Open Access

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

Journal of Inequalities and Applications20162016:307

https://doi.org/10.1186/s13660-016-1248-0

Received: 3 September 2016

Accepted: 15 November 2016

Published: 25 November 2016

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 \((A_{1}, A_{0} \cap A_{1})\).

Keywords

sum spaceintersection space K-functionalthe general K-interpolation methodlimiting K-interpolation methods

MSC

46B70

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). □

Declarations

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Authors’ Affiliations

(1)
Department of Mathematics, Government College University

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