Real interpolation with variable exponent

TL;DR

This paper develops a theory of real interpolation for function spaces with variable exponents, establishing foundational properties and linking it to fixed exponent cases, with applications to variable Besov and Lorentz spaces.

Contribution

It introduces and analyzes the real interpolation method for variable exponent spaces, extending classical theory and providing new tools for variable Besov and Lorentz spaces.

Findings

Established basic properties of real interpolation with variable exponents.

Showed reduction to fixed exponent cases under certain conditions.

Applied the theory to interpolate variable Besov and Lorentz spaces.

Abstract

We present the real interpolation with variable exponent and we prove the basic properties in analogy to the classical real interpolation. More precisely, we prove that under some additional conditions, this method can be reduced to the case of fixed exponent. An application, we give the real interpolation of variable Besov and Lorentz spaces as introduced recently in Almeida and H\"ast\"o (J. Funct. Anal. 258 (5) 1628--2655, 2010) and L. Ephremidze et al. (Fract. Calc. Appl. Anal. 11 (4) (2008), 407--420).

Equations

$$\left|g(x)-g(y)\right|\le \frac{c_{\log }(g)}{\log (e+1/\left|x-y\right|)}$$

$$|g(x)-g(0)|\le \frac{c_{\log }}{\ln (e+1/|x|)}$$

$$\left|g(x)-g_{\infty }\right|\le \frac{c_{\log }}{\log (e+\left|x\right|)}$$

$$\min \left\{{\left({\varrho }_{p\left(\cdot \right)}(f{\chi }_A)\right)}^{\frac{1}{p^{-}}},{\left({\varrho }_{p\left(\cdot \right)}(f{\chi }_A)\right)}^{\frac{1}{p^{+}}}\right\}\le {\Vert f\Vert }_{p\left(\cdot \right)}\le \max \left\{{\left({\varrho }_{p\left(\cdot \right)}(f{\chi }_A)\right)}^{\frac{1}{p^{-}}},{\left({\varrho }_{p\left(\cdot \right)}(f{\chi }_A)\right)}^{\frac{1}{p^{+}}}\right\}.$$

$$\Big{(}\sum\limits_{k=-\infty}^{\infty}\delta_{k}^{q}\Big{)}^{1/q}\leq c\text{ }\Big{(}\sum\limits_{k=-\infty}^{\infty}\varepsilon_{k}^{q}\Big{)}^{1/q}.$$

$$\displaystyle\Big{(}\frac{\gamma}{w(Q)}\int_{Q}\left|f(y)\right|w(y)dy\Big{)}^{p\left(x\right)}$$

$$\omega (m,b)=\min \left(b^m,1\right)\text{, }p\left(y,0\right)=p\left(y\right)\text{ and }g(x,y)=(e+\frac{1}{x}{)}^{-m}+(e+\frac{1}{y}{)}^{-m}$$

$$\omega (m,b)=\min \left(b^m,1\right)\text{, }p\left(y,0\right)=p\left(0\right)\text{ and }g(x,y)=(e+\frac{1}{x}{)}^{-m}{\chi }_{\{x:p(x)<p(0)\}}(x).$$

$$\omega (m,b)=1\text{, }p\left(y,0\right)=p_{\infty }\text{ and }g(x,y)=(e+x{)}^{-m}{\chi }_{\{x:p(x)<p_{\infty }\}}(x),$$

$${\int }_Q{\left|f(y)\right|}^{p\left(y\right)}w(y)dy\le 1$$

$${\eta }_t=t^s{\int }_t^{\infty }{\tau }^{-s}{\epsilon }_{\tau }\frac{d\tau }{\tau }\text{and}{\delta }_t=t^{-s}{\int }_0^t{\tau }^s{\epsilon }_{\tau }\frac{d\tau }{\tau }.$$

$${\Vert {\eta }_t\Vert }_{L^{q(\cdot )}((0,\infty ),\frac{dt}{t})}+{\Vert {\delta }_t\Vert }_{L^{q(\cdot )}((0,\infty ),\frac{dt}{t})}\lesssim {\Vert {\epsilon }_t\Vert }_{L^{q(\cdot )}((0,\infty ),\frac{dt}{t})}.$$

$$A_0,A_1\hookrightarrow Z,$$

$$f=f_0+f_1$$

$${\Vert f\Vert }_{A_0+A_1}=\underset{f=f_0+f_1}{\inf }\left({\Vert f_0\Vert }_{A_0}+{\Vert f_1\Vert }_{A_1}\right).$$

$${\Vert f\Vert }_{A_0\cap A_1}=\underset{f=f_0+f_1}{\max }\left({\Vert f_0\Vert }_{A_0},{\Vert f_1\Vert }_{A_1}\right).$$

$$K(t,f,A_0,A_1)=\underset{f=f_0+f_1}{\inf }\left({\Vert f_0\Vert }_{A_0}+t{\Vert f_1\Vert }_{A_1}\right),f\in A_0+A_1,$$

$$K(t,f,A_0,A_1)\le \max (1,\frac{t}{s})K(s,f,A_0,A_1).$$

$${\Vert f\Vert }_{(A_0,A_1{)}_{\theta ,q(\cdot )}}={\Vert t^{-\theta }K(t,f)\Vert }_{L^{q(\cdot )}((0,\infty ),\frac{dt}{t})}$$

$${\Vert f\Vert }_{(A_0,A_1{)}_{\theta ,\infty }}=\underset{t>0}{\sup }{t}^{-\theta }K(t,f)$$

$$\left\|f\right\|_{(A_{0},A_{1})_{\theta,q(\cdot)}}\approx\Big{(}\sum_{v=-\infty}^{0}2^{-v\theta q(0)}\alpha_{v}^{q(0)}\Big{)}^{\frac{1}{q(0)}}+\Big{(}\sum_{v=1}^{\infty}2^{-v\theta q_{\infty}}\alpha_{v}^{q_{\infty}}\Big{)}^{\frac{1}{q_{\infty}}}.$$

$$\left\|f\right\|_{(A_{0},A_{1})_{\theta,q(\cdot)}}\approx\Big{(}\int_{0}^{1}t^{-\theta q(0)}K(t,f)^{q(0)}\frac{dt}{t}\Big{)}^{\frac{1}{q(0)}}+\Big{(}\int_{1}^{\infty}t^{-\theta q_{\infty}}K(t,f)^{q_{\infty}}\frac{dt}{t}\Big{)}^{\frac{1}{q_{\infty}}}.$$

$$S=\Big{(}\sum_{v=-\infty}^{0}2^{-v\theta q(0)}\alpha_{v}^{q(0)}\Big{)}^{\frac{1}{q(0)}}+\Big{(}\sum_{v=1}^{\infty}2^{-v\theta q_{\infty}}\alpha_{v}^{q_{\infty}}\Big{)}^{\frac{1}{q_{\infty}}}\lesssim\left\|f\right\|_{(A_{0},A_{1})_{\theta,q(\cdot)}}.$$

$$\sum_{v=1}^{\infty}2^{-v\theta q_{\infty}}\big{(}\alpha_{v}\big{)}^{q_{\infty}}\lesssim 1\quad\text{and}\quad\sum_{v=-\infty}^{0}2^{-v\theta q(0)}\big{(}\alpha_{v}\big{)}^{q(0)}\lesssim 1$$

$$2^{-v\theta q_{\infty}}\big{(}\alpha_{v}\big{)}^{q_{\infty}}\leq\int_{2^{v-1}}^{2^{v}}\big{(}t^{-\theta}K(t,f)\big{)}^{q\left(t\right)}\frac{dt}{t}+2^{-v}=\delta$$

$$\big{(}\delta^{-\frac{1}{q_{\infty}}}2^{-v\theta}\alpha_{v}\big{)}^{q_{\infty}}=\Big{(}\frac{1}{\log 2}\int_{2^{v-1}}^{2^{v}}\delta^{-\frac{1}{q_{\infty}}}2^{-v\theta}\alpha_{v}\frac{d\tau}{\tau}\Big{)}^{q_{\infty}}\lesssim 1.$$

$${\int }_{2^{v-1}}^{2^v}{\delta }^{-\frac{1}{q_{\infty }}}{2}^{-v\theta }{\alpha }_v\frac{d\tau }{\tau }\lesssim {\int }_{2^{v-1}}^{2^v}{\delta }^{-\frac{1}{q_{\infty }}}{\tau }^{-\theta }K(\tau ,f)\frac{d\tau }{\tau }.$$

$$c\int_{2^{v-1}}^{2^{v}}\delta^{-\frac{q\left(\tau\right)}{q_{\infty}}}\big{(}\tau^{-\theta}K(\tau,f)\big{)}^{q\left(\tau\right)}\frac{d\tau}{\tau}+1$$

$${\delta }^{-\frac{q\left(\tau \right)}{q_{\infty }}}\approx {\delta }^{-1},\tau \in [2^{v-1},2^v],v\in \mathbb{N}.$$

$$\int_{2^{v-1}}^{2^{v}}\delta^{-\frac{q\left(\tau\right)}{q_{\infty}}}\big{(}\tau^{-\theta}K(\tau,f)\big{)}^{q\left(\tau\right)}\frac{d\tau}{\tau}\lesssim 1.$$

Full text

MSC 2010: 46B70, 46E30, 46E35.

Key Words and Phrases: real interpolation, embeddings, Besov space, Lorentz space, variable exponent.

1 Introduction

It is well known that real interpolation play an important role in several different areas, especially for modern analysis and its theory started early in 1960’s by J-L. Lions and J. Peetre. There are two ways for introducing the real interpolation method. The first is the $K$-method and the second is the $J$-method. But the spaces generated by the $K$-and $J$-methods are the same. For general literature on real interpolation we refer to [1], [4], [17] and references therein.

In recent years, there has been growing interest in generalizing classical spaces such as Lebesgue spaces, Sobolev spaces, Besov spaces, Triebel-Lizorkin spaces to the case with either variable integrability or variable smoothness. The motivation for the increasing interest in such spaces comes not only from theoretical purposes, but also from applications to fluid dynamics, image restoration and PDE with non-standard growth conditions.

From these in this paper we present a variable version of real interpolation. First we study the variable version of $K$-method, where we present some equivalent norms for the space generated by this method and we prove their basic properties in analogy to the fixed exponent. Secondly, we present the same analysis for the variable version of $J$-method and we prove the first main statement of this paper. That is, under some additional conditions the spaces generated by the $K$-and $J$-methods are the same. Since the reiteration theorem is one of the most important general results in interpolation theory, we will give its proof. Finally, we study the real interpolation of variable exponent Besov and Lorentz spaces. Almost all of the material we present is due to [1], and [4]. Allowing the exponent is vary from point to point will raise extra difficulties which, in general, are overcome by imposing regularity assumptions on this exponent.

2 Preliminaries

As usual, we denote by $\mathbb{R}^{n}$ the $n$-dimensional real Euclidean space, $\mathbb{N}$ the collection of all natural numbers and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$. The letter $\mathbb{Z}$ stands for the set of all integer numbers. The expression $f\lesssim g$ means that $f\leq c\,g$ for some independent constant $c$ (and non-negative functions $f$ and $g$), and $f\approx g$ means $f\lesssim g\lesssim f$.

By $c$ we denote generic positive constants, which may have different values at different occurrences. Although the exact values of the constants are usually irrelevant for our purposes, sometimes we emphasize their dependence on certain parameters (e.g. $c(p)$ means that $c$ depends on $p$, etc.). Further notation will be properly introduced whenever needed.

The variable exponents that we consider are always measurable functions $p$ on $\mathbb{R}$ with range in $[1,\infty[$. We denote the set of such functions by $\mathcal{P}({\mathbb{R}})$. We use the standard notation $p^{-}:=\underset{x\in\mathbb{R}^{n}}{\text{ess-inf}}$ $p(x)$,$\quad p^{+}:=\underset{x\in\mathbb{R}^{n}}{\text{ess-sup }}p(x)$.

The variable exponent modular is defined by $\varrho_{p(\cdot)}(f):=\int_{\mathbb{R}^{n}}\varrho_{p(x)}(\left|f(x)\right|)dx$, where $\varrho_{p}(t)=t^{p}$. The variable exponent Lebesgue space $L^{p(\cdot)}$ consists of measurable functions $f$ on $\mathbb{R}^{n}$ such that $\varrho_{p(\cdot)}(\lambda f)<\infty$ for some $\lambda>0$. We define the Luxemburg (quasi)-norm on this space by the formula \left\|f\right\|_{p(\cdot)}:=\inf\Big{\{}\lambda>0:\varrho_{p(\cdot)}\Big{(}\frac{f}{\lambda}\Big{)}\leq 1\Big{\}}. A useful property is that $\left\|f\right\|_{p(\cdot)}\leq 1$ if and only if $\varrho_{p(\cdot)}(f)\leq 1$, see [7], Lemma 3.2.4.

We say that $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is locally log-Hölder continuous, abbreviated $g\in C_{\text{loc}}^{\log}(\mathbb{R}^{n})$, if there exists $c_{\log}(g)>0$ such that

[TABLE]

for all $x,y\in\mathbb{R}^{n}$. If

[TABLE]

for all $x\in\mathbb{R}^{n}$, then we say that $g$ is $\log$-Hölder continuous at the origin (or has a $\log$ decay at the origin). We say that $g$ satisfies the log*-Hölder decay condition*, if there exists $g_{\infty}\in\mathbb{R}$ and a constant $c_{\log}>0$ such that

[TABLE]

for all $x\in\mathbb{R}^{n}$. We say that $g$ is globally-log*-Hölder continuous*, abbreviated $g\in C^{\log}$, if it is* locally log-Hölder continuous and satisfies the log-Hölder decay condition. The constants $c_{\log}(g)$ and $c_{\log}$ are called the locally log-Hölder constant and the log-Hölder decay constant, respectively.* We note that all functions $g\in C_{\text{loc}}^{\log}(\mathbb{R}^{n})$ always belong to $L^{\infty}$.

We refer to the recent monograph [5] for further properties, historical remarks and references on variable exponent spaces.

2.1 Technical lemmas

In this subsection we present some results which are useful for us. The following lemma is from [7].

Lemma 1

Let\ A\subset\mathbb{R}^{n}\and$\ p\in\mathcal{P}(\mathbb{R}^{n})$ with$\ p^{-}<\infty$. If $\varrho_{p\left(\cdot\right)}(f\chi_{A})>0$ or $p^{+}<\infty$, then

[TABLE]

The next lemma is a Hardy-type inequality which is easy to prove.

Lemma 2

Let $0<a<1$ and $0<q\leq\infty$. Let $\left\{\varepsilon_{k}\right\}_{k}$ be a sequences of positive real numbers and denote $\delta_{k}=\sum_{j=-\infty}^{\infty}a^{\left|k-j\right|}\varepsilon_{j}$. Then there exists constant c>0\depending only on $a$ and $q$ such that

[TABLE]

We will make use of the following statement, see [8], Lemma 3.3 for $w:=1$.

Lemma 3

Let $Q=(a,b)\subset\mathbb{R}$ with $0<a<b<\infty$. Let $p\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous at the origin and $w:\mathbb{R}\rightarrow\mathbb{R}^{+}$ be a weight function. Then for every $m>0$ there exists $\gamma=e^{-4mc_{\log}(1/p)}\in\left(0,1\right)$ such that

[TABLE]

hold if $0<w(Q)<\infty$, all $x\in Q\subset\mathbb{R}$ and all $f\in L^{p\left(\cdot\right)}(w)+L^{\infty}$ with $\left\|fw^{1/p\left(\cdot\right)}\right\|_{p\left(\cdot\right)}+\left\|f\right\|_{\infty}\leq 1$, where

[TABLE]

or

[TABLE]

In addition we have the same estimate, where

[TABLE]

if $p\in\mathcal{P}(\mathbb{R})$ satisfies the log-Hölder decay condition, where $\gamma=e^{-4mc_{\log}}$.

The proof of this lemma is given in [10]. Notice that in the proof of this theorem we need only that

[TABLE]

and/or $\left\|f\right\|_{\infty}\leq 1$.

The next lemma is the continuous version of Hardy-type inequality, see [6].

Lemma 4

Let $s>0$. Let $q\in\mathcal{P}(\mathbb{R})$ be $\log$-Hölder continuous both at the origin and at infinity with $1\leq q^{-}\leq q^{+}<\infty$. Let $\left\{\varepsilon_{t}\right\}_{t}$ be a sequence of positive measurable functions. Let

[TABLE]

Then there exists constant c>0\depending only on $s$, $q^{-}$, c${}_{\log}(q)$ and $q^{+}$ such that

[TABLE]

3 The K-Method

The fundamental notion of real interpolation is the $K$-functional, where it is due to J. Peetre.

Definition 1

Let $A_{0}$ and $A_{1}$ be Banach spaces over $\mathbb{K=R}$ or $\mathbb{C}$. We shall say that $A_{0}$ and $A_{1}$ are compatible if there is a Hausdorff topological vector space $Z$ such that

[TABLE]

with continuous embeddings.

Let $A_{0}$ and $A_{1}$ be compatible. We will say that $(A_{0},A_{1})$ is a compatible couple. Then we can form their sum $A_{0}+A_{1}$ and their intersection $A_{0}\cap A_{1}$. The sum consists of all $f\in Z$ such that we can write

[TABLE]

for some $f_{0}\in A_{0}$ and $f_{1}\in A_{1}$. Then $A_{0}+A_{1}$ is a Banach space with norm defined by

[TABLE]

$A_{0}\cap A_{1}$ is a Banach space with norm defined by

[TABLE]

Let $(A_{0},A_{1})$ be a compatible couple. With $t>0$ fixed, put

[TABLE]

is the $K$-functional. For any $f\in A_{0}+A_{1}$, $K(t,f,A_{0},A_{1})$ is a positive, increasing and concave function of $t$. In particular

[TABLE]

If there is no danger of confusion, we shall write $K(t,f)=K(t,f,A_{0},A_{1})$.

Definition 2

Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$. Let $(A_{0},A_{1})$ be a compatible couple. The space $(A_{0},A_{1})_{\theta,q(\cdot)}$ consists of all $f$ in $A_{0}+A_{1}$ for which the functional

[TABLE]

is finite.

Definition 3

Let $\theta\in[0,1]$. Let $(A_{0},A_{1})$ be a compatible couple. The space $(A_{0},A_{1})_{\theta,\infty}$ consists of all $f$ in $A_{0}+A_{1}$ for which the functional

[TABLE]

is finite.

In the next lemma we prove that the first definition can be given in discrete version, where we need additional assumptions on $q$.

Lemma 5

Let $(A_{0},A_{1})$ be a compatible couple and $f\in A_{0}+A_{1}$. Let $\theta\in(0,1)$, $f\in(A_{0},A_{1})_{\theta,q(\cdot)}$ and we put $\alpha_{v}=K(2^{v},f)$, $v\in\mathbb{Z}$. Let $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity. Then

[TABLE]

Moreover,

[TABLE]

**Proof. **We will do the proof in two steps and we need only to prove the first statement.

Step 1. Let us prove that

[TABLE]

By scaling argument, we need only to prove that

[TABLE]

for any $f\in(A_{0},A_{1})_{\theta,q(\cdot)}$ with $\left\|f\right\|_{(A_{0},A_{1})_{\theta,q(\cdot)}}\leq 1$. To prove the first estimate we need to prove that

[TABLE]

for any $v\in\mathbb{N}$. This claim can be reformulated as showing that

[TABLE]

Using the property $\mathrm{\eqref{K-property}}$, we find that

[TABLE]

By Lemma 3 the last expression with power $q\left(t\right)$ is bounded by

[TABLE]

for any $t\in[2^{v-1},2^{v}]$. Since $q$ is *$\log$-*Hölder continuous at infinity, we find that

[TABLE]

Therefore, from the definition of $\delta$, we find that

[TABLE]

Now, let us prove the second estimate. We need to show that

[TABLE]

for any $v\leq 0$. This claim can be reformulated as showing that

[TABLE]

The property $\mathrm{\eqref{K-property}}$, gives that

[TABLE]

Again by Lemma 3,

[TABLE]

for any $t\in[2^{v},2^{v+1}]$ and any $v\leq 0$. We use the logarithmic decay condition at origin of $q$ to show that

[TABLE]

Therefore, from the definition of $\delta$, we find that

[TABLE]

for any $v\leq 0$. Hence, we proved $\mathrm{\eqref{est-step1}}$.

Step 2. Let us prove that

[TABLE]

This claim can be reformulated as showing that

[TABLE]

Now our estimate clearly follows from the inequalities

[TABLE]

for any $v\in\mathbb{N}$ and

[TABLE]

for any $v\leq 0$. The first claim can be reformulated as showing that

[TABLE]

We need only to show that

[TABLE]

for any $v\in\mathbb{N}$ and any $t\in[2^{v-1},2^{v}]$. From $\mathrm{\eqref{K-property}}$, the left-hand side is bounded by

[TABLE]

and from $\mathrm{\eqref{log-at- zero}}$ we find that

[TABLE]

for any $v\in\mathbb{N}$. Similarly we estimate the second claim. Hence the lemma is proved. $\square$

Let $(A_{0},A_{1})$ be a compatible couple and $f\in A_{0}+A_{1}$. Let $\theta\in[0,1]$, $f\in(A_{0},A_{1})_{\theta,q(\cdot)}$ and we put $\alpha_{v}=K(2^{v},f)$, $v\in\mathbb{Z}$*. *Then we have

[TABLE]

We present some important properties of the spaces $(A_{0},A_{1})_{\theta,q(\cdot)}$.

Theorem 1

Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$. Let $(A_{0},A_{1})$ be a compatible couple of Banach spaces. Then $(A_{0},A_{1})_{\theta,q(\cdot)}$ is Banach space and

[TABLE]

for any $s>0$. Moreover we have

[TABLE]

Proof. Let $\{f_{n}\}_{n}$ be a sequence in $A_{0}+A_{1}$ such that

[TABLE]

Since $L^{q(\cdot)}((0,\infty),\frac{dt}{t})$ is a Banach space, the series $\sum_{n=1}^{\infty}t^{-\theta}K(t,f_{n})$ converges in $L^{q(\cdot)}((0,\infty),\frac{dt}{t})$ then we get

[TABLE]

for all $t>0$. Since $A_{0}+A_{1}$ is a Banach space, then

[TABLE]

for all $t>0$. Applying the $L^{q(\cdot)}((0,\infty),\frac{dt}{t})$-norm to each side, we obtain

[TABLE]

which ensure that $(A_{0},A_{1})_{\theta,q(\cdot)}$ is Banach space. By the property $\mathrm{\eqref{K-property}}$ we find that

[TABLE]

Therefore,

[TABLE]

Let us prove that

[TABLE]

We have

[TABLE]

and

[TABLE]

From Lemma 1, we find our claim $\mathrm{\eqref{est-t-theta}}$. Therefore,

[TABLE]

for any $s>0$. Taking $s=1$, we obtain

[TABLE]

Now since

[TABLE]

we find that

[TABLE]

$\square$

Definition 4

Let $(A_{0},A_{1})$ and $(B_{0},B_{1})$ be two compatible couples of Banach spaces and let $T$ be a linear operator defined on $A_{0}+A_{1}$ and taking values in $B_{0}+B_{1}$. $T$ is said be admissible with respect to the couples $(A_{0},A_{1})$ and $(B_{0},B_{1})$ if, for each $i=1,0$ the restriction of $T$ to $A_{i}$ maps $A_{i}$ into $B_{i}$ and furthermore is a bounded operator from $A_{i}$ into $B_{i}:$

[TABLE]

Notice that every admissible operator $T$ with respect to the couples $(A_{0},A_{1})$ and $(B_{0},B_{1})$ is bounded from $A_{0}+A_{1}$ into $B_{0}+B_{1}$.

Theorem 2

Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$. Let $(A_{0},A_{1})$ and $(B_{0},B_{1})$ be two compatible couples of Banach spaces and let $T$ be admissible with respect to the couples $(A_{0},A_{1})$ and $(B_{0},B_{1})$. Then

[TABLE]

and

[TABLE]

for all $f\in(A_{0},A_{1})_{\theta,q(\cdot)}$.

Proof. Suppose that $T:(A_{0},A_{1})\longrightarrow(B_{0},B_{1})$. Then

[TABLE]

by the property $\mathrm{\eqref{K-property}}$. Multiplying by $t^{-\theta}$ and then applying the $L^{q(\cdot)}((0,\infty),\frac{dt}{t})$-norm to each side we obtain the desired estimate. $\square$

Proposition 3

*Let $\theta\in(0,1)$. Let $(A_{0},A_{1})$ be a compatible couples of Banach spaces.

(i) Let $q,r\in\mathcal{P}(\mathbb{R})$ with $1\leq q(\cdot)\leq r(\cdot)<\infty$. Then*

[TABLE]

and

[TABLE]

(ii) Let $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity with $q(0)=q_{\infty}$. Then

[TABLE]

(iii) Let $q,r\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity with $q(0)=r(0)$ and $q_{\infty}=r_{\infty}$. Then

[TABLE]

(iv) If $A_{1}\hookrightarrow A_{0}$, then

[TABLE]

(v) If $A_{0}=A_{1}$, with equal norm, then

[TABLE]

Proof. We prove (i). From Theorem 1, we obtain

[TABLE]

for any $f\in(A_{0},A_{1})_{\theta,q(\cdot)}$, any $s>0$ with $\left\|f\right\|_{(A_{0},A_{1})_{\theta,q(\cdot)}}\neq 0$ and this implies that

[TABLE]

The last term is bounded since

[TABLE]

and hence

[TABLE]

Hence the property (i) is proved. To prove (ii) we use Lemma 5 and the fact that

[TABLE]

and $q(0)=q_{\infty}$. The property (iii) follows by Lemma 5. Now if $A_{1}\hookrightarrow A_{0}$ we have $\left\|f\right\|_{A_{0}}\leq c\left\|f\right\|_{A_{1}}$ for any $f\in A_{0}$ and

[TABLE]

if $t>c$. Then

[TABLE]

and

[TABLE]

Using the fact that

[TABLE]

and $0<\theta\leq\theta_{1}<1$, we obtain

[TABLE]

So, the property (iv) is proved. Now the property (v) is immediate. The proof is complete. $\square$

4 The J-Method

Let $(A_{0},A_{1})$ be a compatible couple. With $t>0$ fixed, put

[TABLE]

Notice that $J(t,f,A_{0},A_{1})$ is an equivalent norm on $A_{0}\cap A_{1}$ for a given $t>0$. If there is no danger of confusion, we shall write $J(t,f)=J(t,f,A_{0},A_{1})$. For any $f\in A_{0}\cap A_{1}$, $J(t,f)$ is a positive, increasing and convex  function of $t$, such that

[TABLE]

and

[TABLE]

Now we define the interpolation space constructed by the $J$-method.

Definition 5

Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$. Let $(A_{0},A_{1})$ be a compatible couple. The space $(A_{0},A_{1})_{\theta,q(\cdot),J}$ consists of all $f$ in $A_{0}+A_{1}$ that are representable in the form

[TABLE]

where $u(t)$ is measurable with values in $A_{0}\cap A_{1}$ and

[TABLE]

where the infimum is taken over all $u$ such that $\mathrm{\eqref{rep}}$ holds.

Definition 6

Let $\theta\in(0,1)$. The space $(A_{0},A_{1})_{\theta,\infty,J}$ consists of all $f$ in $A_{0}+A_{1}$ that are representable in the form $\mathrm{\eqref{rep}}$, where $u(t)$ is measurable with values in $A_{0}\cap A_{1}$ and

[TABLE]

where the infimum is taken over all $u$ such that $\mathrm{\eqref{rep}}$ holds.

Lemma 6

Let $(A_{0},A_{1})$ be a compatible couple and $f\in A_{0}+A_{1}$. Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity. Then $f\in(A_{0},A_{1})_{\theta,q(\cdot),J}$ if and only if there exist $u_{v}\in A_{0}\cap A_{1}$, $v\in\mathbb{Z}$, with

[TABLE]

and such that

[TABLE]

Moreover

[TABLE]

where the infimum is extended over all sequences $(u_{v})_{v}$ satisfying $\mathrm{\eqref{f-representation}}$.

Proof. Let $f\in(A_{0},A_{1})_{\theta,q(\cdot),J}$. Then we have a representation

[TABLE]

where $u(t)$ is measurable with values in $A_{0}\cap A_{1}$ and

[TABLE]

We set

[TABLE]

Then we have

[TABLE]

Let us prove that

[TABLE]

with $\alpha_{v}=J(2^{v},u_{v})$, $v\in\mathbb{Z}$. We need only to prove that

[TABLE]

and

[TABLE]

First let us prove that

[TABLE]

for any $v\in\mathbb{N}$. This claim can be reformulated as showing that

[TABLE]

Using the property $\mathrm{\eqref{Est-J}}$, we find that

[TABLE]

By Lemma 3,

[TABLE]

for any $t\in[2^{v},2^{v+1}]$. Since, $q$ is *$\log$-*Hölder continuous at the infinity we find that

[TABLE]

Therefore, from the definition of $\delta$, we find that the last integral is dominated by a constant independent on $v\in\mathbb{N}$. Now, let us prove that

[TABLE]

for any $v\leq 0$. This claim can be reformulated as showing that

[TABLE]

The property $\mathrm{\eqref{Est-J}}$, gives that

[TABLE]

Again by Lemma 3,

[TABLE]

for any $t\in[2^{v},2^{v+1}]$ and any $v\leq 0$. We use the logarithmic decay condition at origin of $q$ to show that

[TABLE]

Therefore and from the definition of $\delta$, we find that

[TABLE]

Hence the left-hand side of $\mathrm{\eqref{est-J}}$ can be estimated by

[TABLE]

The first term is bounded since

[TABLE]

Now in $\mathrm{\eqref{est-J}}$ taking the infimum, we conclude that

[TABLE]

Conversely, assume that

[TABLE]

and

[TABLE]

Let us prove that

[TABLE]

where the infimum is taking over all sequences $\{w_{v}\}$ satisfying $\mathrm{\eqref{f-representation}}$. Choose

[TABLE]

Then $f=\int_{0}^{\infty}u(t)\frac{dt}{t}$. This claim can be reformulated as showing that

[TABLE]

Now our estimate clearly follows from the inequalities

[TABLE]

for any $v\in\mathbb{N}$ and

[TABLE]

for any $v\leq 0$. The first claim can be reformulated as showing that

[TABLE]

We need only to show that

[TABLE]

for any $v\in\mathbb{N}$ and any $t\in[2^{v},2^{1+v}]$. The left-hand side is bounded by

[TABLE]

From $\mathrm{\eqref{Est-delta}}$ we find that

[TABLE]

for any $v\in\mathbb{N}$. Similarly we estimate the second claim.

We conclude that

[TABLE]

$\square$

We shall prove that the spaces generated by the $K$-and $J$-methods are the same.

Theorem 4

Let $(A_{0},A_{1})$ be a compatible couple. Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity. Then

[TABLE]

with equivalence of norms.

Proof. Let $f\in(A_{0},A_{1})_{\theta,q(\cdot),J}$ with $f=\int_{0}^{\infty}u(s)\frac{ds}{s}$, where $u(t)$ is measurable with values in $A_{0}\cap A_{1}$. By $\mathrm{\eqref{Est-J1}}$ we have

[TABLE]

Applying Hardy inequality, Lemma 4, we get

[TABLE]

For the converse inequality, Lemma 3.3.2 of [4], and using Theorem 1, implies the existence of a representation

[TABLE]

such that

[TABLE]

for any $v\in\mathbb{Z}$, $\varepsilon>0$ and $\gamma$ is a universal constant less than or equal $3$. By Lemmas 5 and 6 we get

[TABLE]

This completes the proof of this theorem.$\quad\square$

Theorem 5

Let $(A_{0},A_{1})$ be a compatible couple. Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity. Then $A_{0}\cap A_{1}$ is dense in $(A_{0},A_{1})_{\theta,q(\cdot)}$.

Proof. Let $f\in(A_{0},A_{1})_{\theta,q(\cdot)}$. From Theorem 4 we have

[TABLE]

where $u_{v}$, $v\in\mathbb{Z}$ is measurable with values in $A_{0}\cap A_{1}$ and

[TABLE]

Then

[TABLE]

Therefore,

[TABLE]

which tends to zero if $N\longrightarrow\infty$.$\quad\square$

Definition 7

*Let $\theta\in[0,1]$. Let $(A_{0},A_{1})$ be a compatible couple of normed vector spaces. Suppose that $X$ is an intermediate space with respect to $(A_{0},A_{1})$. Then we say that

(i) $X$ is of class $\mathscr{C}_{K}(\theta;A_{0},A_{1})$ if $K(t,f;A_{0},A_{1})\leq Ct^{\theta}\left\|f\right\|_{X},\quad f\in X;$

(ii) $X$ is of class $\mathscr{C}_{J}(\theta;A_{0},A_{1})$ if $\left\|f\right\|_{X}\leq Ct^{-\theta}J(t,f;A_{0},A_{1}),\quad f\in A_{0}\cap A_{1}.$

(iii) We say that $X$ is of class $\mathscr{C}(\theta;A_{0},A_{1})$ if $X$ is of class $\mathscr{C}_{K}(\theta;A_{0},A_{1})$ and of class $\mathscr{C}_{J}(\theta;A_{0},A_{1})$.*

Let $q\in\mathcal{P}(\mathbb{R})$*. *From [4, Theorem 3.5.2] and Proposition 3 we see that $(A_{0},A_{1})_{\theta,q(\cdot)}$ is of class $\mathscr{C}(\theta;A_{0},A_{1})$ if $\theta\in(0,1)$.

We are now ready to prove the reiteration theorem, which is one of the most important general results in interpolation theory.

Theorem 6

Let $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity. Let $(A_{0},A_{1})$ and $(X_{0},X_{1})$ be two compatible couples of normed linear spaces, and assume that $X_{i}$ ($i=0,1$) are complete and of class $\mathscr{C}(\theta_{i};A_{0},A_{1})$, where $\theta_{0},\theta_{1}\in[0,1]$ and $\theta_{0}\neq\theta_{1}$. Put

[TABLE]

Then

[TABLE]

with equivalence of norms. In particular, if $\theta_{0},\theta_{1}\in(0,1)$, $q_{0},q_{1}\in\mathcal{P}(\mathbb{R})$ are log-Hölder continuous both at the origin and at infinity and $(A_{0},A_{1})_{\theta_{i},q_{i}(\cdot)}$ are complete then

[TABLE]

where

[TABLE]

**Proof. **We will do the proof in two steps.

Step 1. Let us prove that

[TABLE]

Let $f\in(X_{0},X_{1})_{\eta,q(\cdot)}$. Then

[TABLE]

Since $X_{i}$ ($i=0,1$) are of class $\mathscr{C}(\theta_{i};A_{0},A_{1})$ we have

[TABLE]

Therefore, from Theorem 5, we get

[TABLE]

Putting $s=t^{\theta_{1}-\theta_{0}}$ and observing that $\eta=\frac{\theta-\theta_{0}}{\theta_{1}-\theta_{0}}$ we find that

[TABLE]

which gives $\mathrm{\eqref{First-emb}}$.

Step 2. Let us prove that

[TABLE]

Assume that $f\in(A_{0},A_{1})_{\theta,q(\cdot)}$. We choose a representation

[TABLE]

where $u_{v}$, $v\in\mathbb{Z}$ is measurable with values in $A_{0}\cap A_{1}$ and

[TABLE]

Applying $\mathrm{\eqref{Est-J1}}$, and that $X_{i}$ ($i=0,1$) are of class $\mathscr{C}(\theta_{i};A_{0},A_{1})$ we get for any $j\in\mathbb{Z}$,

[TABLE]

The last term can be rewritten us

[TABLE]

for any $j\in\mathbb{Z}$. We treat the case where $j\geq 0$. The first sum can be rewritten us

[TABLE]

Applying Lemma 2 we get

[TABLE]

Now if $j\leq 0$, the second sum of $\mathrm{\eqref{sum-equi}}$ can be rewritten us

[TABLE]

Applying again Lemma 2 we get

[TABLE]

This prove the embedding $\mathrm{\eqref{Second-emb}}$ by taking the infimum in view of the Theorem 6 and the fact that

[TABLE]

where

[TABLE]

This completes the proof of Theorem 6.$\quad\square$

5 Application

In this section, we give a simple application of the results of the previous sections. We will present various real interpolation formulas in Besov spaces with variable indices. The symbol $\mathcal{S}(\mathbb{R}^{n})$ is used in place of the set of all Schwartz functions on $\mathbb{R}^{n}$. We denote by $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ the dual space of all tempered distributions on $\mathbb{R}^{n}$. The Fourier transform of a Schwartz function $f$ is denoted by $\mathcal{F}f$. To define the variable Besov spaces, we first need the concept of a smooth dyadic resolution of unity. Let $\Psi$ be a function in $\mathcal{S}(\mathbb{R}^{n})$ satisfying $\Psi(x)=1$ for $\left|x\right|\leq 1$ and $\Psi(x)=0$ for $\left|x\right|\geq 2$. We define $\varphi_{0}$ and $\varphi_{1}$ by $\mathcal{F}\varphi_{0}(x)=\Psi(x)$, $\mathcal{F}\varphi_{1}(x)=\Psi(x)-\Psi(2x)$ and

[TABLE]

Then $\{\mathcal{F}\varphi_{j}\}_{j\in\mathbb{N}_{0}}$ is a smooth dyadic resolution of unity, $\sum_{j=0}^{\infty}\mathcal{F}\varphi_{j}(x)=1$ for all $x\in\mathbb{R}^{n}$. Thus we obtain the Littlewood-Paley decomposition $f=\sum_{j=0}^{\infty}\varphi_{j}\ast f$ of all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ $($convergence in $\mathcal{S}^{\prime}(\mathbb{R}^{n}))$.

Let $p,q\in\mathcal{P}(\mathbb{R}^{n})$. The mixed Lebesgue-sequence space $\ell_{>}^{q(\cdot)}(L^{p(\cdot)})$ is defined on sequences of $L^{p(\cdot)}$-functions by the modular

[TABLE]

The (quasi)-norm is defined from this as usual:

[TABLE]

If $q^{+}<\infty$, then we can replace $\mathrm{\eqref{mixed-norm}}$ by the simpler expression $\varrho_{\ell_{>}^{q(\cdot)}(L^{p(\cdot)})}((f_{v})_{v}):=\sum\limits_{v=1}^{\infty}\left\||f_{v}|^{q(\cdot)}\right\|_{\frac{p(\cdot)}{q(\cdot)}}$. The case $p:=\infty$ can be included by replacing the last modular by \varrho_{\ell_{>}^{q(\cdot)}(L^{\infty})}((f_{v})_{v}):=\sum\limits_{v=1}^{\infty}\big{\|}\left|f_{v}\right|^{q(\cdot)}\big{\|}_{\infty}.

We define the following class of variable exponents \mathcal{P}^{\mathrm{log}}(\mathbb{R}^{n}):=\big{\{}p\in\mathcal{P}:\frac{1}{p}\in C^{\log}\big{\}}, were introduced in [8, Section  2]. We define $1/p_{\infty}:=\lim_{|x|\rightarrow\infty}1/p(x)$ and we use the convention $\frac{1}{\infty}=0$. Note that although $\frac{1}{p}$ is bounded, the variable exponent $p$ itself can be unbounded.

We state the definition of the spaces $B_{p(\cdot),q(\cdot)}^{s(\cdot)}$, which introduced and investigated in [2].

Definition 8

Let $\left\{\mathcal{F}\varphi_{j}\right\}_{j=0}^{\infty}$ be a resolution of unity, $s:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and $p,q\in\mathcal{P}(\mathbb{R}^{n})$. The Besov space $B_{p(\cdot),q(\cdot)}^{s(\cdot)}$ consists of all distributions $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that

[TABLE]

Taking $s\in\mathbb{R}$ and $q\in(0,\infty]$ as constants we derive the spaces $B_{p(\cdot),q}^{s}$ studied by Xu in [21]. We refer the reader to the recent papers [9], [12],[13] and [14] for further details, historical remarks and more references on these function spaces. For any $p,q\in\mathcal{P}_{0}^{\log}(\mathbb{R}^{n})$ and $s\in C_{\text{loc}}^{\log}$, the space $B_{p(\cdot),q(\cdot)}^{s(\cdot)}$ does not depend on the chosen smooth dyadic resolution of* *unity $\{\mathcal{F}\varphi_{j}\}_{j\in\mathbb{N}_{0}}$ (in the sense of equivalent quasi-norms) and

[TABLE]

Moreover, if $p,q,s$ are constants, we re-obtain the usual Besov spaces $B_{p,q}^{s}$, studied in detail in [15], [16], [18], [19] and [20].

Applying Lemma 5 and using the same arguments of [3, Theorem 3.1] we obtain.

Theorem 7

Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity with $q(0)=q_{\infty}$. Let $p,q_{0},q_{1}\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ and $\alpha_{0},\alpha_{1}\in C_{\mathrm{loc}}^{\log}$. If $0\neq\alpha_{0}-\alpha_{1}$ is constant, then

[TABLE]

with $\alpha(\cdot)=(1-\theta)\alpha_{0}(\cdot)+\theta\alpha_{1}(\cdot)$. Moerover

[TABLE]

with $r_{0},r_{1}\in[1,\infty]$ and

[TABLE]

Now we present some interpolation results in variable exponent Lorentz spaces $\mathcal{L}^{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$ introduced by [11].

Definition 9

If $f$ is a measurable function on $\mathbb{R}^{n}$, we define the non-increasing rearrangement of $f$ through

[TABLE]

where $m_{f}$ is the distribution function of $f$.

Definition 10

Let $p,q\in\mathcal{P}(\mathbb{R})$. By $\mathcal{L}^{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$ we denote the space of functions f on $\mathbb{R}^{n}$ such that

[TABLE]

We refer to the recent paper [11] for further details on these scales of spaces. We present an equivalent quasi-norm for the space $\mathcal{L}^{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$, where the proof is quite similar to that for Lemma 5.

Lemma 7

Let $p,q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity. Then

[TABLE]

Moreover,

[TABLE]

Applying this lemma and [4, Theorem 5.2.1] we obtain.

Theorem 8

Let $\theta\in(0,1)$ and $q\in\mathcal{P}(\mathbb{R})$ be log-Hölder continuous both at the origin and at infinity with $q(0)=q_{\infty}$. Then

[TABLE]

with $p=\frac{1}{1-\theta}$.