphi _1}_{q_1}} \right] ^{\theta }= \bigcap _{0<a<b<\infty } \left\{ f\in {\mathcal M}^\varphi _q:\chi _{\{a\le |f|\le b\}}f\in \widetilde{{\mathcal M}^\varphi _q} \right\} ,\end{aligned}$$

  (1.16)
  $$\begin{aligned}&\left[ \overline{{\mathcal M}^{\varphi _0}_{q_0}}, \overline{{\mathcal M}^{\varphi _1}_{q_1}} \right] ^{\theta } = {\mathcal M}^\varphi _q. \end{aligned}$$

  (1.17)

To investigate the effect of the finiteness of the ambient spaces, we consider Morrey spaces on bounded connected open set $\Omega \subseteq {\mathbb R}^n$. For $1\le q<\infty$ and $\varphi \in {\mathcal G}_q$, the space ${\mathcal M}^\varphi _q(\Omega )$ is defined to be the set of all functions $f\in L^q(\Omega )$ such that

$$\begin{aligned} \Vert f\Vert _{{\mathcal M}^\varphi _q(\Omega )} :=\sup _{x\in \Omega , 0<r<\mathrm{diam}(\Omega )} \varphi (r) \left( \frac{1}{|B(x,r)|} \int _{B(x,r)\cap \Omega } |f(y)|^q \ dy \right) ^{\frac{1}{q}}<\infty . \end{aligned}$$

Here, we do not require that $\Omega$ is smooth. Let $\overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )$ be the closure of $C^\infty _\mathrm{c}(\Omega )$ in ${\mathcal M}^\varphi _q(\Omega )$. In the special case of $\varphi := 1$, one defines $\overset{\circ }{L}{}^\infty (\Omega ):= \overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )$. Via the mollification, we shall show that $\overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )$ is the closure of $C_\mathrm{c}(\Omega )$ in ${\mathcal M}^\varphi _q(\Omega )$. We remark that, for $U:=L^{0}(\Omega )$, we have $U{\mathcal M}^\varphi _q={\mathcal M}^\varphi _q(\Omega )$.

The interpolation result for these spaces is presented in the following theorem:

Theorem 1.11

Let $\theta \in (0,1)$, $1\le q_0<\infty$, $1\le q_1<\infty$, $\varphi _0 \in {\mathcal G}_{q_0}$, and $\varphi _1 \in {\mathcal G}_{q_1}$. Assume $\varphi _0^{q_0}=\varphi _1^{q_1}$. Define $\frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}$ and $\varphi :=\varphi _0^{1-\theta }\varphi _1^{\theta }$. Then we have

$$\begin{aligned} \left[ \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega ), \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )\right] _{\theta } = \overset{\circ }{{\mathcal M}}{}^{\varphi }_{q}(\Omega ). \end{aligned}$$

The related result in ${\mathbb R}^n$ can be seen in [37, Corollary 1.4].

Let us explain why the interpolations of Morrey spaces are complicated unlike Lebesgue spaces. Noteworthy is the fact that the first complex interpolation functor behaves differently from Lebesgue spaces. This problem comes basically from the fact that the Morrey norm ${\mathcal M}^p_q$ involves the supremum over all balls B(a, r). Due to this fact, we have many difficulties when $1<q<p<\infty$, namely:

• 1. The Morrey space ${\mathcal M}^p_q$ is not reflexive; see [29, Example 5.2] and [36, Theorem 1.3].

• 2. The Morrey space ${\mathcal M}^p_q$ does not have $C^\infty _\mathrm{c}$ as a dense closed subspace; see [33, Proposition 2.16].

• 3. The Morrey space ${\mathcal M}^p_q$ is not separable; see [33, Proposition 2.16].

• 4. The Morrey space ${\mathcal M}^p_q$ is not included in $L^1+L^\infty$; see Sect. 6 for the proof.

The non-density of $C^\infty _\mathrm{c}$ and the failure of reflexivity and separability influence many other related function spaces such as Besov–Morrey spaces, Triebel–Lizorkin–Morrey spaces, Besov-type spaces, and Triebel–Lizorkin type spaces. These spaces are nowadays called smoothness Morrey spaces. We remark that these spaces cover Morrey spaces as a special case as is shown in [17, Proposition 4.1]. We refer to [15, Theorem 9.6] and [23, Corollary 6.2] for the counterpart of generalized Morrey spaces. We refer to [37, 38, 40] for the results of this direction. Since we do not deal with smoothness Morrey spaces in this paper, we content ourselves with listing the papers containing the definition of the function spaces [12, 17, 18, 27, 32, 34, 35] as well as the textbooks [33, 39] without stating the precise definitions. As is pointed out in [37, Remark 1.9], the second author made a careless claim in [28, Theorem 5.4] that (homogeneous) Besov–Morrey spaces are closed under taking the first complex interpolation. However, this mistake comes essentially from the misunderstanding that $[{\mathcal M}^{p_0}_{q_0},{\mathcal M}^{p_1}_{q_1}]_\theta = {\mathcal M}^p_q$.

Despite a counterexample by Blasco et al. [2, 26], the interpolation theory of Morrey spaces progressed so much. As for the real interpolation results, Burenkov and Nursultanov obtained an interpolation result in local Morrey spaces [3]. Nakai and Sobukawa generalized their results to $B_u^w$ setting [22], where $B_u^w$ denotes the weighted $B_\sigma$-space. We made a significant progress in the complex interpolation theory of Morrey spaces. Denote by $[X_0,X_1]_\theta$ the first complex interpolation; see Definition 2.1. In [8, p. 35] Cobos, Peetre and Persson pointed out that

$$\begin{aligned} \left[ {\mathcal M}^{p_0}_{q_0},{\mathcal M}^{p_1}_{q_1}\right] _\theta \subset {\mathcal M}^p_q \end{aligned}$$

as long as $1 \le q_0 \le p_0<\infty$, $1 \le q_1 \le p_1<\infty$, and $1 \le q \le p<\infty$ satisfy

$$\begin{aligned} \frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}, \quad \frac{1}{q}=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}. \end{aligned}$$

(1.18)

As is shown in [13, Theorem 3(ii)], when an interpolation functor F satisfies

$$\begin{aligned} F\left[ {\mathcal M}^{p_0}_{q_0},{\mathcal M}^{p_1}_{q_1}\right] = {\mathcal M}^p_q \end{aligned}$$

under the condition (1.18), then

$$\begin{aligned} \frac{q_0}{p_0}=\frac{q_1}{p_1} \end{aligned}$$

(1.19)

holds. Lemarié-Rieusset showed this assertion by using the counterexample, given by Ruiz and Vega in [26]. Lemarié-Rieusset also proved that we can choose the second complex interpolation functor, introduced by Calderón [4]. Meanwhile, as for the interpolation result under (1.18) and (1.19) by the first complex interpolation functor, also introduced by Calderón [4], Lu et al. [16, Theorem 1.2] obtained Theorem 1.1 (i). They also extended this result by placing themselves in the setting of a metric measure space. Their technique is to calculate the Calderón product; see [16].

We organize the remaining part of this paper as follows: Sect. 2 collects some fundamental facts on complex interpolation functors. Section 3 is dedicated to Morrey spaces and Sect. 4 generalizes what we obtained to generalized Morrey spaces. Generalized Morrey spaces can be a proper subspace of $L^\infty$. This result forces the result in [11] to be decomposed into two cases. Here we can unify them. In Sect. 5 we consider the function spaces on bounded domains.

2 Preliminaries

2.1 Complex Interpolation Functors

We recall the definition of the complex interpolation functors (see [1, 4]). We write $\overline{S}:=\{z\in {\mathbb C}: 0\le \mathrm{Re}(z)\le 1\}$ and let S be its interior.

Definition 2.1

Let $(X_0,X_1)$ be a compatible couple of Banach spaces.

• 1. The space ${\mathcal F}(X_0,X_1)$ is defined as the set of all functions $F :\overline{S} \rightarrow X_0+X_1$ such that

  • (a) F is continuous on $\bar{S}$ and $\sup \limits _{z\in \bar{S}} \Vert F(z)\Vert _{X_0+X_1}<\infty$,

  • (b) F is holomorphic on S,

  • (c) the functions $t \in {\mathbb R}\mapsto F(j+it) \in X_j$ are bounded and continuous on ${\mathbb R}$ for $j=0,1$.

  The space ${\mathcal F}(X_0,X_1)$ is equipped with the norm

  $$\begin{aligned} \Vert F\Vert _{{\mathcal F}(X_0,X_1)} := \max \left\{ \sup \limits _{t\in {\mathbb R}} \Vert F(it)\Vert _{X_0} , \ \sup \limits _{t\in {\mathbb R}} \Vert F(1+it)\Vert _{X_1} \right\} . \end{aligned}$$

• 2. Let $\theta \in (0,1)$. Define the complex interpolation space $[X_0,X_1]_{\theta }$ with respect to $(X_0,X_1)$ to be the set of all functions $f\in X_0+X_1$ such that $f=F(\theta )$ for some $F\in {\mathcal F}(X_0,X_1)$. The norm on $[X_0,X_1]_{\theta }$ is defined by

  $$\begin{aligned} \Vert f\Vert _{[X_0,X_1]_{\theta }} := \inf \left\{ \Vert F\Vert _{{\mathcal F}(X_0,X_1)} : f=F(\theta ) \mathrm {\ for \ some \ } F \in {\mathcal F}(X_0,X_1) \right\} . \end{aligned}$$

Let X be a Banach space. The space $\mathrm{Lip}({\mathbb R},X)$ is defined to be the set of all functions $F:{\mathbb R} \rightarrow X$ for which the quantity

$$\begin{aligned} \Vert F\Vert _{\mathrm{Lip}({\mathbb R},X)} := \sup _{-\infty<s<t<\infty } \frac{\Vert F(t)-F(s)\Vert _X}{|t-s|}<\infty . \end{aligned}$$

Definition 2.2

(Calderón’s second complex interpolation space) Let $(X_0,X_1)$ be a compatible couple of Banach spaces.

• 1. Define ${\mathcal G}(X_0,X_1)$ as the set of all functions $G:\bar{S} \rightarrow X_0+X_1$ such that

  • (a) G is continuous on $\bar{S}$ and $\sup \limits _{z\in \bar{S}} \left\| \frac{G(z)}{1+|z|}\right\| _{X_0+X_1}<\infty$,

  • (b) G is holomorphic on S,

  • (c) the functions

    $$\begin{aligned} t \in {\mathbb R}\mapsto G(j+it)-G(j) \in X_j \end{aligned}$$

    are Lipschitz continuous on ${\mathbb R}$ for $j=0,1$.

  The space ${\mathcal G}(X_0,X_1)$ is equipped with the norm

  $$\begin{aligned} \Vert G\Vert _{{\mathcal G}(X_0,X_1)} := \max \left\{ \Vert G(i\cdot )\Vert _{\mathrm{Lip}({\mathbb R}, X_0)}, \ \Vert G(1+i\cdot )\Vert _{\mathrm{Lip}({\mathbb R}, X_1)} \right\} . \end{aligned}$$

  (2.1)

• 2. Let $\theta \in (0,1)$. Define the complex interpolation space $[X_0,X_1]^{\theta }$ with respect to $(X_0,X_1)$ to be the set of all functions $f\in X_0+X_1$ such that $f=G'(\theta )$ for some $G\in {\mathcal G}(X_0,X_1)$. The norm on $[X_0,X_1]^{\theta }$ is defined by

  $$\begin{aligned} \Vert f\Vert _{[X_0,X_1]^{\theta }} := \inf \left\{ \Vert G\Vert _{{\mathcal G}(X_0,X_1)} : f=G'(\theta ) \mathrm {\ for \ some \ } G \in {\mathcal G}(X_0,X_1) \right\} . \end{aligned}$$

The key tool used for proving our results is the three-lines lemma for Banach space-valued function which we invoke as follows:

Lemma 2.3

[41, Corollary 2.3] Let X be a Banach space. Suppose that $F:\overline{S} \rightarrow X$ is continuous and bounded and also $F|_S:S\rightarrow X$ is holomorphic. Then we have

$$\begin{aligned} \sup _{t\in {\mathbb R}}\Vert F(\theta +it)\Vert _{X} \le \left( \sup _{t\in {\mathbb R}}\Vert F(it)\Vert _{X}\right) ^{1-\theta } \left( \sup _{t\in {\mathbb R}}\Vert F(1+it)\Vert _{X}\right) ^{\theta } \end{aligned}$$

for all $\theta \in (0,1)$.

The following lemma can be seen as a tool to relate the first and the second complex interpolations:

Lemma 2.4

Let $(X_0,X_1)$ be a compatible couple. Suppose that $G\in {\mathcal G}(X_0,X_1)$ and $\theta \in (0,1)$. For $z\in \overline{S}$ and $k\in {\mathbb N}$, define

$$\begin{aligned} H_k(z):=\frac{G(z+2^{-k}i)-G(z)}{2^{-k}i}. \end{aligned}$$

(2.2)

Then $H_k(\theta )\in [X_0,X_1]_{\theta }$.

Proof

Note that, $H_k$ inherits continuity and holomorphicity from G. By Lipschitz-continuity of $t \in {\mathbb R}\mapsto G(it)-G(0) \in X_0$ and $t \in {\mathbb R}\mapsto G(1+it)-G(1) \in X_1$, we have

$$\begin{aligned} \sup _{t\in {\mathbb R}} \Vert H_k(it)\Vert _{X_0+X_1} \le \sup _{t\in {\mathbb R}} \left\| \frac{G((2^{-k}+t)i)-G(it)}{2^{-k}i} \right\| _{X_0} \le \Vert G\Vert _{{\mathcal G}(X_0,X_1)} \end{aligned}$$

and likewise $\sup \limits _{t\in {\mathbb R}} \Vert H_k(1+it)\Vert _{X_0+X_1} \le \Vert G\Vert _{{\mathcal G}(X_0,X_1)}$. By Lemma 2.3, we have

$$\begin{aligned} \Vert H_k(z)\Vert _{X_0+X_1} \le \left( \Vert G\Vert _{{\mathcal G}(X_0, X_1)} \right) ^{1-\mathrm{Re}(z)} \left( \Vert G\Vert _{{\mathcal G}(X_0,X_1)}\right) ^{\mathrm{Re}(z)} \le \Vert G\Vert _{{\mathcal G}(X_0,X_1)}. \end{aligned}$$

(2.3)

This shows that $H_k(z)\in {\mathcal F}(X_0,X_1)$. Thus, $H_k(\theta )\in [X_0,X_1]_\theta$. $\square$

2.2 Some Elementary Facts on Closed Subspaces

Lemma 2.5

Let $1 \le q<\infty$ and $\varphi \in {\mathcal G}_q$. Define

$$\begin{aligned} A:=\left\{ f\in \overline{{\mathcal M}^\varphi _q}:\lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0\right\} . \end{aligned}$$

(2.4)

Then, A is a closed subset of ${\mathcal M}^\varphi _q$.

Proof

Let $\{f_j\}_{j=1}^\infty \subset A$ such that $f_j$ converges to f in ${\mathcal M}^\varphi _q$. Fix $j\in {\mathbb N}$. For every $a>0$, we have

$$\begin{aligned} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}&\le \Vert f-f_j\Vert _{{\mathcal M}^\varphi _q} + \Vert \chi _{\{|f|<a\}\cap \{|f_j|\ge 2a\}}f_j\Vert _{{\mathcal M}^\varphi _q} + \Vert \chi _{\{ |f_j|<2a\}}f_j\Vert _{{\mathcal M}^\varphi _q}. \end{aligned}$$

On the set $\{|f|<a\}\cap \{|f_j|\ge 2a\}$, we have

$$\begin{aligned} |f_j|\le |f_j-f|+|f|<|f_j-f|+a\le |f_j-f|+\frac{1}{2} |f_j|, \end{aligned}$$

hence $|f_j|\le 2|f-f_j|$. Consequently,

$$\begin{aligned} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q} \le 3\Vert f-f_j\Vert _{{\mathcal M}^\varphi _q} +\Vert \chi _{\{|f_j|<2a\}}f_j\Vert _{{\mathcal M}^\varphi _q}. \end{aligned}$$

Since $f_j \in A$, we have

$$\begin{aligned} \limsup _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q} \le 3\Vert f-f_j\Vert _{{\mathcal M}^\varphi _q}. \end{aligned}$$

By taking $j \rightarrow \infty$, we have $\lim \limits _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0$, hence, $f\in A$. $\square$

We prove the following lemma:

Lemma 2.6

Let $1\le q<\infty$ and $\varphi \in {\mathcal G}_q$. If $f\in \overline{{\mathcal M}^\varphi _q}$, then

$$\begin{aligned} \lim _{R\rightarrow \infty } \Vert \chi _{\{|f|>R\}}f\Vert _{{\mathcal M}^\varphi _q} =0. \end{aligned}$$

(2.5)

Proof

Here, we do not assume that $\inf \varphi =0$. For every $\varepsilon >0$, choose $g=g_\varepsilon \in L^\infty \cap {\mathcal M}^\varphi _q$ such that $\Vert f-g\Vert _{{\mathcal M}^\varphi _q}<\varepsilon$. Observe that, for each $R>0$, we have

$$\begin{aligned} |\chi _{\{|f|>R \}}f| \le |\chi _{\{|f|>R\} \cap \{ |g|\le R/2 \}}f|+|f-g|+|\chi _{\{|g| >R/2 \}}g|. \end{aligned}$$

On the set $\{|f|>R\} \cap \{ |g|\le R/2 \}$, we see that $|f|\le |f-g|+\frac{R}{2}\le |f-g|+\frac{|f|}{2}$, so $|\chi _{\{|f|>R\} \cap \{ |g|\le R/2 \}}f| \le 2|f-g|$. Consequently, for $R>2\Vert g\Vert _{L^\infty }$, we have

$$\begin{aligned} |\chi _{\{|f|>R \}}f| \le 3|f-g|. \end{aligned}$$

Hence, $\Vert \chi _{\{|f|>R \}}f\Vert _{{\mathcal M}^\varphi _q}\le 3\Vert f-g\Vert _{{\mathcal M}^\varphi _q}<3\varepsilon$. Thus, we have showed that (2.5) holds. $\square$

Under the conditions in Theorem 1.7, we have the following approximation formula:

Lemma 2.7

Maintain the same conditions as Theorem 1.7. Let $f \in {\mathcal M}^{\varphi _0}_{q_0} \cap {\mathcal M}^{\varphi _1}_{q_1}$. Then, we have $f \in {\mathcal M}^\varphi _q$ and $f=\lim \limits _{a \rightarrow 0^+}\chi _{\{a \le |f| \le a^{-1}\}}f$ in ${\mathcal M}^\varphi _q$.

Proof

Without any loss of generality, we may assume $q_1<q_0$. The proof is immediate from $|f-\chi _{\{a \le |f| \le a^{-1}\}}f| \le a^{\frac{q_0}{q}-1}|f|^{\frac{q_0}{q}} + a^{1-\frac{q_1}{q}}|f|^{\frac{q_1}{q}}.$ $\square$

2.3 Example of $\varphi$ in ${\mathcal M}^\varphi _q$

As we mentioned in the introduction, the case when $\varphi (t)=t^{n/p}$ boils down to ${\mathcal M}^p _q$. However, considering generalized Morrey spaces is not a mere quest to generalization for its own sake. This applies to the point of applications of generalized Morrey spaces and to the context of interpolations. First, we give an example showing that generalized Morrey spaces are useful.

Example 2.8

In this example, we claim that generalized Morrey spaces are useful. In [30, Theorem 5.1] the following estimate is shown:

$$\begin{aligned} \Vert (1-\Delta )^{-\frac{n}{p}}f\Vert _{{\mathcal M}^{\varphi }_1} \le C \Vert f\Vert _{{\mathcal M}^p_q}. \end{aligned}$$

when $1<q \le p<\infty$ and $\varphi (t)=(1+t)^{n/p}/\log (3+t)$ for $t>0$. We know that $\log$ here can not be removed. See [9, Sect. 5] and [23, Proposition 7.3] for more generalizations.

Generalized Morrey spaces seem to reflect the interpolation property as the following two examples show.

Example 2.9

Let $1 \le q<\infty$ and $\varphi _0,\varphi _1 \in {\mathcal G}_q$. Define $\varphi =\varphi _0+\varphi _1$. Then ${\mathcal M}^\varphi _q={\mathcal M}^{\varphi _0}_q \cap {\mathcal M}^{\varphi _1}_q$ with norm equivalence.

Example 2.10

As is seen from Sect. 1, it seems that the first and second complex interpolation functors seem to control the modulus of the function. From this point, it is important to pay attention to the following proposition for $1 \le q<\infty$ and $\varphi \in {\mathcal G}_q$. $L^\infty \subset {\mathcal M}^\varphi _q$ holds if and only if $\inf \varphi >0$. See [21, Proposition 3.3].

3 The Interpolations of Closed Subspaces of Morrey Space

First, we prove the following lemma:

Lemma 3.1

Suppose that $\theta \in (0,1)$, $1\le q_0 \le p_0<\infty$, $1\le q_1 \le p_1<\infty$, and $\frac{p_0}{q_0}=\frac{p_1}{q_1}$. Assume $q_0 \ne q_1$. Define

$$\begin{aligned} \frac{1}{p}:=\frac{1-\theta }{p_0}+\frac{\theta }{p_1} \quad \mathrm{and} \quad \frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}. \end{aligned}$$

Let E be a measurable set such that $\chi _E \in U{\mathcal M}^p_q$. Then $\chi _{E} \in U{\mathcal M}^{p_0}_{q_0}\cap U{\mathcal M}^{p_1}_{q_1}$.

Proof

Let $\chi _E \in U{\mathcal M}^p_q$ and $\varepsilon >0$. Choose $g_\varepsilon \in U\cap {\mathcal M}^p_q$ such that

$$\begin{aligned} \Vert \chi _E - g_\varepsilon \Vert _{{\mathcal M}^p_q} <\varepsilon . \end{aligned}$$

Define $h_\varepsilon :=\chi _{\{g_\varepsilon \ne 0\} \cap E}$. Then

$$\begin{aligned} |\chi _E-h_\varepsilon |=\chi _E-h_\varepsilon \le |\chi _E-g_\varepsilon |. \end{aligned}$$

Consequently, for $j=0,1$, we have

$$\begin{aligned} \Vert \chi _{E}-h_\varepsilon \Vert _{{\mathcal M}^{p_j}_{q_j}} = \Vert \chi _{E}-h_\varepsilon \Vert _{{\mathcal M}^{p}_{q}}^{q/{q_j}}<\varepsilon ^{q/{q_j}}. \end{aligned}$$

This shows that $\chi _E \in U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1}$. $\square$

Lemma 3.2

Keep the assumption in Lemma 3.1. Then we have

$$\begin{aligned} U\bowtie {\mathcal M}^p_q \subseteq \left[ U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}\right] ^\theta . \end{aligned}$$

Proof

Without loss of generality, assume that $q_0>q_1$. Let $f\in U\bowtie {\mathcal M}^p_q$. Since $\chi _{\{a\le |f|\le b\}} \le \frac{1}{a} \chi _{\{a\le |f|\le b\}}|f|$, we have $\chi _{\{a\le |f|\le b\}} \in U{\mathcal M}^p_q$. From Lemma 3.1, we have $\chi _{\{a\le |f|\le b\}} \in U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1}$. For $z\in \overline{S}$, define

$$\begin{aligned} F(z):=\mathrm{sgn}(f)|f|^{\frac{qz}{q_0}+\frac{q(1-z)}{q_1}} {\ \mathrm and \ } G(z):=(z-\theta )\int _0^1 F(\theta +(z-\theta )t) \ dt. \end{aligned}$$

(3.1)

Decompose $G(z)=G_0(z)+G_1(z)$ where $G_0(z):=\chi _{\{|f|\le 1\}}G(z)$. Let $0<\varepsilon <1$. Since $\chi _{\{\varepsilon \le |f|\le 1\}} \in U{\mathcal M}^{p_0}_{q_0}$ and

$$\begin{aligned} \chi _{\{\varepsilon \le |f|\le 1\}}|G_0(z)| \le (1+|z|) (|f|^{q/q_0}+|f|^{q/q_1}) \chi _{\{\varepsilon \le |f|\le 1\}} \le 2(1+|z|) \chi _{\{\varepsilon \le |f|\le 1\}}, \end{aligned}$$

(3.2)

we have $\chi _{\{\varepsilon \le |f|\le 1\}}G_0(z) \in U{\mathcal M}^{p_0}_{q_0}.$ Observe that

$$\begin{aligned} \Vert G_0(z)-\chi _{\{\varepsilon \le |f|\le 1\}}G_0(z) \Vert _{{\mathcal M}^{p_0}_{q_0}}&= \left\| \chi _{\{|f|< \varepsilon \}} \frac{F(z)-F(\theta )}{\left( \frac{q}{q_1}-\frac{q}{q_0} \right) \log |f|} \right\| _{{\mathcal M}^{p_0}_{q_0}} \nonumber \\&\le \left\| \frac{2|f|^{q/q_0}}{\left( \frac{q}{q_1}-\frac{q}{q_0} \right) \log (\varepsilon ^{-1}) } \right\| _{{\mathcal M}^{p_0}_{q_0}} \nonumber \\&\le \frac{2\Vert f\Vert _{{\mathcal M}^p_q}^{q/q_0}}{\left( \frac{q}{q_1}-\frac{q}{q_0}\right) \log \varepsilon ^{-1}} \rightarrow 0 \end{aligned}$$

(3.3)

as $\varepsilon \rightarrow 0^{+}$. Hence $G_0(z) \in U{\mathcal M}^{p_0}_{q_0}$. Similarly, $G_1(z) \in U{\mathcal M}^{p_1}_{q_1}$. Thus $G(z)\in U{\mathcal M}^{p_0}_{q_0}+U{\mathcal M}^{p_1}_{q_1}$. Let $t\in {\mathbb R}$ and $R>1$. Since $\chi _{\{R^{-1}\le |f|\le R\}} \in U{\mathcal M}^{p_0}_{q_0}$ and

$$\begin{aligned} |(G(it)-G(0))|\chi _{\{R^{-1}\le |f|\le R\}} \le (2+|t|)(R^{q/q_0}+R^{q/q_1})\chi _{\{R^{-1}\le |f|\le R\}}, \end{aligned}$$

(3.4)

we have $[G(it)-G(0)]\chi _{\{R^{-1}\le |f|\le R\}} \in U{\mathcal M}^{p_0}_{q_0}$. Note that

$$\begin{aligned} \Vert [G(it)-G(0)]\chi _{{\mathbb R}^n \setminus \{R^{-1}\le |f|\le R\}}\Vert _{{\mathcal M}^{p_0}_{q_0}} \le {\frac{2\Vert f\Vert _{{\mathcal M}^p_q}^{q/q_0}}{\left( \frac{q}{q_1}-\frac{q}{q_0}\right) \log R}} \rightarrow 0 \end{aligned}$$

(3.5)

as $R\rightarrow \infty$. Thus $G(it)-G(0)\in U{\mathcal M}^{p_0}_{q_0}$. Similarly, $G(1+it)-G(1)\in U{\mathcal M}^{p_1}_{q_1}$. Since $G\in {\mathcal G}({\mathcal M}^{p_0}_{q_0},{\mathcal M}^{p_1}_{q_1})$, we have $G\in {\mathcal G}(U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1})$. From $f=G'(\theta )$, it follows that $f\in [U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}]^\theta$. $\square$

Lemma 3.3

Let $G\in {\mathcal G}(U{\mathcal M}^{p_0}_{q_0},U{\mathcal M}^{p_1}_{q_1})$ and $\theta \in (0,1)$. For $z\in \overline{S}$ and $k\in {\mathbb N}$, define $H_k(z)$ by (2.2). Then we have $H_k(\theta )\in \overline{U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}$.

Proof

Let $\varepsilon >0$. By Lemma 2.4, we have $H_k(\theta )\in [U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}]_{\theta }$. Since $U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1}$ is dense in $[U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}]_{\theta }$, we can find $J_k(\theta ) \in U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1}$ such that

$$\begin{aligned} \Vert H_k(\theta )-J_k(\theta )\Vert _{ \left[ U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}\right] _{\theta }} <\varepsilon . \end{aligned}$$

Since $[U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}]_{\theta } \subseteq [{\mathcal M}^{p_0}_{q_0}, {\mathcal M}^{p_1}_{q_1}]_{\theta } \subseteq {\mathcal M}^p_q$, we have

$$\begin{aligned} \Vert H_k(\theta )-J_k(\theta )\Vert _{ {\mathcal M}^p_q} \lesssim \Vert H_k(\theta )-J_k(\theta )\Vert _{ [U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}]_{\theta }} <\varepsilon . \end{aligned}$$

This shows that $H_k(\theta )\in \overline{U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}$. $\square$

Lemma 3.4

Assume the same conditions on the paramaters as in Lemma 3.1. Then $U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1} \subseteq U{\mathcal M}^p_q$.

Proof

Without loss of generality assume that $q_0>q_1$. Let $f\in U{\mathcal M}^{p_0}_{q_0} \cap U{\mathcal M}^{p_1}_{q_1}$. In view of Lemma 2.7, we may assume $f=\chi _{\{a \le |f| \le a^{-1}\}}f$ for some $a>0$. By the lattice property of the spaces $U{\mathcal M}^{p_0}_{q_0}$, $U{\mathcal M}^{p_1}_{q_1}$, and $U{\mathcal M}^{p}_{q}$, we may assume $f=\chi _E$ for some measurable set E. Choose a sequence $\{g_j\}_{j=1}^\infty \subseteq U\cap {\mathcal M}^{p_1}_{q_1}$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert f-g_j\Vert _{{\mathcal M}^{p_1}_{q_1}}=0. \end{aligned}$$

Define $F_j=\{g_j \ne 0\}\cap E$. Hence $|f-\chi _{F_j}|\le 2$ and $|f-\chi _{F_j}|\le |f-g_j|$. Consequently,

$$\begin{aligned} \Vert f-\chi _{F_j}\Vert _{{\mathcal M}^p_q} = \left\| \,|f-\chi _{F_j}|^{1-\frac{q_1}{q}}|f-\chi _{F_j}|^{\frac{q_1}{q}}\,\right\| _{{\mathcal M}^p_q} \le 2^{1-\frac{q_1}{q}} \Vert f-g_j\Vert _{{\mathcal M}^{p_1}_{q_1}}^{\frac{q_1}{q}}. \end{aligned}$$

This shows that $f\in U{\mathcal M}^p_q$. $\square$

Lemma 3.5

Under the assumption of Lemma 3.1,

$$\begin{aligned} {\mathcal M}^p_q \cap \overline{U{\mathcal M}^p_q}^{{\mathcal M}^{p_0}_{q_0}+{\mathcal M}^{p_1}_{q_1}} \subseteq U\bowtie {\mathcal M}^p_q. \end{aligned}$$

Proof

Let $f\in {\mathcal M}^p_q \cap \overline{U{\mathcal M}^p_q}^{{\mathcal M}^{p_0}_{q_0}+{\mathcal M}^{p_1}_{q_1}}$. We may assume that $0<a<1<b<\infty$ for the purpose of showing $\chi _{\{a \le |f| \le b\}}f \in U{\mathcal M}^p_q$ for all $0<a<b<\infty$.

Choose $\{f_j\}_{j=1}^\infty \subseteq U{\mathcal M}^p_q$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty } \Vert f-f_j\Vert _{{\mathcal M}^{p_0}_{q_0}+{\mathcal M}^{p_1}_{q_1}}=0. \end{aligned}$$

Let $\Theta (t)$ be a piecewise linear function such that $\Theta (1)=1$ and that

$$\begin{aligned} \Theta '(t):=\frac{2}{a}\chi _{(a/2,a)}(t)-\frac{1}{b}\chi _{(b,2b)}(t) \end{aligned}$$

(3.6)

except at $t=\frac{a}{2}, a, b, 2b$. According to [11, Lemma 3.3], we have

$$\begin{aligned} \lim _{j\rightarrow \infty } \Vert \chi _{\{a\le |f|\le b\}} \Theta (|f_j|)-\chi _{\{a\le |f|\le b\}} \Theta (|f|)\Vert _{{\mathcal M}^p_q}=0. \end{aligned}$$

Since $\chi _{\{a\le |f|\le b\}}\Theta (|f_j|)\le a^{-1}|f_j|$, we have $\chi _{\{a\le |f|\le b\}}\Theta (|f|) \in U{\mathcal M}^p_q$. From the inequality $\chi _{\{a\le |f|\le b\}}|f|\le b\chi _{\{a\le |f|\le b\}}\Theta (|f|)$, it follows that $\chi _{\{a\le |f|\le b\}}f\in U{\mathcal M}^p_q$. $\square$

Now, the proof of (1.6) is given as follows:

Proof of (1.6)

In view of Lemma 3.2, we only need to show $[U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}]^\theta \subseteq U \bowtie {\mathcal M}^p_q$. Let $f\in [U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1}]^\theta$. Then there exists $G\in {\mathcal G}(U{\mathcal M}^{p_0}_{q_0}, U{\mathcal M}^{p_1}_{q_1})$ such that $G'(\theta )=f$. Define $H_k(z)$ by (2.2) for $z\in \overline{S}$ and $k\in {\mathbb N}$. By Lemmas 3.3 and 3.4, we have $H_k(\theta )\in U{\mathcal M}^p_q$. Since $H_k(\theta )$ converges to $G'(\theta )=f$ in ${\mathcal M}^{p_0}_{q_0}+{\mathcal M}^{p_1}_{q_1}$, it follows from Lemma 3.5 that $f\in U \bowtie {\mathcal M}^p_q$. $\square$

4 The Interpolations of Closed Subspaces of Generalized Morrey Spaces

We remark that the inclusion $U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1} \subseteq U{\mathcal M}^\varphi _q$ is the important part for the first and second complex interpolations of closed subspaces of Morrey spaces.

Lemma 4.1

Suppose that $\theta \in (0,1)$, $1\le q_0<\infty$, $1\le q_1<\infty$, and $\varphi _0^{q_0}=\varphi _1^{q_1}$. Define

$$\begin{aligned} \varphi :=\varphi _0^{1-\theta }\varphi _1^{\theta } \quad \mathrm{and} \quad \frac{1}{q}:=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}. \end{aligned}$$

Then $U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1} \subseteq U{\mathcal M}^\varphi _q$.

Proof

The proof is similar to the proof of Lemma 3.4. $\square$

We prove the generalization of Lemma 3.1 as follows:

Lemma 4.2

Keep the same assumption as in Lemma 4.1. Let E be a measurable set such that $\chi _E \in U{\mathcal M}^\varphi _q$. Then we have

$$\begin{aligned} \chi _{E} \in U{\mathcal M}^{\varphi _0}_{q_0}\cap U{\mathcal M}^{\varphi _1}_{q_1}. \end{aligned}$$

Proof

Let $\chi _E \in U{\mathcal M}^\varphi _q$ and choose $\{g_k\}_{k=1}^\infty \subseteq U\cap {\mathcal M}^\varphi _q$ for which

$$\begin{aligned} \lim \limits _{k\rightarrow \infty } \Vert \chi _E -g_k\Vert _{{\mathcal M}^\varphi _q}=0. \end{aligned}$$

Define $h_k:=\chi _{\{g_k \ne 0 \} \cap E}$. Then, for each $k=0,1$, we have

$$\begin{aligned} \Vert \chi _E-h_k\Vert _{{\mathcal M}^{\varphi _j}_{q_j}} = \Vert \chi _E-h_k\Vert _{{\mathcal M}^{\varphi }_{q}}^{q/q_j} \le \Vert \chi _E-g_k\Vert _{{\mathcal M}^{\varphi }_{q}}^{q/q_j} \rightarrow 0 \end{aligned}$$

as $k\rightarrow \infty$. Thus, $\chi _E \in U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1}$. $\square$

4.1 The First Complex Interpolation Method

We prove Theorem 1.7, which includes Theorem 1.2 as a special case.

Proof

Without loss of generality, assume that $q_0>q_1$. Define A by (2.4). Suppose that $f\in \overline{{\mathcal M}^\varphi _q}$ satisfies

$$\begin{aligned} \lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0. \end{aligned}$$

(4.1)

Note that, for every $a>0$, we have

$$\begin{aligned} \Vert \chi _{\{a\le |f| \}} f\Vert _{{\mathcal M}^{\varphi _1}_{q_1}}&\le \sup _{B=B(x_0,r)} \left[ \frac{\varphi (r)}{|B|^{1/q}}\left( \int _{B\cap \{a\le |f|\}} a^{q_1-q}|f(x)|^q\ dx\right) ^{\frac{1}{q}} \right] ^{\frac{q}{q_1}} \nonumber \\&\le a^{\frac{q_1-q}{q_1}}\Vert f\Vert _{{\mathcal M}^\varphi _q}^{\frac{q}{q_1}}<\infty . \end{aligned}$$

(4.2)

Given $\varepsilon >0$, choose $g_\varepsilon \in L^\infty \cap {\mathcal M}^\varphi _q$ such that

$$\begin{aligned} \Vert f-g_\varepsilon \Vert _{{\mathcal M}^\varphi _q} <\varepsilon . \end{aligned}$$

(4.3)

Since $g_\varepsilon \in L^\infty$, we have

$$\begin{aligned} \Vert g_\varepsilon \Vert _{{\mathcal M}^{\varphi _0}_{q_0}} \le \Vert g_\varepsilon \Vert _{L^\infty }^{\frac{q_0-q}{q_0}} \Vert g_\varepsilon \Vert _{{\mathcal M}^\varphi _q}^{\frac{q}{q_0}}<\infty . \end{aligned}$$

(4.4)

Define

$$\begin{aligned} g_{\varepsilon ,a}:= {\left\{ \begin{array}{ll} \chi _{\{|f|\ge a\}} f, &{}\quad |g_\varepsilon |>|f|,\\ \chi _{\{|f|\ge a\}} g_\varepsilon , &{}\quad |g_\varepsilon |\le |f|. \end{array}\right. } \end{aligned}$$

(4.5)

Since $|g_{\varepsilon ,a}|\le |g_\varepsilon |$ and $|g_{\varepsilon ,a}|\le \chi _{\{|f|\ge a\}}|f|$, by (4.2) and (4.4), it follows that $g_{\varepsilon , a} \in {\mathcal M}^{\varphi _0}_{q_0}\cap {\mathcal M}^{\varphi _1}_{q_1}$. Using the following inequality:

$$\begin{aligned} |f-g_{\varepsilon ,a}| \le |f-\chi _{\{a\le |f|\}}f| + |f-g_\varepsilon |, \end{aligned}$$

(4.6)

we have

$$\begin{aligned} \lim _{\varepsilon ,a \rightarrow 0^{+}} \Vert f-g_{\varepsilon ,a}\Vert _{{\mathcal M}^\varphi _q}=0. \end{aligned}$$

(4.7)

This shows that $f\in \overline{{\mathcal M}^{\varphi _0}_{q_0} \cap {\mathcal M}^{\varphi _1}_{q_1}}^{{\mathcal M}^\varphi _q} =\left[ {\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}\right] _{\theta }$.

Conversely, let $g\in {\mathcal M}^{\varphi _0}_{q_0}\cap {\mathcal M}^{\varphi _1}_{q_1}$. Then $g\in \overline{{\mathcal M}^{\varphi }_q}$ thanks to Lemma 2.7. Since

$$\begin{aligned} \Vert \chi _{\{|g|<a\}}g\Vert _{{\mathcal M}^\varphi _q} \le a^{\frac{q-q_1}{q}} \Vert g\Vert _{{\mathcal M}^{\varphi _1}_{q_1}} \rightarrow 0 \end{aligned}$$

(4.8)

as $a\rightarrow 0^{+}$, we conclude that

$$\begin{aligned} {\mathcal M}^{\varphi _0}_{q_0} \cap {\mathcal M}^{\varphi _1}_{q_1} \subseteq \left\{ f\in \overline{{\mathcal M}^\varphi _q}:\lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0\right\} . \end{aligned}$$

Using [11, Theorem 4.5], we get

$$\begin{aligned} \left[ {\mathcal M}^{\varphi _0}_{q_0},{\mathcal M}^{\varphi _1}_{q_1}\right] _{\theta } = \overline{{\mathcal M}^{\varphi _0}_{q_0} \cap {\mathcal M}^{\varphi _1}_{q_1}}^{{\mathcal M}^\varphi _q} \subseteq \left\{ f\in \overline{{\mathcal M}^\varphi _q}:\lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0\right\} \end{aligned}$$

as desired. $\square$

Next, we prove (1.13).

Proof of (1.13)

Without loss of generality, assume that $q_0>q_1$. By Theorem 1.7, we have

$$\begin{aligned} \left[ U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}\right] _\theta \subseteq \left[ {\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}\right] _\theta = \left\{ f\in \overline{{\mathcal M}^{\varphi }_q}: \lim _{a\rightarrow 0^+}\Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0\right\} . \end{aligned}$$

Let $g\in [U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]_\theta$ and $\varepsilon >0$. Choose $g_\varepsilon \in U{\mathcal M}^{\varphi _0}_{q_0}\cap U{\mathcal M}^{\varphi _1}_{q_1}$ such that

$$\begin{aligned} \Vert g-g_{\varepsilon }\Vert _{[U{\mathcal M}^{\varphi _0}_{q_0},U{\mathcal M}^{\varphi _1}_{q_1}]_{\theta }} <\varepsilon . \end{aligned}$$

(4.9)

Since $U{\mathcal M}^{\varphi _0}_{q_0}\cap U{\mathcal M}^{\varphi _1}_{q_1}\subseteq U{\mathcal M}^\varphi _q$, we have $g_\varepsilon \in U{\mathcal M}^\varphi _q$. From $[U{\mathcal M}^{\varphi _0}_{q_0},U{\mathcal M}^{\varphi _1}_{q_1}]_{\theta } \subseteq [{\mathcal M}^{\varphi _0}_{q_0},{\mathcal M}^{\varphi _1}_{q_1}]_{\theta } \subseteq {\mathcal M}^\varphi _q$, it follows that

$$\begin{aligned} \Vert g-g_\varepsilon \Vert _{{\mathcal M}^\varphi _q} \lesssim \varepsilon , \end{aligned}$$

(4.10)

and hence, $g\in U{\mathcal M}^\varphi _q$.

Conversely, let $f\in \overline{{\mathcal M}^\varphi _q}\cap U{\mathcal M}^\varphi _q$ such that

$$\begin{aligned} \lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q}=0. \end{aligned}$$

From Lemma 2.6, we have

$$\begin{aligned} \lim _{a\rightarrow 0^{+}} \Vert \chi _{\{|f|>a^{-1}\}} f\Vert _{{\mathcal M}^\varphi _q}=0. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert f-\chi _{\{a\le |f|\le a^{-1}\}}f\Vert _{{\mathcal M}^\varphi _q} \le \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q} + \Vert \chi _{\{|f|>a^{-1}\}}f\Vert _{{\mathcal M}^\varphi _q} \rightarrow 0 \end{aligned}$$

as $a\rightarrow 0^{+}$. Observe also that $\chi _{\{a \le |f| \le a^{-1}\}}f \in \overline{{\mathcal M}^\varphi _q} \cap U{\mathcal M}^\varphi _q$ thanks to the lattice property of U. As a result, we may assume that $f=\chi _{\{a\le |f| \le a^{-1}\}}f$ for some $0<a<1$. For every $z\in \overline{S}$, define

$$\begin{aligned} F(z):=\mathrm{sgn}(f)|f|^{q\left( \frac{1-z}{q_0}+\frac{z}{q_1}\right) }. \end{aligned}$$

Decompose F(z) as $F_0(z):=F(z)\chi _{\{|f|\le 1\}}$ and $F_1(z):=F(z)\chi _{\{|f|>1\}}$. Note that, for any $0<b<c<\infty$, we have a pointwise estimate:

$$\begin{aligned} \chi _{\{b\le |f|\le c\}} \le \frac{1}{b}\chi _{\{b\le |f|\le c\}} |f| \le \frac{|f|}{b}, \end{aligned}$$

(4.11)

so $\chi _{\{b\le |f|\le c\}} \in U{\mathcal M}^\varphi _q$. From Lemma 4.2, it follows that $\chi _{\{b\le |f|\le c\}} \in U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1}$. Since

$$\begin{aligned} |F_0(z)|\le \chi _{\{a\le |f|\le 1\}} \quad \mathrm{and} \quad |F_1(z)| \le \left( a^{-\frac{q}{q_0}}+a^{-\frac{q}{q_1}}\right) \chi _{\{1\le |f|\le a^{-1} \}}, \end{aligned}$$

we have $F(z)=F_0(z)+F_1(z)\in U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}$. Moreover, we also have

$$\begin{aligned}&\sup _{z\in \overline{S}} \Vert F(z)\Vert _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}} \le \Vert \chi _{\{a\le |f|\le 1\}}\Vert _{U{\mathcal M}^{\varphi _0}_{q_0}} \\&\quad + \left( a^{-\frac{q}{q_0}}+a^{-\frac{q}{q_1}}\right) \Vert \chi _{\{1\le |f|\le a^{-1}\}}\Vert _{U{\mathcal M}^{\varphi _1}_{q_1}}. \end{aligned}$$

Next, we shall check that $F|_S:S \rightarrow U{\mathcal M}^{\varphi _0}_{q_0} + U{\mathcal M}^{\varphi _1}_{q_1}$ is a holomorphic function. For every $z\in S$, set $H(z):=\left( \frac{q}{q_1}-\frac{q}{q_0}\right) (\log |f|) F(z)$. Then $H(z)\in U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}$ with

$$\begin{aligned} \Vert&H(z)\Vert _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}} \le \left( \frac{q}{q_1}-\frac{q}{q_0}\right) (\log a^{-1})\\&\left( \Vert \chi _{\{a \le |f| \le 1 \}}\Vert _{U{\mathcal M}^{\varphi _0}_{q_0}} +(a^{-q/q_0}+a^{-q/q_1})\Vert \chi _{\{1 \le |f| \le a^{-1}\}}\Vert _{U{\mathcal M}^{\varphi _1}_{q_1}} \right) . \end{aligned}$$

For each $0<\varepsilon \ll 1$, define $S_\varepsilon :=\{z\in S: \varepsilon<\mathrm{Re}(z)<1-\varepsilon \}$. Let $z\in S_\varepsilon$ be fixed and let $w\in S_\varepsilon$ be such that $z+w\in S_\varepsilon$. As a consequence of the following inequalities

$$\begin{aligned} \left| \frac{F(z+w)-F(z)}{w}-H(z) \right|&= \left| \frac{|f|^{w\left( \frac{q}{q_1}-\frac{q}{q_0}\right) }-1-w\left( \frac{q}{q_1}-\frac{q}{q_0}\right) \log |f|}{w} \right| |F(z)|\\&\le \left| \left( \frac{q}{q_1}-\frac{q}{q_0}\right) \log |f|\right| \left( \sum _{k=2}^{\infty } \frac{|w\log |f||^{k-1}}{k!} \right) |F(z)| \\&\le \left( \frac{q}{q_1}-\frac{q}{q_0}\right) \log (a^{-1}) \left( e^{|w| \log (a^{-1})} -1 \right) |F(z)| \end{aligned}$$

and $\Vert F(z)\Vert _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}}<\infty$, we have

$$\begin{aligned}&\left\| \frac{F(z+w)-F(z)}{w}-H(z) \right\| _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}}\\&\quad \lesssim \left( e^{|w| \log (a^{-1})} -1 \right) \Vert F(z)\Vert _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}} \rightarrow 0 \end{aligned}$$

as $w\rightarrow 0$. Hence, $F:S_\varepsilon \rightarrow U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}$ is holomorphic. Since $\varepsilon >0$ is arbitrary, we conclude that $F:S\rightarrow U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}$ is holomorphic.

Observe that for every $w\in S$, we have

$$\begin{aligned} |F'(w)| \le \left( \frac{q}{q_1}-\frac{q}{q_0} \right) \max \left( a^{-\frac{q}{q_0}}, a^{-\frac{q}{q_1}}\right) \log \frac{1}{a} \times \chi _{\{a\le |f|\le a^{-1}\}}. \end{aligned}$$

(4.12)

Then we have

$$\begin{aligned}&\Vert F(z)-F(z')\Vert _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}}\\&\quad =\left\| \int _{z'}^z F'(w) \ dw\right\| _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}}\\&\quad \le \max \left( \frac{q}{q_0},\frac{q}{q_1}\right) \max \left( a^{-\frac{q}{q_0}},a^{-\frac{q}{q_1}}\right) \log \frac{1}{a}\\&\qquad \times \left( \Vert \chi _{\{a\le |f|\le 1\}}+ \chi _{\{1<|f|\le a^{-1}\}} \Vert _{U{\mathcal M}^{\varphi _0}_{q_0}+U{\mathcal M}^{\varphi _1}_{q_1}}\right) |z-z'|\\&\quad \le \max \left( \frac{q}{q_0},\frac{q}{q_1}\right) \max \left( a^{-\frac{q}{q_0}},a^{-\frac{q}{q_1}}\right) \log \frac{1}{a}\\&\qquad \times \left( \Vert \chi _{\{a\le |f|\le 1\}} \Vert _{U{\mathcal M}^{\varphi _0}_{q_0}} + \Vert \chi _{\{1<|f|\le a^{-1}\}} \Vert _{U{\mathcal M}^{\varphi _1}_{q_1}}\right) |z-z'| \end{aligned}$$

for all $z,z' \in \overline{S}$. Thus, $F:\overline{S} \rightarrow U{\mathcal M}^{\varphi _0}_{q_0} + U{\mathcal M}^{\varphi _1}_{q_1}$ is a continuous function.

Note that, for all $t \in {\mathbb R}$ and $j=0,1$, we have

$$\begin{aligned} |F(j+it)|=|f|^{\frac{q}{q_j}} \le a^{-\frac{q}{q_j}}\chi _{\{a\le |f|\le a^{-1}\}}, \end{aligned}$$

so, $F(j+i t) \in U{\mathcal M}^{\varphi _j}_{q_j}$. Furthermore, using (4.12), we get

$$\begin{aligned} \Vert F(j+it)-F(j+it')\Vert _{U{\mathcal M}^{\varphi _j}_{q_j}}&= \left\| \int _{j+it'}^{j+it} F'(w) dw \right\| _{U{\mathcal M}^{\varphi _j}_{q_j}}\\&\le \left( \frac{q}{q_1}-\frac{q}{q_0}\right) \max (a^{-\frac{q}{q_0}},a^{-\frac{q}{q_1}}) \log \frac{1}{a}\\&\quad \times \Vert \chi _{\{a\le |f|\le a^{-1}\}}\Vert _{U{\mathcal M}^{\varphi _j}_{q_j}} |t-t'| \end{aligned}$$

for all $t,t'\in {\mathbb R}$. This shows that $t\in {\mathbb R}\mapsto F(j+it) \in U{\mathcal M}^{\varphi _j}_{q_j}, j=0, 1$ are continuous functions. In total, we have showed that $F\in {\mathcal F}(U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1})$. Since $F(\theta )=f$, we have $f \in [U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]_\theta$ as desired. $\square$

4.2 The Second Complex Interpolation Method

From now on, we shall always use the assumption of Theorem 1.9. To prove Theorem 1.9, we shall invoke and prove several lemmas:

Lemma 4.3

Keep the assumption in Theorem 1.9. Then we have

$$\begin{aligned} U\bowtie {\mathcal M}^\varphi _q \subseteq \left[ U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}\right] ^\theta . \end{aligned}$$

(4.13)

Proof

Assume that $q_0>q_1$. We go through a similar argument as in the proof of Lemma 3.2 to obtain (4.13). $\square$

Lemma 4.4

Let $G\in {\mathcal G}({\mathcal M}^{\varphi _0}_{q_0},{\mathcal M}^{\varphi _1}_{q_1})$ and $\theta \in (0,1)$. For $z\in \overline{S}$ and $k\in {\mathbb N}$, define $H_k(z)$ by (2.2). Then $H_k(\theta )\in \overline{U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1}}^{{\mathcal M}^\varphi _q}$.

Proof

From Lemma 2.4, it follows that $H_k(\theta )\in [U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]_\theta$. Let $\varepsilon >0$. Since $U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1}$ is dense in $[U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]_{\theta }$, we can find $J_k(\theta ) \in U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1}$ such that

$$\begin{aligned} \Vert H_k(\theta )-J_k(\theta )\Vert _{ [U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]_{\theta }} <\varepsilon . \end{aligned}$$

Since $[U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]_{\theta } \subseteq [{\mathcal M}^{\varphi _0}_{q_0}, {\mathcal M}^{\varphi _1}_{q_1}]_{\theta } \subseteq {\mathcal M}^\varphi _q$, we have

$$\begin{aligned} \Vert H_k(\theta )-J_k(\theta )\Vert _{ {\mathcal M}^\varphi _q} \lesssim \Vert H_k(\theta )-J_k(\theta )\Vert _{ [U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]_{\theta }} <\varepsilon . \end{aligned}$$

This shows that $H_k(\theta )\in \overline{U{\mathcal M}^{\varphi _0}_{q_0} \cap U{\mathcal M}^{\varphi _1}_{q_1}}^{{\mathcal M}^\varphi _q}$. $\square$

Lemma 4.5

We use the assumption of Theorem 4.2. Then we have

$$\begin{aligned} {\mathcal M}^\varphi _q \cap \overline{U{\mathcal M}^\varphi _q}^{{\mathcal M}^{\varphi _0}_{q_0}+{\mathcal M}^{\varphi _1}_{q_1}} \subseteq U\bowtie {\mathcal M}^\varphi _q. \end{aligned}$$

Proof

Let $f\in {\mathcal M}^\varphi _q \cap \overline{U{\mathcal M}^\varphi _q}^{{\mathcal M}^{\varphi _0}_{q_0}+{\mathcal M}^{\varphi _1}_{q_1}}$. Assume $0<a<1<b<\infty$ as before. Choose $\{f_j\}_{j=1}^\infty \subseteq U{\mathcal M}^\varphi _q$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty } \Vert f-f_j\Vert _{{\mathcal M}^{\varphi _0}_{q_0}+{\mathcal M}^{\varphi _1}_{q_1}}=0. \end{aligned}$$

Let $\Theta (t)$ be a function defined by (3.6). By a similar argument as in the proof of [11, Lemma 3.3], we have

$$\begin{aligned} \lim _{j\rightarrow \infty } \Vert \chi _{\{a\le |f|\le b\}} \Theta (|f_j|)-\chi _{\{a\le |f|\le b\}} \Theta (|f|)\Vert _{{\mathcal M}^\varphi _q}=0. \end{aligned}$$

Since $\chi _{\{a\le |f|\le b\}}\Theta (|f_j|)\le a^{-1}|f_j|$, we have $\chi _{\{a\le |f|\le b\}}\Theta (|f|) \in U{\mathcal M}^\varphi _q$. From the inequality $\chi _{\{a\le |f|\le b\}}|f|\le b\chi _{\{a\le |f|\le b\}}\Theta (|f|)$, it follows that $\chi _{\{a\le |f|\le b\}}f\in U{\mathcal M}^\varphi _q$. $\square$

Now, we are ready to prove Theorem 1.9.

Proof of (1.15)

In view of Lemma 4.3, we only need to show that

$$\begin{aligned} \left[ U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}\right] ^\theta \subseteq U \bowtie {\mathcal M}^\varphi _q. \end{aligned}$$

Let $f\in [U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1}]^\theta$. Then there exists $G\in {\mathcal G}(U{\mathcal M}^{\varphi _0}_{q_0}, U{\mathcal M}^{\varphi _1}_{q_1})$ such that $G'(\theta )=f$. For $z\in \overline{S}$ and $k\in {\mathbb N}$, define $H_k(z)$ by (2.2). By Lemmas 4.1 and 4.4, we have $H_k(\theta )\in U{\mathcal M}^\varphi _q$. Since $H_k(\theta )$ converges to $G'(\theta )=f$ in ${\mathcal M}^{\varphi _0}_{q_0}+{\mathcal M}^{\varphi _1}_{q_1}$, by Lemma 4.5, it follows that $f\in U \bowtie {\mathcal M}^\varphi _q$. $\square$

We compare Theorem 1.9 with our previous result.

Remark 4.6

Assume that $\inf \varphi >0$. According to [11, Theorem 5.12],

$$\begin{aligned} \left[ \widetilde{{{\mathcal M}}^{\varphi _0}_{q_0}}, \widetilde{{{\mathcal M}}^{\varphi _1}_{q_1}}\right] ^\theta = \bigcap _{0<a<b<\infty } \left\{ f \in {\mathcal M}^\varphi _q\cap \widetilde{L^\infty } \,:\, \chi _{\{a\le |f|\le b\}}f \in \widetilde{{\mathcal M}^\varphi _q} \right\} . \end{aligned}$$

(4.14)

Meanwhile, in the light of Theorem 1.9, we have

$$\begin{aligned}{}[\widetilde{{{\mathcal M}}^{\varphi _0}_{q_0}}, \widetilde{{{\mathcal M}}^{\varphi _1}_{q_1}}]^\theta = \bigcap _{0<a<b<\infty } \left\{ f \in {\mathcal M}^\varphi _q \,:\, \chi _{\{a\le |f|\le b\}}f \in \widetilde{{\mathcal M}^\varphi _q} \right\} . \end{aligned}$$

(4.15)

Thus, the sets in the right-hand side of (4.14) and (4.15) coincide. In fact, this can be verified directly from the fact that ${\mathcal M}^\varphi _q\subset L^\infty$ (see [11, Theorem 5.9]).

5 The Closure of Compactly Supported Functions in Morrey Spaces on Bounded Connected Open Sets

We recall that we do not require that the domain $\Omega$ is smooth. In view of Theorem 5.1 below and the fact that ${\mathcal M}^\varphi _q \supset L^\infty$ if and only if $\inf \varphi >0$; see [21, Proposition 3.3], we shall concentrate on the case $\inf \varphi =0$.

Lemma 5.1

Let $1\le q<\infty$, $\varphi \in {\mathcal G}_q$, and $\Omega$ be bounded. Then we have $L^\infty (\Omega ) \subseteq {\mathcal M}^\varphi _q(\Omega )$. In particular, when $\inf \varphi >0$, we have ${\mathcal M}^\varphi _q(\Omega )=L^\infty (\Omega )$.

Proof

Let $f\in L^\infty (\Omega )$. For $x\in \Omega$ and $0<r<\mathrm{diam}(\Omega )$, we have

$$\begin{aligned} \varphi (r) \left( \frac{1}{|B(x,r)|} \int _{B(x,r)\cap \Omega } |f(y)|^q \ dy \right) ^{\frac{1}{q}} \le \varphi (\mathrm{diam}(\Omega )) \Vert f\Vert _{L^\infty (\Omega )}. \end{aligned}$$

(5.1)

Consequently, $f\in {\mathcal M}^\varphi _q(\Omega )$ with $\Vert f\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \varphi (\mathrm{diam}(\Omega )) \Vert f\Vert _{L^\infty (\Omega )}$. This shows that $L^\infty (\Omega ) \subseteq {\mathcal M}^\varphi _q(\Omega )$. When $\inf \varphi >0$, we combine $L^\infty (\Omega ) \subseteq {\mathcal M}^\varphi _q(\Omega )$ with [21, Proposition 3.3] to obtain ${\mathcal M}^\varphi _q(\Omega )=L^\infty (\Omega )$. $\square$

We shall prove Theorem 1.11. Our proof will use the identification of $\overset{\circ }{{\mathcal M}}{}^{\varphi }_{q}(\Omega )$ as the vanishing generalized Morrey spaces. The definition of these spaces is given as follows (see also [7, 10, 25]).

Definition 5.2

Let $1\le q<\infty$, $\varphi \in {\mathcal G}_q$, and $f\in {\mathcal M}^\varphi _q(\Omega )$. For $0<r<\mathrm{diam}(\Omega )$, define

$$\begin{aligned} \eta _{f,\varphi ,q,\Omega }(r):= \sup _{x\in \Omega , 0<R<r} \frac{\varphi (R)}{|B(x,R)|^{\frac{1}{q}}} \left( \int _{B(x,R)\cap \Omega } |f(y)|^q \ dy \right) ^{\frac{1}{q}}. \end{aligned}$$

The generalized vanishing Morrey space $V{\mathcal M}^\varphi _q(\Omega )$ is defined to be the subset of ${\mathcal M}^\varphi _q(\Omega )$ such that

$$\begin{aligned} \lim _{r\rightarrow 0^{+}} \eta _{f,\varphi ,q,\Omega }(r)=0. \end{aligned}$$

For the setting in ${\mathbb R}^n$, we also define

$$\begin{aligned} \eta _{f,\varphi ,q,{\mathbb R}^n}(r):= \sup _{x\in {\mathbb R}^n, 0<R<r} \frac{\varphi (R)}{|B(x,R)|^{\frac{1}{q}}} \left( \int _{B(x,R)} |f(y)|^q \ dy \right) ^{\frac{1}{q}} \quad (0<r<\infty ) \end{aligned}$$

and $VM^\varphi _q({\mathbb R}^n):=\left\{ f\in {\mathcal M}^\varphi _q({\mathbb R}^n):\lim \limits _{r\rightarrow 0^{+}}\eta _{f,\varphi ,q,{\mathbb R}^n}(r)=0\right\}$.

Before we go further, a helpful remark may be in order.

Remark 5.3

When $\inf \varphi >0$, $V{\mathcal M}^\varphi _q(\Omega )=\{0\}$ by the Lebesgue differentiation theorem.

The fact that vanishing Morrey spaces and the closure of test functions in Morrey spaces coincide can be traced back to [7, Lemma 1.2]. We generalize this fact in the following lemmas:

Lemma 5.4

Let $1\le q<\infty$, $\varphi \in {\mathcal G}_q$, $\inf \varphi =0$, and $f\in V{\mathcal M}^\varphi _q(\Omega )$. Define

$$\begin{aligned} \tilde{f}(x):= {\left\{ \begin{array}{ll} f(x), \quad x\in \Omega ,\\ 0, \quad x\in {\mathbb R}^n\setminus \Omega . \end{array}\right. } \end{aligned}$$

Then we have $\lim \limits _{|h|\rightarrow 0^{+}} \Vert \tilde{f}(\cdot +h)-\tilde{f}\Vert _{{\mathcal M}^\varphi _q({\mathbb R}^n)} =0$.

Proof

Fix $r>0$. Since $\tilde{f}=0$ outside $\Omega$, we have

$$\begin{aligned} \eta _{\tilde{f},\varphi ,q,{\mathbb R}^n}(r) =\sup _{x\in {\mathbb R}^n,0<R<r} \varphi (R)\left( \frac{1}{|B(x,R)|} \int _{B(x,R) \cap \Omega } |f(y)|^q \ dy \right) ^{1/q}. \end{aligned}$$

Let $\Omega _r :=\cup _{z\in \Omega } B(z,r)$. Since $\varphi \in {\mathcal G}_q$, we have

$$\begin{aligned} \eta _{\tilde{f},\varphi ,q,{\mathbb R}^n}(r)&=\sup _{x\in \Omega _r, 0<R<r} \varphi (R)\left( \frac{1}{|B(x,R)|} \int _{B(x,R) \cap \Omega } |f(y)|^q \ dy \right) ^{1/q}\\&\le 3^{n/q} \sup _{x\in \Omega , 0<R<3r} \varphi (R)\left( \frac{1}{|B(x,R)|} \int _{B(x,R) \cap \Omega } |f(y)|^q \ dy \right) ^{1/q}\\&\le 3^{n/q} \eta _{f,\varphi ,q,\Omega }(3r). \end{aligned}$$

Since $f\in V{\mathcal M}^\varphi _q(\Omega )$, we have $\lim \limits _{r\rightarrow 0^{+}}\eta _{f,\varphi ,q,\Omega }(3r)=0$, and hence

$$\begin{aligned} \lim \limits _{r\rightarrow 0^{+}}\eta _{\tilde{f},\varphi ,q,{\mathbb R}^n}(r)=0. \end{aligned}$$

Let $h\in {\mathbb R}^n$. Then we have

$$\begin{aligned}&\Vert \tilde{f}(\cdot +h)-\tilde{f}\Vert _{{\mathcal M}^\varphi _q({\mathbb R}^n)}\\&\quad \le \sup _{x\in {\mathbb R}^n, R\ge r} \varphi (R) \left( \frac{1}{|B(x,R)|} \int _{B(x,R)} |\tilde{f}(y+h)-\tilde{f}(y)|^q \ dy\right) ^{1/q}\\&\qquad + \sup _{x\in \Omega , 0<R< r} \varphi (R)\left( \frac{1}{|B(x,R)|} \int _{B(x,R)} |\tilde{f}(y+h)-\tilde{f}(y)|^q \ dy \right) ^{1/q}\\&\quad \le \frac{\varphi (r)}{|B(x,r)|^{1/q}} \Vert \tilde{f}(\cdot -h)-\tilde{f}\Vert _{L^q({\mathbb R}^n)}\\&\qquad + 2 \sup _{x\in {\mathbb R}^n, 0<R<r} \frac{\varphi (R)}{|B(x,R)|^{1/q}} \left( \int _{B(x,R)} |\tilde{f}(y)|^q \ dy\right) ^{1/q}. \end{aligned}$$

By the $L^q$-continuity of translation, we get

$$\begin{aligned} \limsup _{|h|\rightarrow 0^+} \Vert \tilde{f}(\cdot +h)-\tilde{f}\Vert _{{\mathcal M}^\varphi _q({\mathbb R}^n)} \le 2 \eta _{\tilde{f},\varphi ,q,{\mathbb R}^n}(r). \end{aligned}$$

Finally, taking $r\rightarrow 0^{+}$, we get $\lim \limits _{|h|\rightarrow 0^{+}} \Vert \tilde{f}(\cdot +h)-\tilde{f}\Vert _{{\mathcal M}^\varphi _q({\mathbb R}^n)}=0$.

Lemma 5.5

Let $1\le q<\infty$, $\varphi \in {\mathcal G}_q$, and $f\in {\mathcal M}^\varphi _q({\mathbb R}^n)$ be such that f vanishes almost everywhere outside $\Omega$. If

$$\begin{aligned} \lim \limits _{|y|\rightarrow 0^{+}} \Vert f(\cdot -y)-f\Vert _{{\mathcal M}^\varphi _q(\Omega )} =0, \end{aligned}$$

then $f\in \overline{ C^\infty ({\mathbb R}^n) \cap {\mathcal M}^\varphi _q({\mathbb R}^n)}^{{\mathcal M}^\varphi _q(\Omega )}$.

Proof

By the translation, we may assume that $0 \in \Omega$. Let $r_0>0$ be so small that $B(0,r_0) \subset \Omega$. Let $\psi$ be a smooth function supported on the unit ball $B(0,r_0)$, $0\le \psi \le 1$, and $\Vert \psi \Vert _{L^1(\Omega )}=1$. For every $x\in \Omega$ and $j\in {\mathbb N}$, define $\psi _j(x):=j^n\psi (jx)$. Note that $f*\psi _j \in C^\infty ({\mathbb R}^n)$, since f is locally integrable.

Let $x_0\in \Omega$ and $r>0$ be fixed. By the Minkowski integral inequality, we have

$$\begin{aligned} \Big ( \int _{B(x_0,r)\cap \Omega }&\big |f*\psi _j(x)-f(x)\big |^q \ dx \Big )^{\frac{1}{q}}\\&= \left( \int _{B(x_0,r) \cap \Omega } \left| \int _{B(0,1/j)} (f(x-y) -f(x))\psi _j(y) \ dy \right| ^q \ dx \right) ^{\frac{1}{q}}\\&\le \int _{B(0,1/j)} \psi _j(y) \left( \int _{B(x_0,r)\cap \Omega } |f(x-y)-f(x)|^q \ dx \right) ^{1/q} \ dy\\&\le \frac{|B(0,r)|^{1/q}}{\varphi (r)} \int _{B(0,1/j)} \psi _j(y) \Vert f(\cdot -y)-f\Vert _{{\mathcal M}^\varphi _q(\Omega )} \ dy\\&\le \frac{|B(0,r)|^{1/q}}{\varphi (r)} \sup _{y\in B(0,1/j)} \Vert f(\cdot -y)-f\Vert _{{\mathcal M}^\varphi _q(\Omega )}. \end{aligned}$$

Consequently, r and $x_0$ being arbitrary, we have

$$\begin{aligned} \Vert f-f*\psi _j\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \sup _{y\in B(0,1/j)} \Vert f(\cdot -y)-f\Vert _{{\mathcal M}^\varphi _q(\Omega )}. \end{aligned}$$

Finally, by taking $j\rightarrow \infty$, we get $|y| \rightarrow 0$, and hence $\lim \limits _{j\rightarrow \infty } \Vert f-f*\psi _j\Vert _{{\mathcal M}^\varphi _q(\Omega )}=0$. This shows that $f\in \overline{ C^\infty ({\mathbb R}^n) \cap {\mathcal M}^\varphi _q({\mathbb R}^n)}^{{\mathcal M}^\varphi _q(\Omega )}$ as desired. $\square$

Recall that we are assuming $\inf \varphi =0$. This assumption is necessary when we derive (5.3) from (5.2) below.

Lemma 5.6

Let $1\le q<\infty$ and $\varphi \in {\mathcal G}_q$ be such that $\inf \varphi =0$. Then we have $\overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )=V{\mathcal M}^\varphi _q(\Omega )$.

Proof

As before a translation allows us to assume $B(0,r_0) \subset \Omega$. Let $f\in \overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )$. For any $\varepsilon >0$, choose $g\in C^\infty _c(\Omega )$ such that

$$\begin{aligned} \Vert f-g\Vert _{{\mathcal M}^\varphi _q(\Omega )}<\varepsilon . \end{aligned}$$

Let $0<r<\mathrm{diam}(\Omega )$. Note that, for every $R \in (0,r)$, we have

$$\begin{aligned} \varphi (R)&\left( \frac{1}{|B(x, R)|} \int _{B(x,R)\cap \Omega } |f(y)|^q \ dy\right) ^{1/q}\\&\le \frac{\varphi (R)}{|B(x, R)|^{1/q}} \left[ \left( \int _{B(x,R)\cap \Omega } |f(y)-g(y)|^q \ dy\right) ^{1/q}\right. \\&\quad \left. + \left( \int _{B(x,R)\cap \Omega } |g(y)|^q \ dy \right) ^{1/q}\right] \\&\le \Vert f-g\Vert _{{\mathcal M}^\varphi _q(\Omega )} + \Vert g\Vert _{{L^\infty (\Omega )}}^q\varphi (r)\\&\le \varepsilon +\Vert g\Vert _{L^\infty (\Omega )}\varphi (r). \end{aligned}$$

Consequently,

$$\begin{aligned} \eta _{f,\varphi ,q, \Omega }(r)\le \Vert g\Vert _{L^\infty (\Omega )} \varphi (r). \end{aligned}$$

(5.2)

By taking $r\rightarrow 0^+$, we get

$$\begin{aligned} \lim _{r\rightarrow 0^{+}} \eta _{f,\varphi ,q,\Omega }(r)=0. \end{aligned}$$

(5.3)

This shows that $f\in V{\mathcal M}^\varphi _q(\Omega )$.

Conversely, by assuming $f \in V{\mathcal M}^\varphi _q(\Omega )$, we shall show that $f\in \overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )$. Define

$$\begin{aligned} \tilde{f}(x):={\left\{ \begin{array}{ll} f(x), \quad x\in \Omega ,\\ 0, \quad x\notin \Omega . \end{array}\right. } \end{aligned}$$

By Lemmas 5.4 and 5.5, we can find $\{\tilde{g_j}\}_{j=1}^\infty \subset C^\infty ({\mathbb R}^n) \cap {{\mathcal M}}{}^\varphi _q(\mathbb R^n)$ such that

$$\begin{aligned} \Vert \tilde{f}-\tilde{g_j}\Vert _{\mathcal {M}^\varphi _q(\Omega )} \le \frac{1}{j}. \end{aligned}$$

Define $g_j:=\chi _{\Omega }\tilde{g_j}$. Since $\Omega$ is bounded, we have $\Vert g_j\Vert _{L^\infty (\Omega )} \lesssim 1$. Write $\Omega = \bigcup _{k=1}^{\infty } K_k$ where $\{K_k\}_{k=1}^\infty$ is a collection of compact sets with property $K_k \subseteq \mathrm{int}{K_{k+1}}$. Let $g_{j,k}:=g_j\chi _{K_k}$. Note that $g_{j,k} \in L^{\infty }_\mathrm{c}(\Omega )$. Let $\psi \in C^\infty _\mathrm{c}(\Omega )$ with $\mathrm {supp}(\psi ) \subset B(0,r_0)$, $0\le \psi \le 1$, and $\Vert \psi \Vert _{L^1}=1$. For every $l\in {\mathbb N}$, define

$$\begin{aligned} \psi _l(x):=l^n\psi (lx). \end{aligned}$$

For large $l\in {\mathbb N}$, observe that $g_{j,k}*\psi _l \in C^\infty _\mathrm{c}(\Omega )$ in view of the size of the support of $g_{j,k}$. Note that

$$\begin{aligned}&\Vert f-g_{j,k}*\psi _l\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \Vert f-g_j\Vert _{{\mathcal M}^\varphi _q(\Omega )} + \Vert g_j-g_{j,k}\Vert _{{\mathcal M}^\varphi _q(\Omega )}\\&\quad + \Vert g_{j,k}-g_{j,k}*\psi _l\Vert _{{\mathcal M}^\varphi _q(\Omega )}. \end{aligned}$$

Since $g_{j,k} \in L^\infty _\mathrm{c}(\Omega ) \subseteq VM^\varphi _q(\Omega )$ and

$$\begin{aligned} \Vert g_{j,k}-g_{j,k}*\psi _l\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \sup _{y\in B(0,\frac{1}{l})} \Vert g_{j,k}-g_{j,k}(\cdot -y)\Vert _{{\mathcal M}^\varphi _q(\Omega )}, \end{aligned}$$

we have $\lim \limits _{l\rightarrow \infty }\Vert g_{j,k}-g_{j,k}*\psi _l\Vert _{{\mathcal M}^\varphi _q(\Omega )}=0.$ For any $\varepsilon >0$, choose $\delta >0$ such that $\varphi (r)<\varepsilon$ for every $0<r<\delta$. Since $g_j\in L^\infty _{c}({\mathbb R}^n) \subseteq L^q({\mathbb R}^n)$ and

$$\begin{aligned} \Vert g_j-g_{j,k}\Vert _{{\mathcal M}^\varphi _q(\Omega )}&\le \sup _{x\in \Omega , 0<r<\delta } \varphi (r) \left( \frac{1}{|B(x,r)|} \int _{B(x,r)\cap \Omega } |g_j(y)-g_{j,k}(y)|^q \ dy\right) ^{1/q}\\&\quad + \sup _{x\in \Omega , r\ge \delta } \varphi (r) \left( \frac{1}{|B(x,r)|} \int _{B(x,r)\cap \Omega } |g_j(y)-g_{j,k}(y)|^q \ dy\right) ^{1/q}\\&\le \varepsilon \Vert g_j\Vert _{L^\infty (\Omega )} + \frac{\varphi (\delta )}{|B(x,\delta )|^{1/q}} \Vert \chi _{\Omega \setminus K_k} g_j\Vert _{L^q({\mathbb R}^n)}, \end{aligned}$$

by the dominated convergence theorem, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty } \Vert g_j-g_{j,k}\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \varepsilon \Vert g_j\Vert _{L^\infty (\Omega )}. \end{aligned}$$

Since $\varepsilon >0$ is arbitrary, we have $\lim \limits _{k\rightarrow \infty } \Vert g_j-g_{j,k}\Vert _{{\mathcal M}^\varphi _q(\Omega )} =0$. Consequently,

$$\begin{aligned} \limsup _{k,l \rightarrow \infty } \Vert f-g_{j,k}*\psi _l\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \Vert f-g_j\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \Vert \tilde{f}-\tilde{g_j}\Vert _{{\mathcal M}^\varphi _q({\mathbb R}^n)} \le \frac{1}{j}. \end{aligned}$$

By taking $j\rightarrow \infty$, we see that $f\in \overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )$. $\square$

Before proving Theorem 1.11, we shall prove the following lemmas:

Lemma 5.7

For all $f\in {\mathcal M}^\varphi _q(\Omega )$, we have

$$\begin{aligned} \lim _{a\rightarrow 0^{+}}\Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q(\Omega )}=0. \end{aligned}$$

Proof

For every $x\in \Omega$ and $0<r<\mathrm{diam}(\Omega )$, we have

$$\begin{aligned} \varphi (r) \left( \frac{1}{|B(x,r)|} \int _{B(x,r) \cap \Omega } \chi _{\{|f|<a\}}(y) |f(y)|^q \ dy\right) ^{\frac{1}{q}}\le \varphi (\mathrm{diam}(\Omega )) a. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert \chi _{\{|f|<a\}}f\Vert _{{\mathcal M}^\varphi _q(\Omega )} \le \varphi (\mathrm{diam}(\Omega )) a\rightarrow 0 \end{aligned}$$

as $a\rightarrow 0^{+}$. $\square$

Lemma 5.8

Let $g \in V{\mathcal M}^\varphi _q(\Omega )$ and $|f|\le |g|$. Then we have $f \in V{\mathcal M}^\varphi _q(\Omega )$.

Proof

This is a direct consequence of $\eta _{f,\varphi ,q,\Omega }(r)\le \eta _{g,\varphi ,q,\Omega }(r)$ for every $r>0$. $\square$

Lemma 5.9

Keep using the same assumption as in Theorem 1.11. Let E be a measurable set such that $\chi _E \in V{\mathcal M}^\varphi _q(\Omega )$. Then $\chi _E$ belongs to $V{\mathcal M}^{\varphi _0}_{q_0}(\Omega ) \cap V{\mathcal M}^{\varphi _1}_{q_1}(\Omega )$.

Proof

From our assumption, we have $\varphi _0^{q_0}=\varphi _1^{q_1}=\varphi ^q$. This implies

$$\begin{aligned} \eta _{\chi _{E},\varphi _0, q_0, \Omega }(r) = \eta _{\chi _{E},\varphi , q, \Omega }(r)^{q/q_0} \quad \mathrm{and } \quad \eta _{\chi _{E},\varphi _1, q_1, \Omega }(r) = \eta _{\chi _{E},\varphi , q, \Omega }(r)^{q/q_1}. \end{aligned}$$

By taking $r\rightarrow 0^{+}$, we see that $\chi _E \in V{\mathcal M}^{\varphi _0}_{q_0}(\Omega ) \cap V{\mathcal M}^{\varphi _1}_{q_1}(\Omega )$. $\square$

Finally, we give the proof of Theorem 1.11 as follows.

Proof of Theorem 1.11

Without loss of generality, we may assume that $q_0>q_1$. By a similar argument as in the proof of Theorem 1.7, we have

$$\begin{aligned}&\left[ \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega ), \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )\right] _{\theta } \subseteq \left[ {\mathcal M}^{\varphi _0}_{q_0}(\Omega ), {\mathcal M}^{\varphi _1}_{q_1}(\Omega )\right] _{\theta }\\&\quad \subseteq \overline{L^\infty (\Omega ) \cap {\mathcal M}^\varphi _q(\Omega )}^{{\mathcal M}^\varphi _q(\Omega )} \subseteq V{\mathcal M}^\varphi _q(\Omega ). \end{aligned}$$

Conversely let $f\in \overset{\circ }{{\mathcal M}}{}^\varphi _q(\Omega )$. For every $z\in \overline{S}$, define

$$\begin{aligned} F(z):=\mathrm{sgn}(f)|f|^{q\left( \frac{1-z}{q_0}+\frac{z}{q_1}\right) }, \quad F_0(z):=\chi _{\{|f|\le 1\}}F(z), \quad \mathrm{and} \quad F_1(z):=\chi _{\{|f|> 1\}}F(z). \end{aligned}$$

Since $C^\infty _\mathrm{c}(\Omega ) \subseteq L^\infty (\Omega )$, we can combine Lemmas 2.6 and 5.7 to obtain

$$\begin{aligned} \lim _{a\rightarrow 0^{+}} \Vert f-\chi _{\{a\le |f|\le a^{-1}\}}f\Vert _{{\mathcal M}^\varphi _q(\Omega )}=0. \end{aligned}$$

Therefore, we may assume that

$$\begin{aligned} f=\chi _{\{a\le |f|\le a^{-1}\}}f. \end{aligned}$$

(5.4)

for some $a \in (0,1)$.

By Lemma 5.6, we have $f\in V{\mathcal M}^\varphi _q(\Omega )$. Meanwhile, for any $0<b<c<\infty$, we have (4.11). From Lemmas 5.6, 5.8, and 5.9, it follows that $\chi _{\{b\le |f|\le c\}} \in \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega ) \cap \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )$. Since

$$\begin{aligned} |F_0(z)| =|f|^{q\left( \frac{1-\mathrm{Re}(z)}{q_0}+ \frac{\mathrm{Re}(z)}{q_1} \right) } \chi _{\{|f|\le 1\}} = |f|^{\frac{q}{q_0}} |f|^{q \ \mathrm{Re}(z)\left( \frac{1}{q_1}- \frac{1}{q_0} \right) } \chi _{\{a\le |f|\le 1\}} \le \chi _{\{a\le |f|\le 1\}} \end{aligned}$$

and

$$\begin{aligned} |F_1(z)|&= \left( |f|^{\frac{q}{q_0}}\right) ^{1-\mathrm{Re}(z)} \left( |f|^{\frac{q}{q_1}}\right) ^{\mathrm{Re}(z)} \chi _{\left\{ 1<|f|<a^{-1}\right\} }\\&\le \left( |f|^{q/q_0}+|f|^{q/q_1}\right) \chi _{\left\{ 1<|f|<a^{-1}\right\} }\\&\le \left( a^{-q/q_0}+a^{-q/q_1} \right) \chi _{\left\{ 1<|f|<a^{-1}\right\} }, \end{aligned}$$

we have $F_0(z)\in \overset{\circ }{{\mathcal M}}{} ^{\varphi _0}_{q_0}(\Omega )$ and $F_1(z) \in \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )$, and hence $F(z)\in \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega )+\overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )$. Moreover, we also have

$$\begin{aligned} \sup _{z\in \overline{S}}&\Vert F(z)\Vert _{ \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega ) + \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega ) }\\&\le \sup _{z\in \overline{S}} \Vert F_0(z)\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega )} + \Vert F_1(z)\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )}\\&\le \Vert \chi _{\{a<|f|\le 1\}}\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega )} + (a^{-\frac{q}{q_0}}+a^{-\frac{q}{q_1}}) \Vert \chi _{\{1\le |f|\le a^{-1}\}}\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )}<\infty . \end{aligned}$$

Observe that for all $w\in S$, we have

$$\begin{aligned} |F'(w)| \le \left( \frac{q}{q_1} -\frac{q}{q_0} \right) \left( |f|^{\frac{q}{q_0}}+ |f|^\frac{q}{q_1} \right) \chi _{\{a\le |f|\le a^{-1}\}} |\log |f|| \le C_{a,q,q_0,q_1} \chi _{\{a\le |f|\le a^{-1}\}} \end{aligned}$$

where $C_{a,q,q_0,q_1} := \left( \frac{q}{q_0} -\frac{q}{q_1} \right) \left( a^{-q/q_0}+a^{-q/q_1} \right) \log \frac{1}{a}$. Consequently, for all $z_1, z_2\in \overline{S}$, we have

$$\begin{aligned}&\Vert F(z_2)-F(z_1)\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega )+\overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )} \\&\quad = \left\| \int _{z_1}^{z_2} F'(w) \ dw\right\| _{\overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega )+\overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )} \\&\quad \le C_{a,q,q_0,q_1}\left( \Vert \chi _{\{a\le |f|\le 1\}} \Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega )} + \Vert \chi _{\{1\le |f|\le a^{-1}\}}\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )}\right) |z_2-z_1|. \end{aligned}$$

This shows that $F:\overline{S}\rightarrow \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega )+ \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )$ is a continuous function. Likewise, we also can verify that $F|_{S}:S\rightarrow {\overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}}(\Omega ) +\overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega )$ is a holomorphic function by the same argument as in the proof of (1.13). On the boundary of $\overline{S}$, we have

$$\begin{aligned} |F(j+it)|=|f|^{\frac{q}{q_j}} \le a^{-\frac{q}{q_j}} \chi _{\{a\le |f|\le a^{-1}\}} \end{aligned}$$

for $j=0,1$ and $t\in {\mathbb R}$ from (4.11), so $F(j+it) \in \overset{\circ }{{\mathcal M}}{}^{\varphi _j}_{q_j}(\Omega )$. By a similar argument for showing the continuity of F(z), we also have

$$\begin{aligned} \Vert F(j+it_1)-F(j+it_2)\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _j}_{q_j}(\Omega )} \le C_{a,q,q_0,q_1} \Vert \chi _{\{a\le |f|\le a^{-1}\}}\Vert _{\overset{\circ }{{\mathcal M}}{}^{\varphi _j}_{q_j}(\Omega )}|t_1-t_2| \end{aligned}$$

for all $t_1,t_2 \in {\mathbb R}$. Hence $F\in {\mathcal F}( \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega ), \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega ) )$. Since $F(\theta )=f$, we conclude that $f\in [ \overset{\circ }{{\mathcal M}}{}^{\varphi _0}_{q_0}(\Omega ), \overset{\circ }{{\mathcal M}}{}^{\varphi _1}_{q_1}(\Omega ) ]_{\theta }$ as desired. $\square$

Appendix: a function \(f \in {\mathcal M}^p_q \setminus (L^1+L^\infty )\)

We aim here to present an example of a function $f \in {\mathcal M}^p_q \setminus (L^1+L^\infty )$. Let $n=1$ for simplicity. Define

$$\begin{aligned} f=f_p:=\sum _{j=100}^\infty [\log _2\log _2 j]^{1/p}\chi _{[j!,j!+[\log _2\log _2 j]^{-1}]}. \end{aligned}$$

(6.1)

Lemma 6.1

Let $1 \le q<p<\infty$. Then f given by (6.1) belongs to ${\mathcal M}^p_q$ but does not belong to $L^1+L^\infty$.

Proof

Let (a, b) be an interval which intersects the support of f.

• 1. Case 1 : $b-a<2$. In this case, there exists uniquely $j \in {\mathbb N} \cap [100,\infty )$ such that $[a,b] \cap [j!,j!+[\log _2\log _2 j]^{-1}] \ne \emptyset$. Thus,

  $$\begin{aligned}&(b-a)^{\frac{1}{p}-\frac{1}{q}} \left( \int _a^b f(t)^q\,dt\right) ^{\frac{1}{q}}\\&\quad = (b-a)^{\frac{1}{p}-\frac{1}{q}} \left( \int _{\max (a,j!)}^{\min (b,j!+[\log _2\log _2 j]^{-1})} f(t)^q\,dt \right) ^{\frac{1}{q}}\\&\quad \le (\min (b,j!+[\log _2\log _2 j]^{-1})-\max (a,j!))^{\frac{1}{p}-\frac{1}{q}}\\&\qquad \left( \int _{\max (a,j!)}^{\min (b,j!+[\log _2\log _2 j]^{-1})} f(t)^q\,dt \right) ^{\frac{1}{q}}\\&\quad = [\log _2\log _2 j]^{\frac{1}{p}} (\min (b,j!+[\log _2\log _2 j]^{-1})-\max (a,j!))^{\frac{1}{p}}\\&\quad \le 1. \end{aligned}$$

• 2. Case 2 : $b-a>2$. Set

  $$\begin{aligned} m := \min ([a,b] \cap \mathrm{supp}(f)), \quad M := \max ([a,b] \cap \mathrm{supp}(f)). \end{aligned}$$

  Choose $j_m,j_M \in {\mathbb N} \cap [100,\infty )$ so that $m \in [j_m!,j_m!+j_m{}^{-1}]$ and $M \in [j_M!,j_M!+j_M{}^{-1}]$. If $j_M-j_m \le 2$, then we go through a similar argument as before. Assume $j_M-j_m \ge 3$. Then we have

  $$\begin{aligned} b-a \ge M-m \ge j_M!-j_m!-j_m{}^{-1} \ge j_M!-j_m!-1. \end{aligned}$$

  Thus,

  $$\begin{aligned} (b-a)^{\frac{1}{p}-\frac{1}{q}} \left( \int _a^b f(t)^q\,dt\right) ^{\frac{1}{q}}&\le (j_M!-j_m!-1)^{\frac{1}{p}-\frac{1}{q}} \left( \int _{j_m!}^{j_M!+1} f(t)^q\,dt\right) ^{\frac{1}{q}}\\&\le C j_M!{}^{\frac{1}{p}-\frac{1}{q}} \left( \sum _{j=j_m}^{j_M} (\log _2 \log _2 j)^{\frac{q-p}{p}}\right) ^{\frac{1}{q}}\\&\le C. \end{aligned}$$

Thus, $f \in {\mathcal M}^p_q$.

Now we disprove $f \in L^1+L^\infty$. Let R be fixed. Then a geometric observation shows that

$$\begin{aligned} \Vert f-\min (f,R)\Vert _{L^1} \le \Vert f-h\Vert _{L^\infty } \end{aligned}$$

for any $h \in L^\infty$ with $\Vert h\Vert _{L^\infty }\le R$.

Let $S>2R+2$ be an integer. Then

$$\begin{aligned}&\int _{f=S}(f-\min (f,R)) = |\{f=S\}|(S-R) \ge \frac{S}{2}|\{f=S\}| \\&\quad = \frac{S}{2}\sum _{k=2^{2^S}}^{2^{2^{S+1}}} \frac{1}{k} \ge C S(2^{S+1}-2^S). \end{aligned}$$

Thus, $\Vert f-\min (f,R)\Vert _{L^1}=\infty .$ Hence, $f \notin L^1+L^\infty$.

Remark 6.2

For the case in ${\mathbb R}^n$ with $n>1$, we can consider

$$\begin{aligned} f(x)=f(x_1,\ldots ,x_n):=\prod _{j=1}^n f_p(x_j), \end{aligned}$$

where $f_p(x_j)$ is defined in (6.1).