Embedding theorems for variable exponent fractional Sobolev spaces and an application

1.   Introduction

Let $R$ be an associative ring. Throughout the paper we will denote by $R^{\times}$, the set of all invertible elements of $R$. For $x, y\in R$, we set

A map $f$: $R\rightarrow R$ is said to be additive if

for all $x, y\in R$.

In 1999, Essannouni and Kaidi ^[1] obtained the following result, which generalized a well-known result due to Hua ^[2].

Theorem 1.1. [1,Theorem A] Let $D$ be a division ring with $D\neq F_2$, the field of two elements. Let $R = M_n(D)$ be the ring of $n\times n$ matrices with $n\geq 2$. Let $f$: $R\rightarrow R$ be a bijective additive map satisfying the identity

for every $x\in R^{\times}$. Then, $f$ is either an automorphism or an antiautomorphism.

In 2005, Chebotar et al. ^[3] proved that a bijective additive map $f$ on a division ring $D$ that satisfies

for all $x, y\in D^{\times}$ must have the form

where $\varphi$ is an automorphism or antiautomorphism, and $f(1)$ is a central element of $D$. In 2006, Lin and Wong ^[4] generalized this result to matrix rings.

In 2018, Catalano ^[5] initiated the study of maps preserving products of division rings. More precisely, she proved the following result.

Theorem 1.2. [5,Theorem 5] Let $D$ be a division ring with characteristic different from $2$. Let $Z$ be the center of $R$. With $m, k\in D^{\times}$, let $f$: $D\rightarrow D$ be a bijective additive map satisfying the identity

for every $x, y\in D^{\times}$ such that $xy = k$. Then,

for all $x\in D$, where $\varphi$: $D\rightarrow D$ is either an automorphism or an antiautomorphism. Moreover, we have the following:

(1) If $\varphi$ is an automorphism, then $f(1)\in Z$.

(2) If $\varphi$ is an antiautomorphism, then $f(1) = f(k)^{-1}m$ and $f(k)\in Z$.

In 2019, Catalano et al. ^[6] initiated the study of maps preserving products of matrices. More precisely, they proved the following result.

Theorem 1.3. [6,Theorem 1] Let $D$ be a division ring with characteristic different from $2$. Let $R = M_n(D)$ be the ring of $n\times n$ matrices with $n\geq 2$, and let $Z$ be the center of $R$. With $m, k\in R^{\times}$, let $f$: $R\rightarrow R$ be a bijective additive map satisfying the identity

for every $x, y\in R^{\times}$ such that $xy = k$. Then,

for all $x\in R$, where $\varphi$: $R\rightarrow R$ is either an automorphism or an antiautomorphism. Moreover, we have the following:

(1) If $\varphi$ is an automorphism, then $f(1)\in Z$.

(2) If $\varphi$ is an antiautomorphism, then $f(1) = f(k)^{-1}m$ and $f(k)\in Z$.

The study of maps preserving products of matrices is an active topic. For recent results on maps preserving products of matrices, we refer the reader to ^{[7,8,9,10,11]}.

We point out that the proof of Theorem 1.3 is wrong. Indeed, they used the following identity due to Hua:

where $a, b, b^{-1}-a\in R^{\times}$. By taking $a = 1$ and $b = x\in R^*$ in (1.1), they obtained that

for all $x\in R^{\times}$, where

By using Theorem 1.1 they got that $\varphi$ is an automorphism or an antiautomorphism. However, the condition of $x^{-1}-1\in R^{\times}$ (equivalently, $1-x\in R^{\times}$) should be added in the use of (1.1). Thus, they cannot obtain that

for all $x\in R^{\times}$, where $f = f(1)\varphi$. In fact, they can obtain that

for all $x\in R^{\times}$ with $1-x\in R^{\times}$, which cannot use Theorem 1.1 to get that $\varphi$ is an automorphism or an antiautomorphism.

In the present paper we shall give the following result:

Theorem 1.4. Let $D$ be a division ring such that either char$(D)\neq 2, 3$ or $D$ is not a field and char$(D)\neq 2$. Let $R = M_n(D)$ be the ring of $n\times n$ matrices with $n\geq 2$, and let $Z$ be the center of $R$. With $m, k\in R^{\times}$, let $f, g$: $R\rightarrow R$ be bijective additive maps satisfying the identity

for every $x, y\in R^{\times}$ such that $xy = k$. Then,

for all $x\in R$, where $\varphi$: $R\rightarrow R$ is either an automorphism or an antiautomorphism. Moreover, we have the following:

(1) If $\varphi$ is an automorphism, then $g(x) = \varphi(x)g(1)$ for all $x\in R$.

(2) If $\varphi$ is an antiautomorphism, then $g(x) = f(k)^{-1}f(x)g(k)$ for all $x\in R$.

As a consequence of Theorem 1.4 we shall give the following result, which gives a correct version of Theorem 1.3.

Theorem 1.5. Let $D$ be a division ring such that either char$(D)\neq 2, 3$ or $D$ is not a field and char$(D)\neq 2$. Let $R = M_n(D)$ be the ring of $n\times n$ matrices with $n\geq 2$, and let $Z$ be the center of $R$. With $m, k\in R^{\times}$, let $f$: $R\rightarrow R$ be a bijective additive map satisfying the identity

for every $x, y\in R^{\times}$ such that $xy = k$. Then,

for all $x\in R$, where $\varphi$: $R\rightarrow R$ is either an automorphism or an antiautomorphism. Moreover, we have the following:

(1) If $\varphi$ is an automorphism, then $f(1)\in Z$.

(2) If $\varphi$ is an antiautomorphism, then $f(1) = f(k)^{-1}m$ and $f(k)\in Z$.

We organize the paper as follows: In Section $2$ we shall give the proof of Theorem 1.4. In Section 3 we shall give the proof of Theorem 1.5.

We remark that the method in the proof of Theorem 3 is different from that in the proof of Theorem 2. We believe that the method will play a certain role in the study of maps preserving products of matrices.

2.   Preliminaries

Throughout this section, let $D$ be a division ring and let $R = M_n(D)$ with $n > 1$. By $Z$ we denote the center of $D$. We identify $Z$ with the center of $R$ canonically. For $A\in R$, we denote by $|A|$ the determinant of $A$.

We begin with the following technical result, which will be used in the proof of our main result.

Lemma 2.1. Let $D$ be a division ring with char$(D)\neq 2, 3$. Let $R = M_n(D)$ with $n > 1$. For any $\alpha, \beta\in D$ and $1\leq i, j, k, l\leq n$, we set

We claim that either there exists $\gamma\in\{1, 2, 3\}$ such that

or there exist $\gamma_1, \gamma_2\in R^{\times}$ such that

for all $i = 1, 2$.

Proof. We prove the result by way of the following several cases:

Case 1. Suppose that $i = j = k = l$. We set

It is clear that $\gamma+T, \gamma+1+T\in R^{\times}$.

Case 2. Suppose that $i = j$, $k = l$, and $i\neq k$. We set

It is clear that $\gamma+T, \gamma+1+T\in R^{\times}$.

Case 3. Suppose that $i = j$ and $k\neq l$. We set

It is clear that $\gamma+T, \gamma+1+T\in R^{\times}$.

Case 4. Suppose that $i\neq j$ and $k = l$. We set

It is clear that $\gamma+T, \gamma+1+T\in R^{\times}$.

Case 5. Suppose that $i\neq j$, $k\neq l$, and $(i, j)\neq (l, k)$. It is easy to check that

for all $\gamma\in D$. We set $\gamma = 1.$ It is clear that

Case 6. Suppose that $i\neq j$, $k\neq l$, and $(i, j) = (l, k)$. We may assume that $i < j$. The case of $i > j$ can be discussed analogously. It is easy to check that

for all $a_i\in D$, $i = 1, \cdots, n$. Suppose first that $\alpha\beta\neq 1, 4$. We set $\gamma = 1.$ It follows that

This implies that

and

This implies that

Suppose next that $\alpha\beta = 1$ or $4$. We first discuss the case of $\alpha\beta = 1$. We set $\gamma = 2$. It follows that

This implies that

and

We get that

We now assume that $\alpha\beta = 4$. We set

It is clear that $\gamma_1, \gamma_2\in R^{\times}$. Note that

It is clear that

Note that

It is clear that

Note that

It implies that

This implies that

for $i = 1, 2$. The proof of the result is complete. □

The following technical result will be used in the proof of our main result.

Lemma 2.2. Let $D$ be a division ring, which is not a field. Suppose that char$(D)\neq 2$. Let $R = M_n(D)$ with $n > 1$. For any $\alpha, \beta\in D$ and $1\leq i, j, k, l\leq n$, we set $T = \alpha e_{ij}+\beta e_{kl}$. We claim that either there exists $\gamma\in\{1, 2, 3\}$ such that

or there exist $\gamma_1, \gamma_2\in R^{\times}$ such that

for $i = 1, 2$.

Proof. In view of Lemma 2.1, we may assume that char$(D) = 3$. It is clear that $D$ is a central division algebra over $Z$. It is well known that the dimension of a finite dimensional central division algebra is a perfect square (see [12,Corollary 1.37]). This implies that dim$_Z(D)\geq 4$. We now prove the result by way of the following two cases:

Case 1. Suppose first that $\alpha, \beta, 1$ are linearly independent over $Z$. We get that there exists $\gamma\in D$ such that $\gamma, \alpha, \beta, 1$ are linearly independent over $Z$. We now discuss the following six subcases:

Subcase 1.1. Suppose that $i = j = k = l$. We set

It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that

It is clear that

This implies that

as desired.

Subcase 1.2. Suppose that $i = j$, $k = l$, and $i\neq k$. We set

It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that

It is clear that

This implies that

as desired.

Subcase 1.3. Suppose that $i = j$ and $k\neq l$. We set

It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that

It is clear that

This implies that

as desired.

Subcase 1.4. Suppose that $i\neq j$ and $k = l$. We set

It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that $\gamma_1-\gamma_2 = \alpha$. It is clear that

This implies that

as desired.

Subcase 1.5. Suppose that $i\neq j$, $k\neq l$, and $(i, j)\neq (l, k)$. It is easy to check that

for all $a_i\in D$, $i = 1, \cdots, n$. We set

This implies that

for $i = 1, 2$. It follows that

for $i = 1, 2$, as desired.

Subcase 1.6. Suppose that $i\neq j$, $k\neq l$, and $(i, j) = (l, k)$. We may assume that $i < j$. The case of $i > j$ can be discussed analogously. It is easy to check that

for all $a_i\in D$, $i = 1, \cdots, n$. We set

It is clear that

This implies that

Since char$(D) = 3$ we get that

It is easy to check that

This implies that

for $i = 1, 2$. Note that

We get that

This implies that

Note that

We get that

and

This implies that

as desired.

Case 2. Suppose next that $\alpha, \beta, 1$ are linearly dependent over $Z$. Note that dim$_{Z}D\geq 4$. We get that there exists $\gamma_1'\in D$ such that $\gamma_1'\not\in L(1, \alpha, \beta)$, where $L(1, \alpha, \beta)$ is a subspace of $D$ generalized by $1, \alpha, \beta$. It is clear that

where $L(1, \alpha, \beta, \gamma_1)$ is a subspace of $D$ generalized by $1, \alpha, \beta, \gamma_1'$. We get that there exists $\gamma_2'\in D$ such that $\gamma_2'\not\in L(1, \alpha, \beta, \gamma_1')$. We now discuss the following six subcases.

Subcase 2.1. Suppose that $i = j = k = l$. We set $\gamma_1 = \gamma_1'$ and $\gamma_2 = \gamma_2'$. It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that

This implies that

as desired.

Subcase 2.2. Suppose that $i = j$, $k = l$, and $i\neq k$. We set $\gamma_1 = \gamma_1'$ and $\gamma_2 = \gamma_2'$. It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that

This implies that

as desired.

Subcase 2.3. Suppose that $i = j$ and $k\neq l$. We set $\gamma_1 = \gamma_1'$ and $\gamma_2 = \gamma_2'$. It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that

This implies that

as desired.

Subcase 2.4. Suppose that $i\neq j$ and $k = l$. We set $\gamma_1 = \gamma_1'$ and $\gamma_2 = \gamma_2'$. It is clear that

for $i = 1, 2$. This implies that

for $i = 1, 2$. Note that

This implies that

as desired.

Subcase 2.5. Suppose that $i\neq j$, $k\neq l$ and $(i, j)\neq (l, k)$. We set $\gamma_1 = \gamma_1'$ and $\gamma_2 = \gamma_2'$. It is easy to check that

for all $a_i\in D$, $i = 1, \cdots, n$. It is clear that

for $i = 1, 2$. We get that

for $i = 1, 2$. Note that

for $i = 1, 2$. It follows that

for $i = 1, 2$, as desired.

Subcase 2.6. Suppose that $i\neq j$, $k\neq l$, and $(i, j) = (l, k)$. We may assume that $i < j$. The case of $i > j$ can be discussed analogously. It is easy to check that

for all $a_i\in D$, $i = 1, \cdots, n$. Suppose first that $\alpha\beta\in Z$. We set

It is clear that

This implies that

Since char$(D) = 3$ we get that

It is easy to check that

This implies that

for $i = 1, 2$. Note that

We get that

This implies that

Note that

Since $\alpha\beta\in Z$, we get that

and

This implies that

as desired.

Suppose next that $\alpha\beta\not\in Z$. We set

It is clear that

This implies that

Since char$(D) = 3$ we get that

It is easy to check that

This implies that

for $i = 1, 2$. Note that

Since $\alpha\beta\not\in Z$ we get that

This implies that

Note that

Since $\alpha\beta\not\in Z$, we get that

and

This implies that

as desired. The proof of the result is complete. □

3.   The proof of Theorem 1.4

The following result will be used in the proof of our main result, which is of some independent interests.

Proposition 3.1. Let $D$ be a division ring such that either char$(D)\neq 2, 3$ or $D$ is not a field and char$(D)\neq 2$. Let $R = M_n(D)$, where $n > 1$. Let $\varphi$: $R\rightarrow R$ be a bijective additive map such that $\varphi(1) = 1$ and

for all $x\in R^{\times}$ with $x+1\in R^{\times}$. Then, $\varphi$ is either an automorphism or an antiautomorphism.

Proof. We first claim that $\varphi$ is a Jordan automorphism by way of the following three steps:

Step 1. We claim that

for all $1\leq i, j, k, l\leq n$ and $\alpha, \beta\in D$. In particular, we have

for all $1\leq i, j\leq n$ and $\alpha\in D$. We set

In view of both Lemmas 2.1 and 2.2 we note that either there exists $\gamma\in\{1, 2, 3\}$ such that

or there exist $\gamma_1, \gamma_2\in R^{\times}$ such that

for all $i = 1, 2$. Suppose first that there exists $\gamma\in\{1, 2, 3\}$ such that

By our hypothesis we have

Note that

Expanding the last relation we get

which implies that $\varphi(T^2) = \varphi(T)^2$, as desired.

Suppose next that there exist $\gamma_1, \gamma_2\in R^{\times}$ such that

for all $i = 1, 2$. By our hypothesis we have that

Expanding (3.4) we get

Using (3.3) we get from (3.5) that

By our hypothesis we have that

Expanding (3.8) we get

Using (3.7) we get from (3.9) that

By our hypothesis we have that

Expanding (3.12) we get

Using (3.11) we get from (3.13) that

Subtracting (3.6) from (3.10) we get

It follows from both (3.14) and (3.15) that

as desired.

Step 2. We claim that

for all $\alpha, \beta\in D$ and $1\leq i, j, k, l\leq n$.

On one hand, we get from (3.2) that

On the other hand, we get from (3.1) that

Combining (3.17) with (3.18) we get

for all $\alpha, \beta\in D$ and $1\leq i, j, k, l\leq n$, as desired.

Step 3. We claim that $\varphi(x^2) = \varphi(x)^2$ for all $x\in R$.

For any

we get from both (3.2) and (3.16) that

In view of Step 3, we get that $\varphi$ is a Jordan automorphism. Since char$(D)\neq 2$, we get from [13,Theorem 1] that $\varphi$ is an automorphism or antiautomorphism. This proves the result. □

The following simple result will be used in the proof of our main result:

Lemma 3.1. Let $D$ be a division ring such that either char$(D)\neq 2, 3$ or $D$ is not a field and char$(D)\neq 2$. Let $R = M_n(D)$, where $n > 1$. Let $f$: $R\rightarrow R$ be an additive map such that $f(x) = 0$ for all $x\in R^{\times}$ with $x+1\in R^{\times}$. Then $f = 0$.

Proof. For any $\alpha\in D$ and $1\leq i, j\leq n$, In view of both Lemmas 2.1 and 2.2 we get that there exists $\gamma\in R^{\times}$ such that

By our hypothesis we have

Since $f$ is additive, we get that $f(\alpha e_{ij}) = 0$. For any

we get that $f(x) = 0$, as desired. □

We are in a position to give the proof of our main result.

Proof of Theorem 1.4. For $x\in R^{\times}$ with $x+1\in R^{\times}$, we note that

We set

Since

we have that

It follows that

Note that

It follows from (3.19) that

For any $z\in R^{\times}$, we note that $z(z^{-1}k) = k$. This implies that

and so

We get from (3.20) that

Multiplying (3.21) by $m^{-1}f(x+1)$ on the right hand side, we get

which implies that

for all $x\in R^{\times}$ with $x+1\in R^{\times}$. It follows from (3.22) that

for all $x\in R^{\times}$ with $x+1\in R^{\times}$. We define

for all $x\in R$. Then,

for all $x\in R$. It is clear that $\varphi(1) = 1$. The additivity of $f$ immediately yields the additivity of $\varphi$. It follows from (3.23) that

for all $x\in R^{\times}$ with $x+1\in R^{\times}$. In view of Proposition 3.1, we can conclude that $\varphi$ is an automorphism or antiautomorphism.

For any $x\in R^{\times}$, since

we get that

This implies that

for all $x\in R^{\times}$. In view of Lemma 3.1, we get that

for all $x\in R$. Suppose first that $\varphi$ is an automorphism. We get from (3.24) that

for all $x\in R$. In particular, if $f = g$, we have that

for all $x\in R$. This implies that $f(1)\in Z$. Suppose next that $\varphi$ is an antiautomorphism. We get from (3.24) that

for all $x\in R$. In particular, if $f = g$, we get that

for all $x\in R$. This implies that

for all $x\in R$. Since $f$ is a bijective map we obtain that $f(k)\in Z$. Note that $f(k)f(1) = m$, and so $f(1) = f(k)^{-1}m$. The proof of the result is complete. □

4.   The proof of Theorem 1.5

As a consequence of Theorem 1.4 we give the proof of Theorem 1.5 as follows:

Proof of Theorem 1.5. In view of Theorem 1.4, we have that

for all $x\in R$, where $\varphi$: $R\rightarrow R$ is either an automorphism or an antiautomorphism. Moreover, we have the following:

$(1)$ If $\varphi$ is an automorphism, then $f(x) = \varphi(x)f(1)$ for all $x\in R$.

$(2)$ If $\varphi$ is an antiautomorphism, then $f(x) = f(k)^{-1}f(x)f(k)$ for all $x\in R$.

Suppose first that $\varphi$ is an automorphism. Since

for all $x\in R$, we get that

for all $x\in R$. Since $\varphi$ is an automorphism, we get that $f(1)\in Z$, as desired.

Suppose next that $\varphi$ is an antiautomorphism. Since

for all $x\in R$, we get that

for all $x\in R$. Recall that $f$ is a bijective map. We get from the last relation that $f(k)\in Z$. Since $k1 = k$, we have that

This implies that

as desired. The proof of the result is complete. □

5.   Conclusions

We give a complete description of maps preserving products of matrices over a division $D$ such that either char$(D)\neq 2, 3$ or $D$ is not a field and char$(D)\neq 2$, which gives a correct version of Theorem 1.3. The future study of this field is to give a complete description of maps preserving products of matrices over a division $D$ with char$(D)\neq 2$.

Author contributions

Lan Lu: writing the draft of the manuscript. Yu Wang: correcting some errors in the proof of some results and writing the final manuscript.

Use of AI tools declaration

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

Acknowledgments

The authors are thankful to the anonymous referees for their valuable comments and suggestions. The second author is supported by the science and technology research project of education department of Jilin province (Grant No. JJKH20241000KJ) and the "fourteen five" scientific planning project of Jilin province education society (Grant No. G215000).

Conflict of interest

The authors declare no conflicts of interest.