On the composition operator with variable integrability

1.   Introduction

Let $(X, d, \mu)$ be a metric measure space equipped with a metric $d$ and the Borel regular measure $\mu.$ If $\varphi: X\rightarrow X$ is a non-singular map, i.e.,

then we can define the composition operator

According to (1.1), the measure $(\mu\circ\varphi^{-1})(\cdot): = \mu(\varphi^{-1}(\cdot))$ is absolutely continuous with respect to the measure $\mu$. From the Radon-Nikodym theorem, there exists a measurable function $u_{\varphi}:X\to [0, \infty +]$ such that

The composition operator $C_{\varphi}$ appears naturally in the context of variable change and has recently been applied to dynamical systems, partial differential equations, and data science. For more details, see ^[3,8,16,32]. For seminal applications, refer to ^[23,24].

In $L^{p}$ spaces, characterizing of the boundedness of composition operators is a fundamental problem (e.g., see ^[5,27,28,29]). For other function spaces, see ^{[1,6,7,13,20,25,30,31]}. It is well-known that $C_{\varphi}$ continuously maps $L^{p}(X)$ into itself if and only if the function $u_{\varphi}$ is essentially bounded on $X$. In such a case, $\varphi$ is said to induce a composition operator on $L^p(X)$.

The first question that we address in this article is:

Q.1 What kind of control should be imposed on the variable exponent $p:X\rightarrow \mathbb{R}$ such that the map $\varphi$ induces a composition operator $C_{\varphi}$ on $L^{p(\cdot)}(X)$?

A fruitful analysis of variable integrability spaces $L^{p(\cdot)}(X)$ is obtained when $X$ is a Euclidean domain, e.g., ^[9,11,12]. Unfortunately, the same cannot be said for general metric measure spaces. However, the authors in ^[15] achieved boundedness of the maximal operator with a notion of "dimension" through a local uniformity condition:

where, $B(x, r): = \{z\in X:d(z, x) < r\}, \ r\geq0,$ see also ^[22] by endowing $X$.

An essential difficulty in proving the boundedness of $C_{\varphi}$ over all $L^{p(\cdot)}(X)$ is that Cavalier's principle (see [4, Lemma 1.10]) does not hold. This motivates our question about appropriate control conditions (Q.1). A control on the variable exponent $p(\cdot)$ compatible with a condition like (1.3), which we propose in this article, is:

for every ball $B\subset X.$ Note that when $p(\cdot)$ is constant, the condition (1.4) is trivial. It seems natural to impose a "control" on the local supremum and infimum of the variable exponent as a new ingredient in our analysis.

Another important point is that our boundedness result for $C_{\varphi}$ over $L^{p(\cdot)}(X)$ can be extended in several directions including:

● $n$-dimensional Euclidean domains on $\mathbb{R}^{n}$ with Lebesgue measure and Euclidean distance.

● Complete Riemannian manifolds of positive Riemannian measure and distance.

● Locally compact and separable group equipped with a left-invariant metric and left-invariant Haar measure.

Currently, the boundedness of $C_{\varphi}$ on variable integrability spaces such as $L^{p(\cdot)}(X)$ is not fully understood, even for $n$-dimensional Euclidean domains. However, boundedness results for these operators are known in more regular function spaces, such as variable exponent Bergman spaces (see ^[26]) and, recently, in holomorphic function spaces ^[21]. Additionally, the inequality

does not hold in $L^{p(\cdot)}(X)$ unless $p(\varphi(\cdot)) = p(\cdot)$ almost everywhere in $X.$

Regarding compactness, it is well-known that $L^{p}(X)$ with $p\geq1$ does not support compact composition operators if $X$ has no atoms. In this paper, we show that metric measure spaces satisfying a local uniform property as in $(1.3)$ have no atoms. This allows us to provide a different proof from the constant exponent case, using recent developments on precompact sets on $L^{p(\cdot)}(X)$ obtained in ^[14].

We also study a related class of operators:

This class $T_{\varphi}$ represents the natural extension to the variable exponent case. Recent studies on boundedness, compactness, and closed-range properties for such operators have been conducted in ^[2,10] for bounded exponents defined on complete $\sigma$-finite spaces.

In this paper, we study the boundedness of $T_{\varphi}$ by providing a simple proof on $L^{p(\cdot)}(X)$, where $X$ is a metric measure space with a doubling measure $\mu$ and unbounded exponents $p(\cdot)$. On the other hand, the non-compactness of $T_{\varphi}$ is obtained when $X$ is a connected space. Particularly, in some cases, $T_{\varphi}$ maps $L^{p(\cdot)}(X)$ to $L^{p(\cdot)}(X)$. For example, when

the right-hand embedding implies, in particular, that $p(\varphi(x)) \geq p(x)$ a.e. in $x \in X$. Moreover, this implies the right inequality in (1.4). In this sense, the results concerning the operator $C_{\varphi}$ are obtained with a weaker hypothesis by replacing the embedding with regularity of log-continuous type, which we adapt to the environment of metric measure spaces.

Finally, another property that has not been explored in the framework of spaces with variable integrability is the weak compactness of the operator $T_{\varphi}$, even in the Euclidean case. In this direction, we show that the operator $T_{\varphi}$ behaves well on weakly compact sets in $L^{p(\cdot)}([0, 1])$, and some results are obtained in the non-reflexive setting $p^{-} = 1$. In fact, our partial results allow us to state the following conjecture: Let $p^{-} = 1$,

Let us now describe the organization of the article. In Section 2, we fix the notations and recall the definitions and a few results that will be important in our work. In Section 3, we study continuity and compactness for the operator $C_{\varphi}$. Finally, we study some properties of the operator $T_{\varphi}$ in Section 4.

2.   Preliminaries and notations

We assume throughout the paper that $\mathcal{B}_{X}$ is the $\sigma$-algebra of Borel generated by $\mu$-measurable open sets in $X$, the measure $\mu$ of every open nonempty set is positive, and the measure of every bounded set is finite on $X$.

2.1.   Doubling measure and $Q$-Ahlfors property

We define the property of doubling measure, which endows a metric measure space with good properties; for more details, see ^[4,17].

Definition 1. A measure $\mu$ is said to satisfy the doubling condition if there exists a positive constant $C$ such that

Another fruitful property for a metric measure space is the regular $Q$-Ahlfors property, which in some cases is stronger than the doubling property (see ^[15]).

Definition 2. We say that the measure $\mu$ is lower Ahlfors $Q$-regular if there exists a positive constant $C$ such that $\mu(B) \leq C \ diam(B)^{Q}$ for every ball $B\subset X$ with $diam B\leq 2 \ diam X.$ We say that $\mu$ is upper Ahlfors $Q$-regular if there exists a positive constant $C$ such that $\mu(B) \geq C \ diam(B)^{Q}$ for every ball $B \subset X$ with $diam B \leq 2 \ diam X$. The measure $\mu$ is Ahlfors $Q$-regular if it is upper and lower Ahlfors $Q$-regular, i.e., if

2.2.   The variable exponents class

The class of variable exponents, denoted by $\mathcal{P}(X)$, is defined by

Given $A\subset X$ and $p(\cdot)\in \mathcal{P}(X)$, we put

When the domain is clear, we simply write $p^{+} = p^{+}(X)$ and $p^{-} = p^{-}(X).$ Some properties of regularity at infinity, relative to variable exponent, are useful for studying composition operators within the framework of non-standard functional spaces. For Euclidean domains and metric measures spaces, these properties are provided in ^[9,11] and ^[14,15], respectively.

Definition 3. Let $p(\cdot)\in \mathcal{P}(X)$. We say that

1. $p(\cdot)$ is locally log-Hölder continuous, denoted by $p(\cdot)\in LH_{0}(X)$, if there exists a constant $K_0$ such that for all $x, y\in X, \ d(x, y) < \frac{1}{2}$,

2. $p(\cdot)$ is log-Hölder continuous at infinity with point base $x_0 \in X$, denoted by $p(\cdot) \in LH_{\infty}(X)$, if there are constants $K_{\infty}$ and $p_{\infty}$ such that for all $x\in X$,

When $p(\cdot)$ is log-Hölder continuous both locally and at infinity, we denote this by $p(\cdot)\in LH(X).$

In this work, we introduce a class of exponents associated with the $\varphi$-map. Denote by $\mathcal{B}_{0}$ the set of all open balls in $X.$

Definition 4. Let $\varphi: X\to X$ be a Borel measurable map. We define the following kinds of exponents:

where

In the following example, we show that $[\varphi]_{p^+}\geq 1$ for countable sub-covers, in fact, this is sufficient for the continuity result that we will obtain below (see Theorem 2).

Example 1. Let $X: = \mathbb{R}^{+}$ and define $p:\mathbb{R}^{+}\to[1, +\infty)$ as variable exponent given by

It is not difficult to show that

That is, $p(\cdot)\in LH_{\infty}(\mathbb{R}^{+}).$ In addition, it is easy to see that $p(\cdot)\in LH_{0}(\mathbb{R}^{+});$ for $x\in(0, 1)$ and $y\geq1$ such that it follows that

Similarly, for $x\geq 1$ and $y\in(0, 1).$ This is sufficient to show that $p(\cdot)$ is Lipschitz continuous on $\mathbb{R}^{+}$ and, therefore, $p(\cdot)\in LH_0(\mathbb{R}^{+}).$ Now, consider $\varphi:\mathbb{R}^{+}\to \mathbb{R}^{+}$ to be a non-singular map such that $0\leq\varphi(x)\leq x$ for all $x\in\mathbb{R}^{+}.$ On the other hand, let $\epsilon\in(0, +\infty)$ and we consider

Let $\{q_j\}_{j}\subset\mathbb{Q}^{+}$ be such that

Then, by using $LH_{\infty}$-regularity, we have

Hence, since $q_j\rightarrow +\infty$ as $j\to +\infty$

Therefore,

Finally, in particular, note that $L^{p(\cdot)}(\mathbb{R}^{+})$ is not embedded in $L^{p_{\varphi}(\cdot)}(\mathbb{R}^{+})$ and $p(\cdot)\in \mathcal{P}_{\varphi^+}^{\log}(\mathbb{R}^{+})$ (in the sense of accounting coverages).

2.3.   Variable integrability spaces

The space $L^{p(\cdot)}(X)$ is the classical variable Lebesgue space on $X$. Some of its basic properties have been studied, for instance, in ^[15].

Definition 5. Let $p(\cdot)\in \mathcal{P}(X)$ and $L_{0}(X)$ be the set of measurable functions on $X.$ The Lebesgue space with variable exponent $L^{p(\cdot)}(X)$ is defined by

where

is called the associated modular with $p(\cdot)$. It is known that $L^{p(\cdot)}(X)$ is a Banach space equipped with the Luxemburg-Nakano norm

Additionally, when $p(\cdot)$ is bounded, we obtain the inequality that provides a relation between the norm and the modular:

for $f\in L^{p(\cdot)}(X).$ Also, in the case $p^{+} < +\infty$, the space $L^{p(\cdot)}(X)$ can be defined as all measurable functions $f:X\rightarrow \mathbb{R}$ such that $\rho(f) < +\infty$, and the dual space is identified with $L^{p'(\cdot)}(X)$, where $p'(\cdot)$ is the conjugate exponent relative to $p(\cdot)$, that is,

It is well-know that the space of essentially bounded functions with compact support $L^{p(\cdot)}_{c}(X)$ is dense in $L^{p(\cdot)}(X).$

3.   Some properties for $C_{\varphi}$

In this section, we start analyzing the behavior and properties of the operator $C_{\varphi}$ on $L^{p(\cdot)}(X)$ based on the following results in the space $L^{p}(X):$

P.1 $C_{\varphi}:L^{p}(X)\rightarrow L^{p}(X)$ is bounded if and only if $u_{\varphi}\in L^{\infty}(X).$ In particular,

In the second part, we completely characterize the continuity and compactness of operator $C_{\varphi}$ on $L^{p(\cdot)}(X).$

3.1.   On the property P.1 in $L^{p(\cdot)}(X)$

Our first result shows that the integral inequality does not hold in general for any variable exponent, unless the exponent is both bounded and invariant under dilations or contractions induced by $\varphi.$ The variable exponent induced by $\varphi$ is denoted by $p_{\varphi}(\cdot): = p(\varphi(\cdot)).$ Additionally, we associate $\varphi$ with the number

Theorem 1. Assume $p(\cdot)\in\mathcal{P}(X)$ with $p^{+} < +\infty$ and let $\varphi:X\rightarrow X$ be a non-singular measurable transformation. Then, the following statements are equivalent:

M.1 There exists a constant $C > 0$ such that

M.2 The function $u_{\varphi}:X\rightarrow \mathbb{R}$ is essentially bounded on $X$, and

Proof. It is clear that $(M2) \Rightarrow (M1)$ with $C: = \|u_{\varphi}\|_{\infty}.$ To prove $(M1) \Rightarrow (M2),$ first suppose that $p^{+} < \infty;$ since $\varphi$ is non-singular, we have $p\circ\varphi\in L^{\infty}(X)$ so

Assume that $(M.1)$ holds, so for $B\in \mathcal{B}_{0},$ with $\mu(B) < +\infty.$ Considering the function $f = \mathcal{X}_{B} \in L^{p(\cdot)}(X),$ from inequality (3.1), we deduce that

Now, let $\Omega_{\varphi}: = \{x\in X: p(x)\neq p_{\varphi}(x)\}$ and suppose that $\mu(\Omega_{\varphi}) > 0.$ Thus, if $E_{\varphi}: = \{x\in X:p(x) > p_{\varphi}(x)\}\in\mathcal{B}$, then by $\sigma$-additivity, we have

So, we see only the case $\mu(E_{\varphi}) > 0$ because the case $\mu(\Omega_{\varphi} \setminus E_{\varphi}) > 0$ is analogue with minor settings. In fact, since $\mu(E_{\varphi}) > 0$, then by [17, Lemma 3.3.31], we can take $y\in E_{\varphi}$ such that it satisfies

Hence, since $\varphi^{-1}(X) = X$ from (3.2), we can choose $B_{y}\in\mathcal{B}_{0}$ (sufficiently large rate) such that

Hence, using (3.1) with the functions $f_{n}(\cdot) = n^{\frac{1}{p(\cdot)}} \mathcal{X}_{B_{y}}(\cdot)\in L^{p(\cdot)}(X),$ we obtain that

Thus,

By the classical Jensen's inequality we have

so, taking $n\rightarrow +\infty$

Therefore, $\mu(E_{\varphi}\cap\varphi^{-1}(B_{y})) = 0,$ which is in contradiction with (3.3). Consequently, $\mu(E_{\varphi}) = 0.$ In addition, $u_{\varphi}\in L^{+\infty}(X)$ follows from (3.2) with $B = E\in \mathcal{B}_{X}.$ □

3.2.   Continuity for $C_{\varphi}$

In this section, we extend the classical continuity result in standard $L^{p}$ spaces to function spaces with variable integrability $L^{p(\cdot)}(X)$. The proof strategy is inspired by the argument given by Cruz-Uribe and Fiorenza on the boundedness of the maximal operator in $L^{p(\cdot)}(X)$ when $X$ is a Euclidean domain (see [9, Theorem 3.16]). For this purpose, we adapt [9, Lemmas 3.26 and 3.24] to the setting of metric measure spaces with Ahlfors regularity; see also ^[15].

Lemma 1. Let $\varphi:X\rightarrow X$ be a non-singular Borel map. Then,

(i) If $p(\cdot)\in LH_{0}(X)$ and $\mu$ is lower Ahlfors $Q$-regular, then there exists a positive constant $C$ such that

(ii) If $p(\cdot)\in LH_{\infty}(X)$ and $\mu$ is upper Ahlfors $Q$-regular, then there are positive constants $C_1, C_2$ such that for every function $f$ with $0\leq f\leq 1$ on $E\in \mathcal{B}$, we have

Similarly,

Proof. $(i)$ Since $p(\varphi(x))-p^{+}_{B}\leq0$ for $x\in \varphi^{-1}(B)$, it suffices to check $(i)$ for balls $B: = B(v, r)$ with $r\leq 1/2.$ For $y_{0}\in B$ such that

from $p(\cdot)\in LH(X)$, it follows that

So,

Therefore, since $\mu$ is lower Ahlfors $Q$-regular and $p(\varphi(x))-p_{B}^{+}\leq 0$, we have

$(ii)$ For $x_{0}\in X,$ define the function $h_{x_0}: X\rightarrow \mathbb{R}$ given by $h_{x_0}(x): = (e+d(x, x_{0}))^{-r}, \ \ x\in X.$ First, we show that

Indeed, for the base point $x_0\in X$, we consider the countable collection $\{C_{j}: j\in\mathbb{N}\}$ where,

Hence,

On the other hand, decompose $\varphi^{-1}(E): = F_{1}\cup F_{2}$ where

On $F_{1}$ we have,

By the $LH_{\infty}$-regularity,

Finally, since $f(\varphi(x))\leq 1$, we get

□

With the previous lemma, the main result of this work in response to (Q.1) is the following:

Theorem 2. Assume that $\mu$ is a $Q$-Ahlfors regular measure on $X$, let $\varphi: X \rightarrow X$ be a non-singular map. Then:

$(C1)$ Let $p(\cdot) \in \mathcal{P}_{\varphi^{+}}^{log}(X).$ If the map $u_{\varphi}$ is essentially bounded on $X$, then the operator $C_{\varphi}$ maps $L^{p(\cdot)}(X)$ into itself.

$(C2)$ Let $p(\cdot) \in \mathcal{P}_{\varphi^{-}}^{log}(X).$ If the operator $C_{\varphi}$ maps $L^{p(\cdot)}(X)$ into itself, then there exists a positive constant $C$ such that $\mu(\varphi^{-1}(B)) \leq C \, \mu(B)$ for every ball $B \in \mathcal{B}_{0}$, i.e.,

$(C3)$ Assume $p(\cdot)\in \mathcal{P}_{\varphi}^{log}(X).$ If, in addition, $\mu$ is a doubling measure on $X$, then $(C1)$ and $(C2)$ are equivalent.

Proof. $(C1).$ Suppose that $u_{\varphi}\in L^{\infty}(X).$ Since $p^{+} < +\infty$, by density it is sufficient to show that

Indeed, let $f\in L_{c}^{\infty}(X)\cap L^{p(\cdot)}(X)$ with $\|f\|_{p(\cdot)}\leq 1$ and consider the decomposition $|f| = f_1+f_2$ where $f_1 : = |f|\chi_{\{|f| > 1\}}$ and $f_2: = |f|\chi_{\{|f|\leq 1\}}.$ We will divide the test into two steps:

Step 1. Let us see that there exists a constant $C > 0$ independent of $f_{1}$ such that

Since $f_{1}\in L^{\infty}(\mathbb{R})$ and $X$ is separable, it follows that the open cover of $X$

with $K: = \text{ess} \sup_{x\in X}(f_{1}\circ\varphi)(x) > 1$ and $r_{Q}: = K^{-\frac{1}{Q}},$ admits a pairwise disjoint countable open sub-cover $\{B^{j}_{5r_{Q}}: j\in\mathbb{N}\}$ such that

From the $\mu$-regularity, $\mu(B^{j}_{r_{Q}})\leq C_{Q} \ K^{-1}$ for all $j\in\mathbb{N}.$ In addition, since $p(\cdot)\in LH_{0}(X)$, by using Lemma 1 there exists $M > 0$ such that for $B\in \mathcal{B}_{0}$ and $x\in\varphi^{-1}(B)$, we have $\mu(B)^{p_{\varphi}(x)-p^{+}(B)}\leq M$; also, by the fact that $K\in(1, +\infty)$ taking $r(\cdot): = p(\cdot)-p_{\varphi}(\cdot)$ for $x\in X$, we get

So,

From $[\varphi]_{p^{+}}\geq 1,$

Hence,

Therefore, the inequality (3.4) is obtained with $C_{1}: = (M+1)\mathcal{U}(\varphi).$ Now, we estimate the size of $C_{\varphi}f_{2}.$

Step 2. There exists a constant $C_2 > 0$ independent of $f_{2}$ such that

Since $p(\cdot)\in LH_{\infty}(X),$ we get $f_{2}\in L^{p_{\infty}}(X)$ and $u_{\varphi}\in L^{\infty}(X)$ implies also that $C_{\varphi}f_{2}\in L^{p_{\infty}}(X).$ Hence, for some $C > 0$ independent of $f_{2}$, we obtain

this last integral is finite by Lemma 1. Therefore, from (3.4) and (3.5), we obtain

$(C2).$ Suppose that

By the inequality (2.1) and the relation given by (1.2), it is sufficient to consider the case when $B\in\mathcal{B}_{0}$ is such that

In this case, since $p(\cdot)\in LH_{0}(X)$ and $[\varphi]_{p^{-}}\leq 1$, applying Lemma 1, we can find a constant $C > 0$ such that,

Finally, in order to obtain $(C3)$ by assuming the doubling property of $\mu$, note that

Hence, by differentiation, it is easy to see that $\mathcal{U}(\varphi)$ is finite if and only if $u_{\varphi}$ is in $L^{\infty}(X).$ □

Remark 1. According to the proof of the above theorem:

● Another hypothesis to obtain $(C1)$ is $p_{\varphi}(\cdot)\geq p(\cdot)$ a.e. in $X$ and $p(\cdot)\in LH_{\infty}(X)$. Note that in our result, we replace $p_{\varphi}(\cdot)\geq p(\cdot)$ by hypothesis $p(\cdot)\in LH_{0}(X)$ and $[\varphi]_{p^{+}}\leq 1$, which also replaces the embedding $L^{p_{\varphi}(\cdot)}(X)\hookrightarrow L^{p(\cdot)}(X)$ proposed in [2, Theorem 3.4].

● In the proof of the above theorem, note that the condition (1.4) applies to a uniform Vitali cover of the space. In this sense, control over the inductor map $\varphi$ and the exponent $p(\cdot)$ can be relaxed as shown in Example 1. In fact, note that for a suitable $\varphi$ (e.g., $\varphi(x): = Ax$, $x\in\mathbb{R}^{+}$), it suffices to note that

3.3.   Compactness for $C_{\varphi}$

We start this section by showing that $L^{p(\cdot)}(X)$ does not support non-trivial compact composition operators $C_{\varphi}$. In addition, we approach recent results related to weak compactness in variable Lebesgue spaces and provide some properties for $C_{\varphi}.$

Lemma 2. Let $\mu$ be a $Q$-Ahlfors regular measure on $X$, $\varphi: X \rightarrow X$ be a non-singular Borel map, and $p(\cdot) \in LH_{0}(X)$ such that $[\varphi]_{p^{-}} = 1$. There exists a positive constant $C_{0}$ such that, given $A \in \mathcal{B}_{0}$ with $\mu(A) > 0$, if for any ball $B$ with $A \cap B \neq \emptyset$ and $\mu(B) < 1$, then

Proof. On one hand,

On the other hand, since $[\varphi]_{p^{-}} = 1$

Therefore, for $x\in \varphi^{-1}(A\cap B)$, it follows that $\mu(B)^{\frac{p(x)}{p_{\varphi}(x)}}\leq \mu(B)^{\frac{p^{-}(B)}{p^{+}(B)}}.$ Hence, since $p(\cdot)\in LH_{0}(X)$ by [15, Lemma 3.6], there exists $C_{0} > 0$ such that

□

Theorem 3. Let $\mu$ be a Ahlfors $Q$-regular and doubling measure on $X$. The non-trivial bounded composition operator $C_{\varphi}$ is not compact on $L^{p(\cdot)}(X).$

Proof. Note initially that the space $(X, d, \mu)$ does not contain atoms. Indeed, if $E\in \mathcal{B}_{X}$ is an atom, then $\mu(E) > 0$, so there exists $a\in E$ such that $\mu(E\cap B(a, r)) > 0$ for all $r > 0.$ Hence, if

then the atomicity of $E$ implies that $\mu(E\cap B(a, r)) = 0,$ a contradiction. On the other hand, if $\mu(E\cap B(a, r)) = \mu(E)$ for all $r > 0$, then by using the $Q$-Ahlfors property of $\mu$, we obtain $0 < \mu(E)\leq C (2r)^{Q}$, thus $\mu(E) = 0$ as $r\rightarrow0^{+}$ which is also a contradiction. Besides, suppose that $C_{\varphi}$ is compact on all $L^{p(\cdot)}(X)$ and we consider for $\epsilon\in(0, +\infty)$ the set

If $\mu(U_{\epsilon}) > 0$ for some $\epsilon > 0$, then from the non-atomicity of $\mu$ it follows that there exists a decreasing sequence $\{U_{n}\}$ such that

Let's construct a bounded sequence in $L ^{p(\cdot)}$-norm that is not equi-integrable in $L^{p(\cdot)};$ given $\delta > 0,$ since $\varphi:X\rightarrow X$ is non-singular, choose $\rho > 0$ such that

For this $\rho > 0$, there exists $N > 0$ which $1/N\leq \rho.$ Now, since $\mu(U_{n}) > 0$ for all $n\in\mathbb{N}$ by [17, Lemma 3.3.31] we can fix $x_{n}\in U_{n}$ satisfying $\mu(U_{n}\cap B(x_{n}, \delta)) > 0.$ Consider the set $A_{\delta}\in \mathcal{B}_{0},$ given by

it is clear that $\mu(A_{\delta}) < \delta.$ Moreover, by using Lemma 2, the function $f_{N}:X\rightarrow \mathbb{R}$ given by

is such that

By differentiation,

Therefore, taking $\delta\in (0, +\infty)$ small enough, we can suppose that we have the following lower bound:

Thus,

By switching to a subsequence if necessary, we have that the sequence $\{C_{\varphi} f_n\}$ is not equi-integrable in $L^{p(\cdot)}(X)$. Thus, by virtue of [14, Theorem 1], this contradicts the compactness of $C_{\varphi}$. Consequently, $\mu(U_{\epsilon}) = 0$ for all $\epsilon\in(0, +\infty)$ or equivalently $u_{\varphi}(x) = 0$ a.e. in $x\in X.$ This implies that,

that is, $C_{\varphi}f = 0$ on $\varphi^{-1}(B)$ for all $B\in\mathcal{B}_{0}$ and it is enough for to get $C_{\varphi} = 0.$ □

4.   Some properties for $T_{\varphi}$

4.1.   $L^{p(\cdot)}$-boundedness of $T_{\varphi}$

In this second part, we provide a complete characterization of the continuity for composition operators in the framework of variable Lebesgue spaces.

Theorem 4. Let $\mu$ with doubling property on $X$, $p(\cdot)\in \mathcal{P}(X)$, and $\varphi:X\rightarrow X$ be a non-singular Borel measurable map. Then, the composition operator $T_{\varphi}$ maps space $L^{p(\cdot)}(X)$ into $L^{p_{\varphi}(\cdot)}(X)$ if and only if the function $x\mapsto u_{\varphi}(x)^{1/p(x)}$ is essentially bounded, that is, $u_{\varphi}(\cdot)^{1/p(\cdot)}\in L^{\infty}(X).$ Moreover,

Remark 2. Theorem 4 generalizes the well-known result in the framework of Lebesgue spaces with constant exponent. More precisely, when $p(\cdot) = p\geq1$, then both the induced exponent and the space induced by the measurable map $\varphi$ remain invariant, that is,

Remark 3. Since $f\in L^{p(\cdot)}(X)$ if and only if $f^{p(\cdot)-1}\in L^{p'(\cdot)}(X)$ (the same is true for $p'(\cdot)$), by Theorem 4, it follows that $T_{\varphi}$ maps $L^{p(\cdot)}(X)$ into $L^{p_{\varphi}(\cdot)}(X)$ if and only if $T_{\varphi}$ maps $L^{p'(\cdot)}(X)$ into $L^{p'_{\varphi}(\cdot)}(X).$ An interesting question is whether this is true for the operator $C_{\varphi}$ acting on $L^{p'(\cdot)}$ and $L^{p(\cdot)}.$

Remark 4. In the case that

Theorem 4 provided a sufficient condition for the operator $T_{\varphi}$ to map $L^{p(\cdot)}(X)$ into itself. In fact, the embedding (4.1) provided a class of maps $\varphi:X\rightarrow X$ induced composition operators on $L^{p(\cdot)}(X).$

Remark 5. Note that from Remark 4, the embedding condition (4.1) can be modified by assuming that the variable exponent decays to infinity; for example, assume $p_{\varphi}(x)\geq p(x)$ a.e. in $x\in X$ (e.g., see [9, Theorem 2.45]) and $p(\cdot)\in LH_{\infty}(X).$

Remark 6. If the function $x\mapsto u_{\varphi}(x)$ is bounded, then for any $p(\cdot)\in\mathcal{P}(X)$ the function $x\rightarrow u_{\varphi}(x)^{1/p(x)}$ is also bounded. So, by Theorem 4, we obtain the following weak inequality: there exists $C > 0$ depending of $\varphi$ such that for each $t > 0$ and $f\in L^{p(\cdot)}(X),$

Proof. (Proof of Theorem 4) Suppose that $T_{\varphi}$ maps $L^{p(\cdot)}(X)$ into $L^{p_{\varphi}(\cdot)}(X)$. From closed graph theorem there exists $C > 0$ such that $\|T_{\varphi}\|\leq C.$ Let us show that $u_{\varphi}(\cdot)^{1/p(\cdot)}\in L^{\infty}(X);$ note first that $u_{\varphi}(\cdot)^{1/p(\cdot)}\in L^{1}_{loc}(X)$, so let $B\in \mathcal{B}_{0}$ and define the function $f:X\rightarrow \mathbb{R}$ by

Hence, it is clear that $\rho(f/\lambda)\leq 1$ for all $\lambda\geq 1$ so $f\in L^{p(\cdot)}(X)$ and $\|f\|_{p(\cdot)} = 1$. This implies $f\circ\varphi\in L^{p_{\varphi}(\cdot)}(X)$ and $\|f\circ\varphi\|_{p_{\varphi}(\cdot)}\leq C,$ that is, there exists $\lambda_{0} > 0$, which $\lambda_{0} < C$ and

Consequently,

by differentiation, it follows that the map $x\mapsto C^{-p(x)} u_{\varphi}(x)$ is essentially bounded. Reciprocally, denote by $M_{\varphi}$ the multiplication operator with symbol $u_{\varphi}(\cdot)^{1/p(\cdot)}$ so if $u_{\varphi}(\cdot)^{1/p(\cdot)}\in L^{\infty}(X)$, then for each $f\in L^{p(\cdot)}(X),$

Since $L^{p(\cdot)}(X)$ is a lattice,

Hence,

that is, $T_{\varphi}$ maps $L^{p(\cdot)}(X)$ into $L^{p_{\varphi}(\cdot)}(X)$ and $\|T\|\leq \operatorname{ess} \sup\nolimits_y \left\{u_{\varphi}(y)^{1/p(y)}\right\},$ computing with the normalized functions given in the equality of the norm (4.2). □

4.2.   Compactness for $T_{\varphi}$

In the study of the compactness of $C_{\varphi}$ over $L^{p(\cdot)}$, the presence of non-atomic sets was a consequence of the regular Ahlfors structure of the space, which was necessary to have at least one continuous composition operator. For the case of the operator $T_{\varphi}$, continuity only requires that the space admits a doubling measure, which is a weaker hypothesis than the Ahlfors regularity. Therefore, to guarantee no atomic sets, we assume that the space is connected.

Lemma 3. Every connected doubling metric measure space $(X, d, \mu)$ does not contain atoms.

Proof. Assume that $E\in \mathcal{B}_{0}$ is an atom, then $\mu(E) > 0$, which implies $\mu(E\cap B(w, r)) > 0$ for some $w\in E$ and all $r > 0.$ Hence, in the case $\mu(E\cap B(w, r)) = \mu(E)$ by using [4, Lemma 3.7], there are constants $C, \sigma > 0$ such that

from here that, $\mu(E) = 0$ as $r\rightarrow0^{+},$ which is a contradiction. On another case, $\mu(E\cap B(w, r)) < \mu(E)$, but the atomicity of $E$ implies that $\mu(E\cap B(w, r)) = 0,$ which is a contradiction. □

As a consequence of Lemma 3, the following result is well-know in the framework of non-atomic metric measure spaces (see, [2, Theorem 4.2] and [10, Theorems 5.2 and 5.3]).

Theorem 5. Assume that $(X, d, \mu)$ is a metric measure space with doubling measure, $r(\cdot)\in \mathcal{P}(X)$ such that $1\leq p^{-}\leq p^{+} < +\infty,$ and $\varphi:X\rightarrow X$ Borel non-singular map. If $X$ is connected, then the space $L^{r(\cdot)}(X)$ does not admit compact composition operators $T_{\varphi}.$

4.3.   Weak compactness of $T_{\varphi}$ on $L^{p(\cdot)}([0, 1])$

In the case $1 < p^{-}\leq p^{+} < +\infty$, it is well-know that $L^{p(\cdot)}$ is a reflexive space, so every composition operator on $L^{p(\cdot)}$ is weakly compact. The non-reflective case $(p^{-} = 1)$ is different and has been explored in ^[29] for the constant exponent case. In the following theorem, we provide some results in this direction.

Denote by $\lambda$ the Lebesgue measure on $[0, 1]$ or $\mathbb{R}$ and $\Omega_{1}: = p^{-1}(\{1\})$ for an exponent $p(\cdot)$.

Theorem 6. Let $r(\cdot) \in \mathcal{P}([0, 1])$ with $\lambda(\Omega_{1}) = 0$ and $r^{+} < \infty$. Let $\varphi: [0, 1] \rightarrow [0, 1]$ be a non-singular map such that

Then, the following properties hold:

(w.1) $T_{\varphi}$ maps relatively weakly compact subsets into relatively weakly compact subsets.

(w.2) Let $v_{\varphi}(\cdot) : = r(\cdot)^{1/p(\cdot)}$. The operator $T_{\varphi}$ is relatively weakly compact if and only if the multiplication operator $M_{v_{\varphi}}$ is relatively weakly compact.

(w.3) Let $r^{-} = 1$. If

then the operator $T_{\varphi}: L^{r(\cdot)}([0, 1]) \rightarrow L^{r_{\varphi}(\cdot)}([0, 1])$ is not weakly compact.

Proof. To prove $(w1)$, assume that $M > 0$; then since $r^{-} = 1$, we may choose a sequence $(z_n)\subset[0, 1]\setminus\Omega_{1}$ satisfying $r(z_n)\rightarrow1$ when $n\rightarrow +\infty.$ So, denote by

Let us reason by contradiction that if $T_{\varphi}$ is weakly compact, then the subset $T_{\varphi}(B)$ is relatively weakly compact in $L^{r_{\varphi}(\cdot)}([0, 1]).$ Hence, by [18, Theorem 4.3] we have

Since,

by (4.3) we get

Now, let $\mathcal{I}_{0}': = \{[a, b]\in\mathcal{I}_0:[a, b]\subset[0, 1]\}$ and we consider the functions $f:\Omega\rightarrow \mathbb{R}$ given by

So, it is clear that $\{f_{ab}:[a, b]\in\mathcal{I'}_{0}\}\subset B$ and thus from (4.5), given $\epsilon > 0$, there exists $\lambda_0 > 0$ small such that

by differentiation,

In particular, taking $z = z_n$ and $n\rightarrow +\infty$, we obtained $M < \liminf u_{\varphi}(z_n) = 0$ because $\epsilon$ is taken arbitrarily so $M < 0$, which is a contradiction. The property $(w2)$ follows from (4.4), [18, Theorem 4.3] and from the fact $\mathcal{U}(\varphi) < \infty.$ Finally, the property $w3$ follows from 4.4 and [18, Theorem 4.3]. □

Remark 7. The proof of Theorem 6 can be easily extended to $L^{r(\cdot)}(\mathbb{R})$ by applying results recently obtained in [19, Section 5] under the restriction that $\Omega_1$ be a null-set.

The case $\lambda(\Omega_1) > 0$ and $r^{-} = 1$: Suppose that $\varphi^{-1}(\Omega_1)\subset\Omega_1,$ let us choose $z_0\in\mathbb{R}$ such that $p(z_0) = 1$ and for each $n\in\mathbb{N}$ define $I_n: = (z_0-1/2^{n}, z_0+1/2^{n})$ so $I_n\cap\Omega_1\neq \emptyset$ for each $n\in\mathbb{N}.$ We show that $T_{\varphi}(B)$ is not relatively weakly compact provided that $M > 0$; define the sequence of measurable functions $\{f_n\}$ by

It is clear that $f_n\in B$ for every $n\in\mathbb{N}.$ However, from $u_{\varphi}(x)\geq M$ a.e. $x\in\mathbb{R}$, we obtain

Taking a subsequence if necessary, let us make $\lambda(I_n)\rightarrow0$, this means $\lambda(\varphi^{-1}(I_n))\rightarrow0$ and by differentiation

Therefore,

Applying [19, Proposition 5.11], we yields the assertion for $T_{\varphi}(B).$

5.   Conclusions

Lebesgue spaces with variable integrability have proven to be an excellent framework for partial differential equations with non-standard growth. In particular, eigenvalue problems have been analyzed in Sobolev spaces with a constant exponent, where the composition operator has revealed connections with eigenvalue estimates. In this direction, one can not only study the applicability of the composition operator on Sobolev spaces of variable integrability in an $n$-dimensional Euclidean domain but also extend these techniques when defining such spaces on a complete Riemannian manifold. Therefore, in this work, we outline a possible direction of study within the framework of non-standard function spaces.

Author contributions

Javier Henríquez-Amador and Carlos Álvarez: Writing-original draft, formal analysis and commenting; Eiver Rodríguez and John Millán: Commenting and review. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors thank the anonymous referees for the useful suggestions to improve this article. Carlos F. Álvarez and Jhon Millán were partially supported by inner project BASEX-PD/2024-02 of University of Sinu, Cartagena.

Conflict of interest

We declare there are no conflicts of interest associated with this work