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.

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).