# On the general K-interpolation method for the sum and the intersection

## Abstract

Let $$(A_{0}, A_{1})$$ be a compatible couple of normed spaces. We study the interrelation of the general K-interpolation spaces of the couple $$(A_{0}+A_{1}, A_{0} \cap A_{1})$$ with those of the couples $$(A_{0}, A_{1} )$$, $$(A_{0}+A_{1}, A_{0} )$$, $$(A_{0}+A_{1}, A_{1} )$$, $$(A_{0}, A_{0} \cap A_{1})$$, and $$(A_{1}, A_{0} \cap A_{1})$$.

## 1 Introduction

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, i.e. we assume that both $$A_{0}$$ and $$A_{1}$$ are continuously embedded in a topological vector space $$\mathscr{A}$$. The sum of $$A_{0}$$ and $$A_{1}$$, denoted by $$A_{0}+ A_{1}$$, is the set of elements $$f \in \mathscr{A}$$ that can be represented as $$f=f_{0}+f_{1}$$ where $$f_{0} \in A_{0}$$ and $$f_{1} \in A_{1}$$. The norm on the sum space $$A_{0}+ A_{1}$$ is given by

$$\|f\|_{A_{0}+A_{1}}=\inf\bigl\{ \|f_{0}\|_{A_{0}}+\|f_{1} \|_{A_{1}}: f_{0} \in A_{0}, f_{1} \in A_{1}, f=f_{0}+f_{1}\bigr\} .$$

The norm on the intersection space $$A_{0} \cap A_{1}$$ is given by

$$\|f\|_{A_{0} \cap A_{1}}=\max\bigl\{ \|f\|_{A_{0}}, \|f\|_{A_{1}}\bigr\} .$$

The Peetre’s K-functional is defined, for each $$f\in A_{0}+A_{1}$$ and $$t>0$$, by

$$K(t,f;A_{0},A_{1})=\inf\bigl\{ \|f_{0} \|_{A_{0}}+t\|f_{1}\|_{A_{1}}: f_{0} \in A_{0}, f_{1} \in A_{1}, f=f_{0}+f_{1} \bigr\} .$$

Let Φ be a normed space of Lebesgue measurable functions, defined on $$(0,\infty)$$, with monotone norm: $$|g|\leq |h|$$ implies $$\|g\|_{\Phi}\leq \|h\|_{\Phi}$$. Further assume that

$$t\longmapsto\min\{1,t\}\in\Phi.$$
(1.1)

By definition, the general K-interpolation space $$(A_{0},A_{1})_{\Phi}$$ is a subspace of $$A_{0}+A_{1}$$ having the following norm:

$$\|f\|_{(A_{0},A_{1})_{\Phi}}=\bigl\| K(t,f;A_{0},A_{1})\bigr\| _{\Phi}.$$

Here Φ is often termed the parameter of the K-interpolation method. We refer to  for a complete treatment of the general K-interpolation method.

Set

$$\Gamma= \bigl([0,1] \times [1,\infty] \bigr)\setminus \bigl( \{0,1\}\times [1, \infty) \bigr).$$

Let $$(\theta,p)\in \Gamma$$, then the classical scale of K-interpolation spaces $$(A_{0}, A_{1})_{\theta,q}$$ (see  or ) is obtained when Φ is taken to be the weighted Lebesgue space $$L_{q}(t^{-\theta})$$ defined by the norm

$$\Vert g\Vert _{\Phi} =\left \{ \textstyle\begin{array} {l@{\quad}l} ( \int_{0}^{\infty}t^{-\theta} \vert g(t)\vert ^{p}\frac{dt}{t} ) ^{1/p}, & (\theta, p)\in (0,1)\times [1,\infty),\\ \sup_{0< t< \infty} t^{-\theta} \vert g(t)\vert , & (\theta,p)\in [0,1]\times\{\infty\}. \end{array}\displaystyle \right .$$

The following identity was proved by Maligranda :

$$(A_{0}+A_{1}, A_{0}\cap A_{1})_{\theta, p} =\left \{ \textstyle\begin{array} {l@{\quad}l}(A_{0},A_{1})_{\theta,p}+(A_{0},A_{1})_{1-\theta,p}, & (\theta,p) \in \Gamma_{1},\\ (A_{0},A_{1})_{\theta,p}\cap(A_{0},A_{1})_{1-\theta,p}, & (\theta,p) \in \Gamma_{2}, \end{array}\displaystyle \right .$$
(1.2)

where

$$\Gamma_{1}= \bigl([0,1/2\bigr) \times [1,\infty] )\setminus \bigl( \{0 \}\times [1,\infty) \bigr)$$

and

$$\Gamma_{2}= \bigl([1/2,1] \times [1,\infty] \bigr)\setminus \bigl( \{1 \}\times [1,\infty) \bigr).$$

Subsequently, Maligranda  considered the K-interpolation spaces $$(A_{0}, A_{1})_{\varrho, p}$$, which are obtained when Φ is given by

$$\Vert g\Vert _{\Phi} =\left \{ \textstyle\begin{array} {l@{\quad}l} ( \int_{0}^{\infty} (\frac{\vert g(t)\vert }{\varrho(t)} ) ^{p}\frac{dt}{t} ) ^{1/p}, & 1\leq p< \infty,\\ \sup_{0< t< \infty}\frac{\vert g(t)\vert }{\varrho(t)}, & p=\infty, \end{array}\displaystyle \right .$$

and extended the identity (1.2) by imposing certain monotonicity conditions on the parameter function ϱ. Another related identity, proved by Persson , states that

$$(A_{0}+A_{1}, A_{0}\cap A_{1})_{\varrho, p}=(A_{0}+A_{1}, A_{0})_{\varrho, p}\cap (A_{0}+A_{1}, A_{1})_{\varrho, p}.$$

Recently, Haase  has completely described how the classical K-interpolation spaces for the couples $$(A_{0}, A_{1} )$$, $$(A_{0}+A_{1}, A_{0} )$$, $$(A_{0}+A_{1}, A_{1} )$$, $$(A_{0}, A_{0} \cap A_{1})$$, $$(A_{1}, A_{0} \cap A_{1})$$, and $$({A_{0}+A_{1}}, A_{0} \cap A_{1})$$ interrelate. The assertions (1.5)-(1.12) in , Theorem 1.1, concern the spaces $$(A_{0}+A_{1}, A_{0}\cap A_{1})_{\theta, p}$$, and the goal of this paper is to extend these assertions by means of replacing the classical scale $$(A_{0}, A_{1})_{\theta, p}$$ by the general scale $$(A_{0}, A_{1})_{\Phi}$$.

The main ingredient of our proofs will be the estimate in Proposition 2.4 (see below) which relates the K-functional of the couple $$(A_{0}+A_{1}, A_{0}\cap A_{1})$$ with that of the original couple $$(A_{0}, A_{1})$$, whereas this estimate has not been used in . Consequently, our arguments of the proofs are different from those in .

We will also apply our general results to the limiting K-interpolation spaces $$(A_{0}, A_{1})_{0,p;K}$$ and $$(A_{0}, A_{1})_{1,p;K}$$ recently introduced by Cobos, Fernández-Cabrera, and Silvestre . Namely, if the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ are given by the norms

$$\|g\|_{\Phi_{0}} = \biggl( \int_{0}^{1} \bigl|g(s)\bigr|^{p}\frac{ds}{s} \biggr)^{\frac{1}{p}} + \sup_{s>1}\bigl|g(s)\bigr|$$
(1.3)

and

$$\|g\|_{\Phi_{1}} = \sup_{0< s< 1} \frac{|g(s)|}{s} + \biggl( \int_{1}^{\infty}\biggl(\frac{|g(s)|}{s} \biggr)^{p}\frac{ds}{s} \biggr)^{\frac{1}{p}},$$
(1.4)

where $$1\leq p <\infty$$, then $$(A_{0}, A_{1})_{\Phi_{0}}= (A_{0}, A_{1})_{0,p;K}$$ and $$(A_{0}, A_{1})_{\Phi_{1}}= (A_{0}, A_{1})_{1,p;K}$$. Note that, for limiting values $$\theta=0,1$$, the space $$(A_{0}, A_{1})_{\theta, p}$$ is trivial (containing only zero element) when p is finite. The space $$(A_{0}, A_{1})_{0,p;K}$$ corresponds to the limiting value $$\theta=0$$, and the space $$(A_{0}, A_{1})_{1,p;K}$$ corresponds to the limiting value $$\theta=1$$. We will, for convenience, write $$(A_{0}, A_{1})_{\{0\}, p}$$ for $$(A_{0}, A_{1})_{0,p;K}$$, and $$(A_{0}, A_{1})_{\{1\}, p}$$ for $$(A_{0}, A_{1})_{1,p;K}$$.

The paper is organised as follows. In Section 2, we establish all necessary background material, whereas Section 3 contains the main results.

## 2 Background material

In the following we will use the notation $$A \lesssim B$$ for non-negative quantities to mean that $$A\leq c B$$ for some positive constant c which is independent of appropriate parameters involved in A and B. If $$A \lesssim B$$ and $$B \lesssim A$$, we will write $$A\approx B$$. Moreover, we will use the symbol $$X \hookrightarrow Y$$ to show that X is continuously embedded in Y.

The elementary but useful properties of the K-functional are collected in the following proposition.

### Proposition 2.1



Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces. Then $$K(t,f; A_{0},A_{1})$$ is non-decreasing in t, and $$K(t,f;A_{0},A_{1})/t$$ is non-increasing in t. Moreover, we have

\begin{aligned}& K(t,f;A_{0},A_{1})\leq \|f\|_{A_{0}},\quad f \in A_{0}, t>0; \end{aligned}
(2.1)
\begin{aligned}& K(t,f;A_{0},A_{1})\leq t\|f\|_{A_{1}},\quad f\in A_{1}, t>0; \end{aligned}
(2.2)
\begin{aligned}& K(t,f;A_{0},A_{1})=tK\bigl(t^{-1},f;A_{1},A_{0} \bigr),\quad f\in A_{0}+A_{1}, t>0; \end{aligned}
(2.3)
\begin{aligned}& K(t,f+g;A_{0},A_{1})\leq K(t,f;A_{0},A_{1})+K(t,g;A_{0},A_{1}),\quad f,g\in A_{0}+A_{1}, t>0. \end{aligned}
(2.4)

In the next three propositions, we describe some formulas which relate the K-functional of the couples $$(A_{0}+A_{1}, A_{1})$$, $$(A_{0},A_{0}\cap A_{1})$$, and $$(A_{0}+A_{1}, A_{0}\cap A_{1})$$ with that of the original couple $$(A_{0},A_{1})$$.

### Proposition 2.2

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$f\in A_{0} + A_{1}$$. Then

$$K(t,f;A_{0}+ A_{1},A_{1} ) = K(t,f;A_{0},A_{1}), \quad 0< t < 1.$$

### Proof

In view of (2.3), the proof follows immediately from the following relation:

$$K(t,f;A_{0},A_{0} + A_{1})= K(t,f;A_{0},A_{1}),\quad t>1,$$
(2.5)

which has been derived in , Lemma 2.1. □

For the proof of the next result, we refer to , Lemma 2.3.

### Proposition 2.3

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$f\in A_{0}$$. Then

$$K(t,f;A_{0},A_{0}\cap A_{1}) \lesssim K(t,f;A_{0},A_{1}) + t \|f\|_{A_{0}},\quad 0 < t < 1.$$

The next result is derived in , Theorem 3.

### Proposition 2.4

Let $$(A_{0},A_{1})$$ be a compatible couple of normed space, and let $$f\in A_{0}+A_{1}$$. Then

$$K(t,f;A_{0} + A_{1},A_{0}\cap A_{1}) \approx K(t,f;A_{0}, A_{1}) + tK\bigl(t^{-1},f;A_{0} , A_{1}\bigr), \quad 0< t< 1.$$

In our proofs, we will make use of the fact that, for a parameter space Φ, both $$\|s\chi_{(0,1)}(s)\|_{\Phi}$$ and $$\|\chi_{(1,\infty)}\|_{\Phi}$$ are finite. This fact is a simple consequence of (1.1). Moreover, in view of the monotonicity of the norm $$\|\cdot\|_{\Phi}$$ and the fact that $$K(t,f;A_{0},A_{1})=\|f\|_{A_{0}+A_{1}}$$, we have

$$\|f\|_{(A_{0},A_{1})_{\Phi}}\approx\bigl\| \chi_{(0,1)}(t)K(t,f;A_{0},A_{1})\bigr\| _{\Phi}+\bigl\| \chi_{(1,\infty)}(t)K(t,f;A_{0},A_{1})\bigr\| _{\Phi}.$$
(2.6)

We will make use of the next result, without explicitly mentioning it, in our proofs.

### Proposition 2.5

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that $$A_{1} \hookrightarrow A_{0}$$. Then

$$\|f\|_{(A_{0},A_{1})_{\Phi}} \approx \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi}.$$

### Proof

It will suffice to derive

$$\|f\|_{(A_{0},A_{1})_{\Phi}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi},$$
(2.7)

as the converse estimate is trivial. Using (2.6) and (2.1), we get

\begin{aligned} \|f\|_{(A_{0},A_{1})_{\Phi}} \lesssim& \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi} + \|f\|_{A_{1}} \|\chi_{(1,\infty)}\|_{\Phi} \\ \approx& \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi} + \|f\|_{A_{1}}, \end{aligned}

as our assumption $$A_{1} \hookrightarrow A_{0}$$ implies that $$\|f\|_{A_{0}}\approx \|f\|_{A_{0}+A_{1}}$$, so

$$\|f\|_{(A_{0},A_{1})_{\Phi}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi} + \|f\|_{A_{0}+A_{1}}.$$
(2.8)

Since $$K(t,f;A_{0},A_{1})/t$$ is non-increasing in t, we obtain

$$\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi}\geq K(1,f;A_{0},A_{1})\|s \chi_{(0,1)(s)}\|_{\Phi},$$

which gives

$$\|f\|_{A_{0}+A_{1}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi}.$$
(2.9)

Now (2.7) follows from (2.8) and (2.9). The proof is complete. □

## 3 Main results

### Theorem 3.1

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces. Then, for an arbitrary parameter space Φ, we have with equivalent norms

$$(A_{0} + A_{1},A_{0})_{\Phi} \cap (A_{0} + A_{1},A_{1})_{\Phi} = (A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi}.$$

### Proof

Put $$B_{0}= (A_{0} + A_{1},A_{0})_{\Phi}$$, $$B_{1}= (A_{0} + A_{1},A_{1})_{\Phi}$$, and $$B=(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi}$$. Let $$f \in A_{0}+A_{1}$$. Then by Proposition 2.4

$$\|f\|_{B} \approx \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi} + \bigl\| \chi_{(0,1)}(s)sK\bigl(s^{-1},f;A_{0},A_{1} \bigr)\bigr\| _{\Phi},$$

next making use of (2.3), we arrive at

$$\|f\|_{B} \approx \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi} + \bigl\| \chi_{(0,1)}(s)K(s,f;A_{1},A_{0}) \bigr\| _{\Phi}.$$

Finally, appealing to Proposition 2.2, we get

$$\|f\|_{B} \approx \|f\|_{B_{0}} + \|f\|_{B_{1}},$$

which concludes the proof. □

### Remark 3.2

The result of Theorem 3.1 generalizes the assertion (1.5) in , Theorem 1.1.

### Theorem 3.3

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces. Then, for an arbitrary parameter space Φ, we have with equivalent norms

$$(A_{0}, A_{0}\cap A_{1})_{\Phi} + (A_{1}, A_{0}\cap A_{1})_{\Phi} = (A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi}.$$

### Proof

Put $$B_{0}=(A_{0}, A_{0}\cap A_{1})_{\Phi}$$, $$B_{1}=(A_{1}, A_{0}\cap A_{1})_{\Phi}$$ and $$B=(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi}$$. Let $$f \in B_{0}+B_{1}$$, and take an arbitrary decomposition $$f = f_{0} + f_{1}$$ with $$f_{0} \in B_{0}$$ and $$f_{1} \in B_{1}$$. Then by (2.4), we have

\begin{aligned} \|f\|_{B} \lesssim & \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0} + A_{1},A_{0}\cap A_{1})\bigr\| _{\Phi} \\ &{}+\bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0} + A_{1},A_{0}\cap A_{1})\bigr\| _{\Phi}, \end{aligned}

now applying the simple fact that

$$K(t,f_{j};A_{0} + A_{1},A_{0}\cap A_{1})\leq K(t,f_{j};A_{j},A_{0}\cap A_{1}) \quad (j=0,1), t>0,$$

we obtain

$$\|f\|_{B} \lesssim \|f_{0}\|_{B_{0}}+ \|f_{1}\|_{B_{1}},$$

from which the estimate $$\|f\|_{B}\lesssim \|f\|_{B_{0}+ B_{1}}$$ follows as the decomposition $$f = f_{0} + f_{1}$$ is arbitrary. In order to establish the converse estimate, we take $$f \in B$$ and note that there exists (by definition of the norm on $$A_{0}+A_{1}$$) a particular decomposition $$f=f_{0}+f_{1}$$ with $$f_{0} \in A_{0}$$ and $$f_{1} \in A_{1}$$ such that

$$\|f_{0}\|_{A_{0}}+\|f_{1} \|_{A_{1}}\lesssim \|f\|_{A_{0}+A_{1}}.$$
(3.1)

By Proposition 2.3,

\begin{aligned} \|f_{0}\|_{B_{0}} \lesssim& \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi} + \bigl\| s\chi_{(0,1)}(s)\bigr\| _{\Phi} \|f_{0} \|_{A_{0}} \\ \approx& \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi} + \|f_{0}\|_{A_{0}}, \end{aligned}

since $$f_{0}=f-f_{1}$$, we get by (2.4)

$$\|f_{0}\|_{B_{0}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0}, A_{1})\bigr\| _{\Phi} + \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0}, A_{1})\bigr\| _{\Phi} + \|f_{0}\|_{A_{0}},$$

next we use (2.2) to obtain

\begin{aligned} \|f_{0}\|_{B_{0}} \lesssim& \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0}, A_{1})\bigr\| _{\Phi} + \bigl\| s\chi_{(0,1)}(s)\bigr\| _{\Phi} \|f_{1}\|_{A_{1}} + \|f_{0}\|_{A_{0}} \\ \approx&\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0}, A_{1}) \bigr\| _{\Phi} + \|f_{1}\|_{A_{1}} + \|f_{0} \|_{A_{0}} \end{aligned}

and, using (3.1), we get

$$\|f_{0}\|_{B_{0}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0}, A_{1})\bigr\| _{\Phi} + \|f\|_{A_{0}+A_{1}},$$

in accordance with (2.9), we deduce that

$$\|f_{0}\|_{B_{0}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0}, A_{1})\bigr\| _{\Phi}.$$

Analogously, we can obtain

$$\|f_{1}\|_{B_{1}} \lesssim \bigl\| \chi_{(0,1)}(s)sK \bigl(s^{-1},f;A_{0}, A_{1}\bigr)\bigr\| _{\Phi}.$$

Therefore, combining the previous two estimates, we find that

$$\|f_{0}\|_{B_{0}}+\|f_{1}\|_{B_{1}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0}, A_{1}) \bigr\| _{\Phi} + \bigl\| \chi_{(0,1)}(s)sK\bigl(s^{-1},f;A_{0}, A_{1}\bigr)\bigr\| _{\Phi},$$

from which, in view of Proposition 2.4, it follows that

$$\|f\|_{B_{0}+ B_{1}}\lesssim \|f\|_{B},$$

which completes the proof. □

### Remark 3.4

The result of Theorem 3.3 generalizes the assertion (1.6) in , Theorem 1.1.

In order to formulate the further results, we need the following conditions on the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$:

(C1):

$$\|\chi_{(0,1)}(s)g(s)\|_{\Phi_{0}}\lesssim \|\chi_{(0,1)}(s)g(s)\|_{\Phi_{1}}$$.

(C2):

$$\|\chi_{(0,1)}(s)g(s)\|_{\Phi_{1}}\lesssim \|\chi_{(0,1)}(s)g(s)\|_{\Phi_{0}}$$.

(C3):

$$\|\chi_{(1,\infty)}(s)g(s)\|_{\Phi_{1}}\lesssim\|\chi_{(1,\infty)}(s)g(s)\|_{\Phi_{0}}$$.

(C4):

$$\|\chi_{(1,\infty)}(s)g(s)\|_{\Phi_{0}}\approx \|\chi_{(0,1)}(s)sg(1/s)\|_{\Phi_{1}}$$.

(C5):

$$\|\chi_{(1,\infty)}(s)g(s)\|_{\Phi_{1}}\approx \|\chi_{(0,1)}(s)sg(1/s)\|_{\Phi_{0}}$$.

### Remark 3.5

Let $$(\theta, p)\in \Gamma$$, and assume that $$\Phi_{0}$$ and $$\Phi_{1}$$ are given by the norms

$$\Vert g\Vert _{\Phi_{0}} =\left \{ \textstyle\begin{array} {l@{\quad}l} ( \int_{0}^{\infty}t^{-\theta} \vert g(t)\vert ^{p}\frac{dt}{t} ) ^{1/p}, & (\theta, p)\in (0,1)\times [1,\infty),\\ \sup_{0< t< \infty} t^{-\theta} \vert g(t)\vert , & (\theta,p)\in [0,1]\times\{\infty\}, \end{array}\displaystyle \right .$$
(3.2)

and

$$\Vert g\Vert _{\Phi_{1}} =\left \{ \textstyle\begin{array} {l@{\quad}l} ( \int_{0}^{\infty}t^{1-\theta} \vert g(t)\vert ^{p}\frac{dt}{t} ) ^{1/p}, & (\theta, p)\in (0,1)\times [1,\infty),\\ \sup_{0< t< \infty} t^{1-\theta} \vert g(t)\vert , & (\theta,p)\in [0,1]\times\{\infty\}. \end{array}\displaystyle \right .$$
(3.3)

Then it is easy to see that (C1) and (C3) hold for $$(\theta,p) \in \Gamma_{1}$$, and (C2) holds for $$(\theta,p) \in \Gamma_{2}$$. The conditions (C4) and (C5) hold trivially for all $$(\theta,p) \in \Gamma$$.

### Remark 3.6

Let $$1 \leq p < \infty$$, and assume that $$\Phi_{0}$$ and $$\Phi_{1}$$ are given by (1.3) and (1.4). Then we note that (C1), (C3), (C4), and (C5) hold.

### Theorem 3.7

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C1), (C3) and (C4). Then we have with equivalent norms

$$(A_{0},A_{1})_{\Phi_{0}}\cap(A_{0},A_{1})_{\Phi_{1}}=(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}.$$

### Proof

Put $$B_{0}=(A_{0},A_{1})_{\Phi_{0}}$$, $$B_{1}=(A_{0},A_{1})_{\Phi_{1}}$$ and $$B= (A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}$$. Let $$f\in A_{0}+ A_{1}$$. Then

\begin{aligned} \|f\|_{B_{0}} + \|f\|_{B_{1}} \approx& \bigl\| \chi_{(0,1)}(s)K(s, f; A_{0}, A_{1})\bigr\| _{\Phi_{0}} + \bigl\| \chi_{(1,\infty)}(s)K(s, f; A_{0}, A_{1})\bigr\| _{\Phi_{0}} \\ &{} + \bigl\| \chi_{(0,1)}(s)K(s, f; A_{0}, A_{1}) \bigr\| _{\Phi_{1}} + \bigl\| \chi_{(1,\infty)}(s)K(s, f; A_{0}, A_{1})\bigr\| _{\Phi_{1}}, \end{aligned}

which, in view of (C1) and (C3), reduces to

$$\|f\|_{B_{0}} + \|f\|_{B_{1}} \approx \bigl\| \chi_{(0,1)}(s)K(s,f; A_{0}, A_{1})\bigr\| _{\Phi_{1}} + \bigl\| \chi_{(1,\infty)}(s)K(s, f; A_{0}, A_{1})\bigr\| _{\Phi_{0}},$$

at this point we use (C4) to obtain

$$\|f\|_{B_{0}} + \|f\|_{B_{1}} \approx \bigl\| \chi_{(0,1)}(s)K(s,f; A_{0}, A_{1})\bigr\| _{\Phi_{1}} + \bigl\| \chi_{(0,1)}(s)K \bigl(s^{-1}, f; A_{0}, A_{1}\bigr) \bigr\| _{\Phi_{1}},$$

finally, applying Proposition 2.4, we conclude that

$$\|f\|_{B_{0}} + \|f\|_{B_{1}} \approx \|f\|_{B}.$$

The proof is complete. □

### Remark 3.8

Applying Theorem 3.7 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.2) and (3.3), we get back the result (1.7) in , Theorem 1.1, for $$(\theta,p)\in \Gamma_{1}$$. Note that the case when $$(\theta,p)\in \Gamma_{2}$$ follows from the case when $$(\theta,p)\in \Gamma_{1}$$ by replacing θ by $$1- \theta$$.

### Corollary 3.9

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0},A_{1})_{\{0\},p}\cap(A_{0},A_{1})_{\{1\},p}=(A_{0} + A_{1},A_{0}\cap A_{1})_{\{1\},p}.$$

### Proof

The proof follows by applying Theorem 3.7 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (1.3) and (1.4). □

### Theorem 3.10

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C1), (C3), and (C5). Then we have with equivalent norms

$$(A_{0},A_{1})_{\Phi_{0}}+(A_{0},A_{1})_{\Phi_{1}}=(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{0}}.$$

### Proof

Put $$B_{0}=(A_{0},A_{1})_{\Phi_{0}}$$, $$B_{1}=(A_{0},A_{1})_{\Phi_{1}}$$ and $$B= (A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{0}}$$. Let $$f \in B_{0} + B_{1}$$, and write $$f = f_{0} + f_{1}$$, where $$f_{0} \in B_{0}$$ and $$f_{1}\in B_{1}$$. Now by Proposition 2.4, we have

$$\|f\|_{B} \approx \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{0}} + \bigl\| \chi_{(0,1)}(s)sK\bigl(s^{-1},f;A_{0},A_{1} \bigr)\bigr\| _{\Phi_{0}},$$

using (C5) gives

$$\|f\|_{B} \approx \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{0}} + \bigl\| \chi_{(1,\infty)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}},$$

since $$f=f_{0}+f_{1}$$, so by (2.4), we have

\begin{aligned} \|f\|_{B} \lesssim& \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{0}} + \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0},A_{1}) \bigr\| _{\Phi_{0}} \\ &{} + \bigl\| \chi_{(1,\infty)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \bigl\| \chi_{(1,\infty)}(s)K(s,f_{1};A_{0},A_{1}) \bigr\| _{\Phi_{1}}, \end{aligned}

by (C1) and (C3), we arrive at

\begin{aligned} \|f\|_{B} \lesssim& \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0},A_{1}) \bigr\| _{\Phi_{0}} \\ &{} + \bigl\| \chi_{(1,\infty)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \bigl\| \chi_{(1,\infty)}(s)K(s,f_{1};A_{0},A_{1}) \bigr\| _{\Phi_{0}}, \end{aligned}

which gives

$$\|f\|_{B} \lesssim \|f_{0}\|_{B_{0}}+ \|f_{1}\|_{B_{1}},$$

from which the estimate $$\|f\|_{B}\lesssim \|f\|_{B_{0}+ B_{1}}$$ follows. To derive the other estimate, take $$f \in B$$, and choose a particular decomposition $$f=f_{0}+f_{1}$$, with $$f_{0} \in A_{0}$$ and $$f_{1} \in A_{1}$$, satisfying (3.1). Then

\begin{aligned} \|f_{0}\|_{B_{0}} \approx & \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{0}} + \bigl\| \chi_{(1,\infty)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{0}} \\ \lesssim & \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{0}} + \bigl\| \chi_{(1,\infty)}\|_{\Phi_{0}}\|f_{0} \bigr\| _{A_{0}} \\ \approx & \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{0}} + \|f_{0} \|_{A_{0}}, \end{aligned}

where we have used (2.1). Next proceeding in the same way as in the proof of Theorem 3.3, we obtain

$$\|f_{0}\|_{B_{0}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1})\bigr\| _{\Phi_{0}}.$$
(3.4)

Also, we can show that

$$\|f_{1}\|_{B_{1}}\lesssim \bigl\| \chi_{(1,\infty)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}},$$

which, in view of (C5), becomes

$$\|f_{1}\|_{B_{1}}\lesssim \bigl\| \chi_{(0,1)}(s)sK \bigl(s^{-1},f;A_{0},A_{1}\bigr)\bigr\| _{\Phi_{0}},$$

which, combined with (3.4), yields

$$\|f_{0}\|_{B_{0}}+ \|f_{1}\|_{B_{1}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{0}} + \bigl\| \chi_{(0,1)}(s)sK\bigl(s^{-1},f;A_{0},A_{1} \bigr)\bigr\| _{\Phi_{0}},$$

which, in view of Proposition 2.4, gives

$$\|f_{0}\|_{B_{0}}+ \|f_{1}\|_{B_{1}} \lesssim \|f\|_{B},$$

from which the desired estimate $$\|f\|_{B_{0}+B_{1}}\lesssim \|f\|_{B}$$ follows. The proof of the theorem is finished. □

### Remark 3.11

Theorem 3.10, applied to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.2) and (3.3), gives back (1.8) in , Theorem 1.1.

### Corollary 3.12

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0},A_{1})_{\{0\},p} + (A_{0},A_{1})_{\{1\},p}=(A_{0} + A_{1},A_{0}\cap A_{1})_{\{0\},p}.$$

### Proof

Apply Theorem 3.10 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (1.3) and (1.4). □

### Theorem 3.13

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C1). Then we have with equivalent norms

$$(A_{0},A_{0} \cap A_{1})_{\Phi_{0}} \cap(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}=(A_{0},A_{0} \cap A_{1})_{\Phi_{1}}.$$

### Proof

Denote $$B_{0}=(A_{0},A_{0} \cap A_{1})_{\Phi_{0}}$$, $$B_{1}=(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}$$, and $$B=(A_{0},A_{0}\cap A_{1})_{\Phi_{1}}$$. Let $$f \in A_{0}$$. The estimate $$\|f\|_{B_{0}}+\|f\|_{B_{1}}\lesssim \|f\|_{B}$$ follows thanks to the condition (C1) and the following simple inequality:

$$K(t,f;A_{0}+A_{1}, A_{0} \cap A_{1})\leq K(t,f;A_{0}, A_{0} \cap A_{1}),\quad t>0.$$
(3.5)

To derive the converse estimate, we apply Proposition 2.3 to obtain

$$\|f\|_{B} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \|f\|_{A_{0}}.$$
(3.6)

Next, since $$K(t,f;A_{0},A_{1})/t$$ is non-increasing in t, observe that

$$\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0}, A_{0}\cap A_{1})\bigr\| _{\Phi_{0}}\geq K(1,f;A_{0},A_{0} \cap A_{1})\bigl\| s \chi_{(0,1)}(s)\bigr\| _{\Phi_{0}},$$

noting $$K(1,f;A_{0},A_{0} \cap A_{1})=\|f\|_{A_{0}}$$, we have

$$\|f\|_{A_{0}} \lesssim \|f\|_{B_{0}}.$$
(3.7)

By Proposition 2.4, we also have

$$\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}} \lesssim \|f\|_{B_{1}}.$$
(3.8)

Finally, combining (3.6), (3.7), and (3.8), we obtain $$\|f\|_{B}\lesssim \|f\|_{B_{0}}+\|f\|_{B_{1}}$$. The proof is finished. □

### Remark 3.14

By applying Theorem 3.13 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.2) and (3.3), we get back (1.9) in , Theorem 1.1, for $$(\theta, p)\in \Gamma_{1}$$.

### Corollary 3.15

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0}, A_{0} \cap A_{1})_{\{0\},p} \cap(A_{0} + A_{1},A_{0}\cap A_{1})_{\{1\},p}=(A_{0},A_{0} \cap A_{1})_{\{1\},p}.$$

### Proof

Apply Theorem 3.13 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (1.3) and (1.4). □

### Theorem 3.16

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C2). Then we have with equivalent norms

$$(A_{0},A_{0} \cap A_{1})_{\Phi_{0}} \cap(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}=(A_{0},A_{0} \cap A_{1})_{\Phi_{0}}.$$

### Proof

It will suffice to establish that $$(A_{0},A_{0}\cap A_{1})_{\Phi_{0}}\hookrightarrow (A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}$$. Let $$f\in (A_{0},A_{0}\cap A_{1})_{\Phi_{0}}$$, then by (3.5) we have

$$\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} + A_{1},A_{0} \cap A_{1})\bigr\| _{\Phi_{1}}\leq\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} ,A_{0}\cap A_{1})\bigr\| _{\Phi_{1}},$$

consequently, in view of condition (C2), we obtain

$$\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} + A_{1},A_{0} \cap A_{1})\bigr\| _{\Phi_{1}}\lesssim\bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} ,A_{0}\cap A_{1})\bigr\| _{\Phi_{0}},$$

which concludes the proof. □

### Remark 3.17

For $$(\theta, p)\in \Gamma_{2}$$, the result (1.9) in , Theorem 1.1, follows from Theorem 3.16, applied to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.2) and (3.3).

### Corollary 3.18

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0}, A_{0} \cap A_{1})_{\{1\},p} \cap(A_{0} + A_{1},A_{0}\cap A_{1})_{\{0\},p}=(A_{0},A_{0} \cap A_{1})_{\{1\},p}.$$

### Proof

Apply Theorem 3.16 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by the norms

$$\|g\|_{\Phi_{0}} = \sup_{0< s< 1} \frac{|g(s)|}{s} + \biggl( \int_{1}^{\infty}\biggl(\frac{|g(s)|}{s} \biggr)^{p}\frac{ds}{s} \biggr)^{\frac{1}{p}}$$
(3.9)

and

$$\|g\|_{\Phi_{1}} = \biggl( \int_{0}^{1} \bigl|g(s)\bigr|^{p}\frac{ds}{s} \biggr)^{\frac{1}{p}} + \sup_{s>1}\bigl|g(s)\bigr|.$$
(3.10)

□

### Theorem 3.19

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C 2). Then we have with equivalent norms

$$(A_{0}+A_{1}, A_{1})_{\Phi_{0}}+(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}=(A_{0}+A_{1}, A_{1})_{\Phi_{1}}.$$

### Proof

Put $$B_{0}=(A_{0}+A_{1}, A_{1})_{\Phi_{0}}$$, $$B_{1}=(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}$$, and $$B=(A_{0}+A_{1}, A_{1})_{\Phi_{1}}$$. Let $$f \in B_{0}+ B_{1}$$, and take an arbitrary decomposition $$f = f_{0} + f_{1}$$ with $$f_{0} \in B_{0}$$ and $$f_{1} \in B_{1}$$. Then by (2.4)

$$\|f\|_{B} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0} + A_{1},A_{1})\bigr\| _{\Phi_{1}} + \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0} + A_{1},A_{1})\bigr\| _{\Phi_{1}},$$

using condition (C2) and the fact that

$$K(t,f_{1};A_{0} + A_{1},A_{1})\leq K(t,f_{1};A_{0} + A_{1}, A_{0} \cap A_{1}), \quad t>0,$$

we obtain

$$\|f\|_{B} \lesssim \|f_{0}\|_{B_{0}}+ \|f_{1}\|_{B_{1}},$$

whence, since $$f=f_{0}+ f_{1}$$ is an arbitrary decomposition, we get $$\|f\|_{B} \lesssim \|f\|_{B_{0}+B_{1}}$$. For the converse estimate, let $$f \in B$$, and choose a particular decomposition $$f=f_{0}+f_{1}$$, with $$f_{0} \in A_{0}$$ and $$f_{1} \in A_{1}$$, satisfying (3.1). By Proposition 2.4,

$$\|f_{0}\|_{B_{1}}\approx \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \bigl\| \chi_{(0,1)}(s)sK\bigl(s^{-1},f_{0};A_{0},A_{1} \bigr)\bigr\| _{\Phi_{1}},$$

using (2.1), we obtain

$$\|f_{0}\|_{B_{1}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \|f_{0}\|_{A_{0}},$$

which, since $$f_{0}=f-f_{1}$$, gives

$$\|f_{0}\|_{B_{1}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \|f_{0}\|_{A_{0}},$$

now using (2.2), it follows that

$$\|f_{0}\|_{B_{1}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1})\bigr\| _{\Phi_{1}} + \|f_{1}\|_{A_{1}} + \|f_{0}\|_{A_{0}}.$$
(3.11)

Using (2.2) also gives

\begin{aligned} \|f_{1}\|_{B_{0}} \approx & \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0} + A_{1},A_{1})\bigr\| _{\Phi_{0}} \\ \lesssim & \|f_{1}\|_{A_{1}}\bigl\| \chi_{(0,1)}(s) \bigr\| _{\Phi_{0}} \\ \approx &\|f_{1}\|_{A_{1}}, \end{aligned}

which, together with (3.11), leads to

$$\|f_{0}\|_{B_{1}} + \|f_{1}\|_{B_{0}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \|f_{0}\|_{A_{0}} + \|f_{1} \|_{A_{1}},$$

whence, in view of (3.1), it follows that

$$\|f_{0}\|_{B_{1}} + \|f_{1}\|_{B_{0}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}} + \|f\|_{A_{0}+A_{1}},$$

according to 2.9, we arrive at

$$\|f_{0}\|_{B_{1}} + \|f_{1}\|_{B_{0}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{1}},$$

appealing to Proposition 2.2 yields

$$\|f_{0}\|_{B_{1}} + \|f_{1}\|_{B_{0}} \lesssim \|f\|_{B}$$

from which the desired estimate $$\|f\|_{B_{0}+ B_{1}} \lesssim \|f\|_{B}$$ follows. The proof is complete. □

### Remark 3.20

We recover (1.10) in , Theorem 1.1, for $$(\theta, p)\in \Gamma_{2}$$, by an application of Theorem 3.19 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.2) and (3.3).

### Corollary 3.21

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0} + A_{1}, A_{1})_{\{1\},p} + (A_{0} + A_{1},A_{0}\cap A_{1})_{\{0\},p}=(A_{0}+ A_{1}, A_{1})_{\{0\},p}.$$

### Proof

Apply Theorem 3.19 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.9) and (3.10). □

### Theorem 3.22

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C1). Then we have with equivalent norms

$$(A_{0}+A_{1}, A_{1})_{\Phi_{0}}+(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}=(A_{0}+A_{1}, A_{1})_{\Phi_{0}}.$$

### Proof

It suffices to show that

$$(A_{0} + A_{1}, A_{0}\cap A_{1})_{\Phi_{1}} \hookrightarrow (A_{0} + A_{1},A_{1})_{\Phi_{0}}.$$

Let $$f\in (A_{0} + A_{1}, A_{0}\cap A_{1})_{\Phi_{1}}$$. Then, using condition (C1) and the elementary fact that

$$K(t,f;A_{0}+ A_{1},A_{1})\leq K(t,f;A_{0}+ A_{1}, A_{0} \cap A_{1}), \quad t>0,$$

we have

\begin{aligned} \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} + A_{1},A_{1}) \bigr\| _{\Phi_{0}} \lesssim& \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} + A_{1},A_{1})\bigr\| _{\Phi_{1}} \\ \leq& \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} + A_{1},A_{0} \cap A_{1})\bigr\| _{\Phi_{1}}, \end{aligned}

which finishes the proof. □

### Remark 3.23

Theorem 3.22, applied to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.2) and (3.3), gives back (1.10) in , Theorem 1.1, for $$(\theta, p)\in \Gamma_{1}$$.

### Corollary 3.24

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0} + A_{1}, A_{1})_{\{0\},p} + (A_{0} + A_{1},A_{0}\cap A_{1})_{\{1\},p}=(A_{0}+ A_{1}, A_{1})_{\{0\},p}.$$

### Proof

Apply Theorem 3.22 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (1.3) and (1.4). □

### Theorem 3.25

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C1). Then we have with equivalent norms

$$(A_{0}, A_{0}\cap A_{1})_{\Phi_{0}}+(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}=(A_{0}, A_{1})_{\Psi},$$

where

$$\|g\|_{\Psi}=\bigl\| \chi_{(0,1)}(s)g(s)\bigr\| _{\Phi_{0}}+\bigl\| \chi_{(0,1)}(s)s g(1/s)\bigr\| _{\Phi_{1}}.$$

### Proof

Set $$B_{0}=(A _{0}, A_{0}\cap A_{1})_{\Phi_{0}}$$ and $$B_{1}=(A_{0} + A_{1}, A_{0} \cap A_{1})_{\Phi_{1}}$$. Let $$f \in B_{0} +B_{1}$$, and write $$f = f_{0} + f_{1}$$ with $$f_{0} \in B_{0}$$ and $$f_{1} \in B_{1}$$. Making use of (2.4), we have

$$\|f\|_{(A_{0},A_{1})_{\Psi}}\lesssim I_{1} +I_{2},$$
(3.12)

where

$$I_{1} = \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0}, A_{1})\bigr\| _{\Phi_{0}} + \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0}, A_{1})\bigr\| _{\Phi_{0}}$$

and

$$I_{2} = \bigl\| \chi_{(0,1)}(s)sK\bigl(s^{-1},f_{0};A_{0}, A_{1}\bigr)\bigr\| _{\Phi_{1}} + \bigl\| \chi_{(0,1)}(s)sK \bigl(s^{-1},f_{1};A_{0}, A_{1}\bigr) \bigr\| _{\Phi_{1}}.$$

The condition (C1), along with the following simple inequality:

$$K(t,f_{0};A_{0}, A_{1})\leq K(t,f_{0};A_{0}, A_{0} \cap A_{1}),\quad t>0,$$

implies that

$$I_{1} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0}, A_{0} \cap A_{1})\bigr\| _{\Phi_{0}} + \bigl\| \chi_{(0,1)}(s)K(s,f_{1};A_{0}, A_{1}) \bigr\| _{\Phi_{1}}.$$
(3.13)

Next we observe that $$f_{0} \in A_{0}$$ as $$B_{0} \subset A_{0}$$. Therefore, we can apply (2.1) to arrive at

$$I_{2} \lesssim \|f_{0}\|_{A_{0}} + \bigl\| \chi_{(0,1)}(s)sK\bigl(s^{-1},f_{1};A_{0}, A_{1}\bigr)\bigr\| _{\Phi_{1}}.$$
(3.14)

The proof of the estimate

$$\|f_{0}\|_{A_{0}}\lesssim \bigl\| \chi_{(0,1)}(s)K(s,f_{0};A_{0}, A_{0} \cap A_{1})\bigr\| _{\Phi_{0}}$$
(3.15)

is the same as that of (3.7). Finally, inserting estimates (3.13) and (3.14) in (3.12) and then using (3.15) and Proposition 2.4, we get

$$\|f\|_{(A_{0},A_{1})_{\Psi}}\lesssim \|f_{0}\|_{B_{0}}+ \|f_{1}\|_{B_{1}},$$

which gives the estimate $$\|f\|_{(A_{0},A_{1})_{\Psi}}\lesssim \|f\|_{B_{0}+B_{1}}$$. In order to prove the other estimate, we take $$f\in (A_{0},A_{1})_{\Psi}$$, and select a particular decomposition $$f = f_{0} + f_{1}$$, with $$f_{0} \in A_{0}$$ and $$f_{1} \in A_{1}$$, satisfying condition (3.1). Then proceeding in the same way as in the proof of Theorem 3.3, we obtain

$$\|f_{0}\|_{B_{0}} \lesssim \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0},A_{1}) \bigr\| _{\Phi_{0}}.$$

Also, we have

$$\|f_{1}\|_{B_{1}} \lesssim \bigl\| \chi_{(0,1)}(s)sK \bigl(s^{-1},f;A_{0},A_{1}\bigr)\bigr\| _{\Phi_{1}}.$$

Therefore, these estimates, along with the definition of Ψ, imply that

$$\|f_{0}\|_{B_{0}} + \|f_{1}\|_{B_{1}} \lesssim \|f\|_{(A_{0},A_{1})_{\Psi}},$$

whence we get $$\|f\|_{B_{0}+B_{1}}\lesssim \|f\|_{(A_{0},A_{1})_{\Psi}}$$. The proof is finished. □

### Remark 3.26

Take $$\Phi_{0}$$ and $$\Phi_{1}$$ to be given by (3.2) and (3.3), then we see that $$\Psi= \Phi_{0}$$. Thus, we recover the result (1.11) in , Theorem 1.1, for $$(\theta,p)\in \Gamma_{1}$$. Since the case when $$(\theta,p)\in \Gamma_{2}$$ follows from the case when $$(\theta,p)\in \Gamma_{1}$$, Theorem 3.25 provides a generalization of the assertion (1.11) in , Theorem 1.1.

### Corollary 3.27

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0}, A_{0}\cap A_{1})_{\{0\},p} + (A_{0} + A_{1},A_{0}\cap A_{1})_{\{1\},p} = (A_{0},A_{1})_{\{0\},p}.$$

### Proof

Apply Theorem 3.25 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (1.3) and (1.4). □

### Theorem 3.28

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and assume that the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ satisfy (C2). Then we have with equivalent norms

$$(A_{0}+A_{1}, A_{0})_{\Phi_{0}} \cap(A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}=(A_{0}, A_{1})_{\Psi},$$

where

$$\|g\|_{\Psi}=\bigl\| \chi_{(0,1)}(s)g(s)\bigr\| _{\Phi_{1}}+\bigl\| \chi_{(0,1)}(s)s g(1/s)\bigr\| _{\Phi_{0}}.$$

### Proof

Set $$B_{0}=(A_{0}+A_{1}, A_{0})_{\Phi_{0}}$$ and $$B_{1}= (A_{0} + A_{1},A_{0}\cap A_{1})_{\Phi_{1}}$$. Let $$f\in A_{0} + A_{1}$$. Applying Proposition 2.2 to the compatible couple $$(A_{1},A_{0})$$, we get

$$\|f\|_{B_{0}} \approx \bigl\| \chi_{(0,1)}(s)K(s,f; A_{1},A_{0}) \bigr\| _{\Phi_{0}},$$

using (2.3), we have

$$\|f\|_{B_{0}} \approx \bigl\| \chi_{(0,1)}(s)sK \bigl(s^{-1},f; A_{0},A_{1}\bigr)\bigr\| _{\Phi_{0}}.$$
(3.16)

By Proposition 2.4,

$$\|f\|_{B_{1}} \approx \bigl\| \chi_{(0,1)}(s)K(s,f;A_{0} , A_{1})\bigr\| _{\Phi_{1}} + \bigl\| \chi_{(0,1)}(s)sK \bigl(s^{-1},f;A_{0} , A_{1}\bigr) \bigr\| _{\Phi_{1}},$$

combining this with (3.16) and making use of (C2), we arrive at

$$\|f\|_{B_{0}} + \|f\|_{B_{1}}\approx \|f\|_{(A_{0}, A_{1})_{\Psi}},$$

which completes the proof. □

### Remark 3.29

Theorem 3.28 generalizes the result (1.12) in , Theorem 1.1.

### Corollary 3.30

Let $$(A_{0},A_{1})$$ be a compatible couple of normed spaces, and let $$1\leq p<\infty$$. Then we have with equivalent norms

$$(A_{0}+ A_{1}, A_{0})_{\{1\},p} + (A_{0} + A_{1},A_{0}\cap A_{1})_{\{0\},p} = (A_{0},A_{1})_{\{0\},p}.$$

### Proof

Apply Theorem 3.28 to the parameter spaces $$\Phi_{0}$$ and $$\Phi_{1}$$ given by (3.9) and (3.10). □

## References

1. Brudnyi, YA, Krugljak, NY: Interpolation Functors and Interpolation Spaces, vol. 1. North-Holland, Amsterdam (1991)

2. Bergh, J, Löfström, J: Interpolation Spaces. An Introduction. Springer, New York (1976)

3. Bennett, C, Sharpley, R: Interpolation of Operators. Academic Press, New York (1988)

4. Maligranda, L: The K-functional for symmetric spaces. In: Interpolation Spaces and Allied Topics in Analysis. Lecture Notes in Math., vol. 1070, pp. 170-182. Springer, Berlin (1984)

5. Maligranda, L: Interpolation between sum and intersection of Banach spaces. J. Approx. Theory 47, 42-53 (1986)

6. Persson, LE: Interpolation with a parameter function. Math. Scand. 59, 199-222 (1986)

7. Haase, M: Identification of some real interpolation spaces. Proc. Am. Math. Soc. 134, 2349-2358 (2006)

8. Cobos, F, Fernández-Cabrera, LM, Silvestre, P: New limiting real interpolation methods and their connection with the methods associated to the unit square. Math. Nachr. 286, 569-578 (2013)

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