Interpolation of generalized Morrey spaces

Abstract

In this paper, we shall establish a theory of interpolation of generalized Morrey spaces. We use the complex interpolation methods. Our results extend the interpolation results for Morrey spaces which are discussed by Lu et al. (Can Math Bull 57:598–608, 2014), and also Lemarié-Rieusset (2014). We establish the interpolation of generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also consider the closure of the functions which are essentially bounded and have compact support. The second interpolation of such spaces will yield a class of closed spaces; we describe the second interpolation of the closure of the functions which are essentially bounded and have compact support. This result will carry over to generalized Morrey spaces, generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also give several examples that explain the subtlety of proving the interpolation of Morrey spaces.

Appendix: Examples of functions

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

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

The bidual of $\widetilde{\mathcal M}^p_q$ is known to be ${\mathcal M}^p_q$ [38, Theorem 1.3]. However, we can show that it is a closed proper subspace of the closure ${\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}$ as the following example shows:

Example 1

For simplicity, we assume $(p,p_0,p_1,q,q_0,q_1)=(4,2,6,2,1,3).$ However, we can readily pass to the general case. Define $f_\mathbf{e}(x) := \frac{3}{4}\mathbf{e} + \frac{1}{4} x$, where $\mathbf{e} \in \{0, 1\}^n$. Note that each $f_\mathbf{e}$ maps $[0, 1]^n$ to $[0, 1]^n$ injectively. Define $g^k_\mathbf{e}(x) := 4^{k+1}f_\mathbf{e}(4^{-k}x)$. Let $E_0 = [0, 1]^n$ and define $E_k$ inductively by

$$\begin{aligned} E_{k+1}:= \bigcup _{\mathbf{e} \in \{0, 1\}^n } g^{k+1}_\mathbf{e}(E_k) \subset [0,4^{k+1}]^n. \end{aligned}$$

Note that $E_k$ is a subset of $E_{k+1}$ and that $E_k$ is made up of $2^{kn}$ disjoint cubes of volume 1. Thus,

$$\begin{aligned} \Vert \chi _{E_k}\Vert _{{\mathcal M}^4_2} \sim \Vert \chi _{E_k}\Vert _{{\mathcal M}^6_3} \sim \Vert \chi _{E_k}\Vert _{{\mathcal M}^8_4} \sim 1. \end{aligned}$$

Let $f=\lim \nolimits _{k \rightarrow \infty }\chi _{E_k}$. Since g is an unbounded set made up of disjoint union of cubes having volume 1,

$$\begin{aligned} \Vert f-g\Vert _{{\mathcal M}^6_3}\ge 1 \end{aligned}$$

for all $g \in L^\infty _\mathrm{c}$. Thus $f \notin \widetilde{\mathcal M}^6_3$.

Let us obtain an intrinsic description of $\overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}$.

Lemma 36

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

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

If $f \in \overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}$, then there exists a function $\{g_j\}_{j=1}^\infty \subset L^\infty$ such that $\{\chi _{\{g_j \ne 0\}}\log |g_j|\}_{j=1}^\infty \subset L^\infty$ and that $g_j \rightarrow f$ in ${\mathcal M}^p_q$. In particular, the space $\overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}$ does not depend on $p_0,q_0,p_1$ and $q_1$.

Proof

From the definition of the closure, we may assume that $f \in {\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}$. In this case, we can take $g_j=\chi _{\{2^{-j} \le |f| \le 2^j\}}f$ for $j \in {\mathbb N}$.

Before proving Proposition 1, let us recall the relation between seven function spaces and then let us state the plan of the proving Proposition 1. Note that we have five different spaces in view of (14) and (16).

• 1. In view of (9) and (11), we prove that $\overset{\diamond }{\mathcal M}{}^p_q$ is different from any other space.

• 2. $[\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta$ is different from any other space.

• 3. We check (13).

Remark 2

Proposition 1 is proved as follows:

• 1. We shall show that $[\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta$ is different from other four function spaces and that inclusions (13) is strict. In view of (11), (13), and (17), it suffices to compare $[\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta$ with $\overset{*}{\mathcal M}{}^p_q$ and $\overset{\diamond }{\mathcal M}{}^p_q$.

  • (a) If we define $f(x):=|x|^{-n/p}\chi _{B(0,1)}(x)$, then f violates [42, (2.8)]. This means that $f \notin \overset{\diamond }{\mathcal M}{}^p_q$. By Theorem 1, we have $f \in [\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta$. If we mollify f in Example 1, then we obtain a function in $\overset{\diamond }{\mathcal M}{}^p_q$. But this function is not in $[\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta$ from the criterion of Theorem 1.

  • (b) Let us prove (13). To this end, we may assume $f \in L^\infty _\mathrm{c}$ because the both function spaces have a common equivalent norm ${\mathcal M}^p_q$. Assuming $f \in L^\infty _\mathrm{c}$, we have $\chi _{[a,b]}(|f|)f \in L^\infty _\mathrm{c}$. Thus, in the light of the criterion of Theorem 1, we have $f \in [\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta$. We need to show that the inclusion is strict. If we define $f(x):=|x|^{-n/p}(1-\chi _{B(0,1)}(x))$, then f violates [42, (2.9)]. This means that $f \notin \overset{*}{\mathcal M}{}^p_q$. Meanwhile by Theorem 1, we have $f \in [\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta$.

  • (c) We learn $f \in \overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q} {\setminus } [\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta \ne \emptyset$, where f is the function in Example 1 in the present paper. Meanwhile, Lemma 36 shows that $f(x)=|x|^{-n/p} \in [\widetilde{\mathcal M}^{p_0}_{q_0},\widetilde{\mathcal M}^{p_1}_{q_1}]^\theta {\setminus } \overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}.$

• 2. By using these results, we can check that we do not have any other inclusion and that the inclusions are strict.

  • (a) We have no inclusionship $\overset{\diamond }{\mathcal M}{}^p_q$ and $\overset{*}{\mathcal M}{}^p_q$ according to [42, Lemma 2.35]. Hence, $\overset{\diamond }{\mathcal M}{}^p_q$ is different from any other spaces in view of (17) and the same applies to $\overset{*}{\mathcal M}{}^p_q$ in view of the function defined in [42, (2.12)].

  • (b) From the function f in Example 1, $\overline{{\mathcal M}^{p_0}_{q_0} \cap {\mathcal M}^{p_1}_{q_1}}^{{\mathcal M}^p_q}$ is different from $\widetilde{\mathcal M}^p_q$.

The Morrey space ${\mathcal M}^p_q$ contains $|x|^{-n/p}$ when $1 \le q<p<\infty$. If we start with this function, we learn that the complex function $|x|^{\frac{-n-it}{p}}$ does not belong to the desired space as the following proposition shows:

Proposition 4

Let $1 \le q<p<\infty$ and $f(t,x):=|x|^{\frac{-n-it}{p}}$. Then $t \in {\mathbb R} \mapsto f(t,\cdot ) \in {\mathcal M}^p_q$ is not continuous.

Proof

It suffices to disprove continuity at $t=0$. Let $|t|<p$. Note that

$$\begin{aligned} |f(t,x)-f(0,x)|=2|x|^{-\frac{n}{p}} \left| \sin \left( \frac{t}{2p}\log |x|\right) \right| \end{aligned}$$

and hence if $R=\exp (t^{-1}p)$, then

$$\begin{aligned}&\Vert f(t,\cdot )-f(0,\cdot )\Vert _{{\mathcal M}^p_q}\\&\quad \ge 2 |B(0,2R)|^{\frac{1}{p}-\frac{1}{q}} \left( \int _{B(0,2R) {\setminus } B(0,R)}|x|^{-\frac{nq}{p}} \left| \sin \left( \frac{t}{2p}\log |x|\right) \right| ^q\,dx \right) ^{\frac{1}{q}}\\&\quad \ge 2 \sin \frac{1}{2}\cdot |B(0,2R)|^{\frac{1}{p}-\frac{1}{q}} \left( \int _{B(0,2R) {\setminus } B(0,R)}|x|^{-\frac{nq}{p}}\,dx \right) ^{\frac{1}{q}}\\&\quad \ge 2 \sin \frac{1}{2}\cdot |B(0,R)|^{\frac{1}{p}-\frac{1}{q}} \left( \int _{B(0,2R) {\setminus } B(0,R)}(2R)^{-\frac{nq}{p}}\,dx \right) ^{\frac{1}{q}}\\&\quad \ge C_{p,q}>0, \end{aligned}$$

as was to be shown.

Proposition 5

Let $1 \le q<p<\infty$ and $f(t,x):=|x|^{\frac{-n-it}{p}}$. Define

$$\begin{aligned} F(t,x):=\int _0^t f(s,x)\,ds. \end{aligned}$$

Then $t \in {\mathbb R} \mapsto F(t,\cdot ) \in {\mathcal M}^p_q$ is Lipschitz continuous but nowhere differentiable.

Proof

Lipschitz continuity of the function F follows from Lemma 12. Let us disprove that f is differentiable. We calculate

$$\begin{aligned} \frac{F(t_2,x)-F(t_1,x)}{t_2-t_1}-f(t_1,x) = \frac{|x|^{-\frac{n+i t_1}{p}}}{t_2-t_1} \int _{t_1}^{t_2} |x|^{-\frac{i(s-t_1)}{p}}-1\,ds. \end{aligned}$$

Note that

$$\begin{aligned} \mathrm{Im}\left( |x|^{-\frac{i(s-t_1)}{p}}-1\right) =- \sin \left( \frac{s-t_1}{p}\log |x|\right) \end{aligned}$$

and hence for $|x|>1$, and $t_2>t_1$

$$\begin{aligned}&\left| \frac{F(t_2,x)-F(t_1,x)}{t_2-t_1}-f(t_1,x)\right| \\&\quad \ge \frac{|x|^{-\frac{n}{p}}}{t_2-t_1} \left| \int _{t_1}^{t_2} \sin \left( \frac{s-t_1}{p}\log |x|\right) \,ds \right| \\&\quad = \frac{p|x|^{-\frac{n}{p}}}{(t_2-t_1)\log |x|} \left( 1-\cos \frac{t_2-t_1}{p}\log |x|\right) . \end{aligned}$$

Fix $t_1,t_2 \in {\mathbb R}$ so that $0<t_2-t_1<10^{-1}p$. Let D be the annulus given by

$$\begin{aligned} D:=\left\{ x \in {\mathbb R}^n:\frac{1}{2}<\frac{t_2-t_1}{p}\log |x|<1 \right\} , \end{aligned}$$

which does not intersect the unit ball $|x|<1$. Note that $\sup _D |x| \le 3\inf _D |x|.$ Therefore,

$$\begin{aligned}&\left\| \frac{F(t_2,\cdot )-F(t_1,\cdot )}{t_2-t_1}-f(t_1,\cdot ) \right\| _{{\mathcal M}^p_q}\\&\quad \ge \left\| \frac{p|\cdot |^{-\frac{n}{p}}}{(t_2-t_1)\log |\cdot |} \left( 1-\cos \frac{t_2-t_1}{p}\log |\cdot |\right) \chi _D \right\| _{{\mathcal M}^p_q}\\&\quad \ge p\left( 1-\cos \frac{1}{2}\right) \left\| |\cdot |^{-\frac{n}{p}}\chi _D \right\| _{{\mathcal M}^p_q} \ge c_p>0. \end{aligned}$$

Therefore,

$$\begin{aligned} \lim _{t_2 \rightarrow t_1} \frac{F(t_2)-F(t_1)}{t_2-t_1}=f(t_1) \end{aligned}$$

(94)

does not take place in ${\mathcal M}^p_q$. However, we know that (94) holds in the topology of ${\mathcal S}'$. Therefore, if F were differentiable at $t=t_1$, then the derivative would be $f(t_1)$. This is a contradiction.

Finally, we conclude this paper with a remedy of this problem.

Proposition 6

Let $1<q_0 \le p_0<\infty$ and $1<q_1 \le p_1<\infty$. If $f \in {\mathcal G}({\mathcal M}^{p_0}_{q_0},{\mathcal M}^{p_1}_{q_1})$, then the limit

$$\begin{aligned} f'(j+it)=\lim _{h \rightarrow 0}\frac{f(j+i(t+h))-f(j+it)}{ih} \end{aligned}$$

exists for almost all t in the weak-* topology of ${\mathcal M}^{p_j}_{q_j}$ for $j=0,1$.

Proof

We just combine that the predual space of the Morrey space ${\mathcal M}^{p_j}_{q_j}$ is separable and the Rademacher theorem.