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 [1] 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 [2] or [3]) 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 [4]:

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

(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 [5] 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 [6], 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 [7] 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 [7], 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 [7]. Consequently, our arguments of the proofs are different from those in [7].

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 [8]. 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

[3]

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 [7], Lemma 2.1. □

For the proof of the next result, we refer to [7], 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 [4], 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 [7], 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 [7], 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 [7], 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 [7], 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 [7], 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 [7], 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 [7], 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 [7], 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 [7], 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 [7], 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 [7], 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). □