The properties of generalized John domains in metric spaces

1.   Introduction and main results

John ^[17] and Martio and Sarvas ^[24] were the first who introduced and studied John domains and uniform domains, respectively. Now, there are plenty of alternative characterizations for uniform and John domains; see ^{[5,6,20,23,28,29,30,31,32]}. Additionally, its importance along with some special domains throughout the function theory is well documented; see ^{[5,7,13,15,20,25,26,35,36,37]}. Moreover, John domains and uniform domains in $\mathbb{R}^n$ enjoy with numerous geometric and function theoretic features in many areas of modern mathematical analysis, see ^{[1,2,3,6,18,19,21,22,31,34]}. As in ^[10], Guo and Koskela have introduced the class of $\varphi$-John domains, which form a natural generalization of John domains. The motivation for this paper stems from the discussions in ^[16,33], where the effect of the removal of a finite set of points and union of generalized John domain was examined. The main result of this paper shows that $D$ is a $\varphi$-John domain if, and only if, $D\backslash P$ is a $\varphi'$-John domain, where $P$ is a subset of $D$ containing finitely many points of $D$, $\varphi$ and $\varphi'$ depend on each other, and, finally, we prove that the union of $\varphi$-John domains is $\varphi''$-John domain.

Throughout the paper, unless otherwise stated, we always assume that $D$ is a proper subdomain of the metric space $X$ and $B(x, r) = \{y\in X: |x-y| < r\}$ denotes the metric ball at $x$ of radius $r$. For a set $D$ in $X$, we use $\overline{D}$ to denote the metric completion of $D$, and we let $\partial D = \overline{D}\backslash D$ be its metric boundary. We write

for the closed annular ring center at $x$ with inner and outer radii $r$ and $R$, respectively.

From now on, for notational convenience, we use notation $|x-y|$ to indicate the distance between $x$ and $y$ in any metric space $X$.

Definition 1.1. A domain (open and connected) $D$ in $X$ is said to be a $C$-uniform domain if there exists a constant $C\geq 1$ with the property that each pair of points $z_1, z_2$ in $D$ can be joined by a rectifiable arc $\gamma$ in $D$ satisfying

(1) (Double cone condition) $\min\left\{l(\gamma[z_1, z]), l(\gamma[z_2, z])\right\} \leq C\delta_D(z)$ for all $z \in\gamma$, and

(2) (Quasiconvex condition) $l(\gamma)\leq C|z_1-z_2|$,

where $l(\gamma)$ denotes the arc-length of $\gamma$, $\gamma[z_i, z]$ is the subcurve of $\gamma$ between $z_i$ and $z$, and $\delta_{D}(z)$ denotes the distance ${\mathrm{dist}}(z, \partial D)$. At this time, $\gamma$ is said to be a double $C$-uniform curve.

If the condition (1) is satisfied, not necessarily (2), then $D$ is said to be a $C$-John domain and the arc $\gamma$ is called a $C$-John curve.

The classes of John domains and of uniform domains in Euclidean space enjoy an important role in many areas of modern mathematical analysis; see ^[14,24,27]. Inspired by the study on generalized quasidisks ^[12], Guo and Koskela ^[11] generalized the definition of John domain as follows.

Definition 1.2. Let $D\subseteq X$ be a bounded domain, let $\varphi: [0, \infty)\rightarrow [0, \infty)$ be a continuous, increasing function with $\varphi(0) = 0$ and $\varphi(t)\geq t$ for all $t > 0$, and let $C\geq 1$ be a constant, $z_0\in D$. We say that $D$ is $\varphi$-John domain, if for any $z\in D$, there exists a rectifiable curve $\gamma:z \curvearrowright z_0$, such that

for all $u\in \gamma$. The concept of $\varphi$-dist and $\varphi$-diam John domains are defined analogously. A corresponding curve is called a $\varphi$- length(dist, diam) John curve.

The notion of $\varphi$-John domains allows us to formulate a second definition of $\varphi$-John domains, and the next definition and Definition 1.2 are equivalent; please see ^[8].

Definition 1.3. Let $D\subseteq X$ be a bounded domain. We say that $D$ is $\varphi$-John domain if there exist constant $C\geq 1$ and function $\varphi$ with the property that each pair of points $z_1, z_2$ in $D$ can be joined by a rectifiable arc $\gamma$ in $D$ satisfying

for all $u\in \gamma$. Here, $\varphi$ is a continuous, increasing function with $\varphi(0) = 0$ and $\varphi(t)\geq t$ for all $t > 0$.

We remark that, in general, the generalized John domain means the $\varphi$-John domain. Obviously, the $\varphi$-John domain is a generalization of the $C$-John domain since the $C$-John domain coincides with the $\varphi$-John domain with $\varphi(Ct) = Ct$. In this paper, we always simplify $C$-John domain by John domain and $\varphi$-John domain by generalized John domain.

In ^[16], Huang et. al. showed that a domain $D$ in $\mathbb{R}^n$ is a John domain if, and only if, $D\backslash P$ is a John domain, where $P$ is a subset of $D$ containing finitely many points of $D$.

Theorem 1.4. (See ^[16], Theorem 1.4) A domain $D\subseteq \mathbb{R}^n$ $(n\geq 2)$ is a John domain if, and only if, $G = D \backslash P$ is also a John domain, where $P = \{p_1, p_2, \cdots, p_m\}$ and $p_i\in D$ $(i = 1, 2, \cdots, m)$.

To state our results, we introduce the following definition.

Definition 1.5. Let $c\geq 1$. Let $X$ be a rectifiable connected and locally compact metric space, and $D \subseteq X$. Then, $D$ is called

(1) $c$-quasiconvex, if for any $x, y\in D$ there is a curve $\gamma$ joining $x$ and $y$ in $D$ satisfying $l(\gamma)\leq c|x-y|$. We also call this $\gamma$ a $c$-quasiconvex curve;

(2) $c$-annular quasiconvex, if for every $x\in D$ and for all $r > 0$, each pair of points $y, z\in A(x; r, 2r)\subseteq D$ can be joined with a curve $\gamma$ in $A(x; r/c, 2cr)\subseteq D$ such that $l(\gamma)\leq c|y-z|$.

Remark 1.6. It was proved by Buckley et al. in ^[4] that if $X$ is connected and $c$-annular quasiconvex at some point $\omega \in X$, then $X$ is $9c$-quasiconvex. Therefore, the annular quasiconvexity implies quasiconvexity.

Our first purpose is to show that a domain $D$ in metric space $X$ is a $\varphi$-John domain if and only if $D\backslash P$ is a $\varphi'$-John domain, where $P$ is a subset of $D$ containing finitely many points of $D$. Our proof is based on a refinement of the method of Huang et. al. ^[16]. We obtain a general result as follows.

Theorem 1.7. Suppose that $X$ is a rectifiably connected and locally compact metric space, and that domain $D\subseteq X$ is a $c$-annular quasiconvex. Then, the following are quantitatively equivalent:

(1) $D$ is a $\varphi$-John domain;

(2) $G = D\backslash P$ is a $\varphi'$-John domain, where $P = \{p_1, p_2, \cdots, p_m\}$ and $p_i\in D$ $(i = 1, 2, \cdots, m)$.

Here, $\varphi$ and $\varphi'$ depend on each other and $c$.

In ^[33], Väisälä studied the union of John domains in Euclidean spaces, and showed that the union of John domains also is a John domain. In ^[8], Guan proved that the union of John domains also is a John domain in Banach spaces. Under affable geometric conditions, we obtain a general result as follows.

Theorem 1.8. Let $X$ be a metric space, and let $D_1, D_2\subseteq X$ be two $c$-quasiconvex domains, where $c\geq 1$. Suppose that $D_1$ and $D_2$ are two $\varphi$-John domains in $X$, and that $z_0\in D_1\cap D_2$ and $r > 0$ with

where $c_0\geq 1$ and ${\mathrm{diam}}(D_i)$ is the diameter of $D_i$, $i = 1, 2$. Then, $D_1\cup D_2$ is a $\varphi''$-John domain with $\varphi''$ depending only on $c$, $c_0$ and $\varphi$. Note that the function $\varphi''$ is a continuous, increasing function with $\varphi''(0) = 0$ and $\varphi''(t)\geq t$ for all $t > 0$.

The rest of this paper is organized as follows. In Section 2, we show that $D$ is a generalized John domain if, and only if, $D\backslash P$ is a generalized John domain, where $P$ is a subset of $D$ containing finitely many points of $D$. The goal of Section 3 is to show that the union of generalized John domains is a generalized John domain.

2.   The decomposition properties of generalized John domains

In this section, we always assume that $X$ is a rectifiably connected and locally compact metric space, and that domain $D\subseteq X$ is a $c$-annular quasiconvex. Furthermore, we suppose that $P = \{p_1, p_2, \cdots, p_m\}$ and $p_i\in D$ $(i = 1, 2, \cdots, m)$.

In what follows, we continue to investigate the decomposition properties of generalized John domain in metric space. The following results play a key role in the proof of Theorem 1.7. Based on ^[10] and ^[16], we will prove the following results.

Lemma 2.1. Under the assumptions of Theorem 1.7. If $D$ is a $\varphi$-John domain, then $G = D\backslash P$ is also a $\varphi_1$-John domain with $\varphi_1$ depending only on $\varphi$.

Proof. By assumption, we show that $G = D\backslash P$ is also a $\varphi_1$-John domain with $\varphi_1$ depending only on $\varphi$. Without loss of generality, in order to prove Lemma 2.1, we need only to consider the case $P = \{p_1\}$. For convenience, we let

For any points $z_1, z_2\in G = D\backslash \{p_1\}$. Now, we divide the discussions into three cases:

Case 1. $z_1, z_2\in D \backslash B_r$.

Since $D$ is a $\varphi$-John domain, then there exist constant $C\geq 1$ and function $\varphi$ with the property that each pair of points $z_1, z_2$ in $D$ can be joined by a rectifiable arc $\gamma$ in $D$ satisfying

for all $z\in \gamma$.

We now consider two subcases:

Subcase 1.1. $\gamma \subseteq D\backslash B_r$.

If $\gamma \subseteq D\backslash B_r$, then we take $\beta = \gamma$, and it is clear that $\beta \subseteq G$.

To prove this subcase, we have the following claim.

Claim 1. Let $z$ be any point in $\beta\subseteq D\backslash B_r$, and we have

Since $\beta = \gamma$ and $\beta\subseteq D\backslash B_r \subseteq G$, for any $z\in \beta$, we get

If $\delta_D(z)\leq |z- p_1|$, by using the definitions of $\delta_D(z)$ and $\delta_G(z)$, we have

If $\delta_D(z) > |z- p_1|$, it follows that

According to the triangle inequality and $|z- p_1| > r$, we deduce that

So, from (2.2) and (2.3), Claim 1 is obtained.

Since $D$ is a $\varphi$-John domain, and $\beta = \gamma\subseteq D\backslash B_r \subseteq G$, for any $z\in \beta$, from the conclusion of Claim 1 and (2.1), it follows that

Subcase 1.2. $\gamma \cap B_r\neq \emptyset$.

We let $z'_1$ be the first intersection point of $\gamma$ from $z_1$ to $z_2$, with $\partial B_r$ and $z'_2$ as the last intersection point of $\gamma$ from $z_1$ to $z_2$ with $\partial B_r$. Let $U_r$ be the disk determined by $z'_1$, $z'_2$ and $p_1$ in $\overline{B_r}$ with center $p_1$ and radius $r$. Then, $z_1'$, $z_2'$ divide $\partial U_r$ into subarcs, and we denote the subarc with shorter arclength by $\alpha$ (if they have the same arclength, then we choose one of them to be $\alpha$), that is,

Set

Claim 2. $l(\alpha)\leq \frac{\pi}{2}|z'_1-z'_2|$.

In disk $U_r$, according to the chord arc formula and the properties of trigonometric function, it follows that

where $\theta\in [0, \pi]$ is the center angle of two points $z'_1$ and $z'_2$. According to (2.5) and $l(\alpha) = \theta\cdot r$, we get that

Therefore, the proof of the Claim 2 is now complete.

By the definition of $z'_1$, $z'_2$ and disk $U_r$, we have

Together with (2.6) and (2.7), it follows that

If $z\in \gamma[z_1, z_1']$ or $z\in \gamma[z_2', z_2]$, by symmetry, we only prove $z\in \gamma[z_1, z_1']$; the proof of $z\in \gamma[z_2', z_2]$ uses the same argument for $z\in \gamma[z_1, z_1']$.

For any $z\in \gamma[z_1, z_1']\subseteq D\backslash B_r$. It follows immediately from Claim 1 that $\delta_D(z)\leq 3\delta_G(z)$.

Hence, together with Claim 2 and $\delta_D(z)\leq 3\delta_G(z)$, it follows that

If $z\in \alpha\subseteq \partial U_r$, then we have

According to the inequality (2.4) and (2.8), it follows immediately from the definition of the $\varphi$-John domain that

where $C\geq 1$ is a constant.

Case 2. $z_1, z_2\in \overline{B_r} \backslash \{p_1\}$.

Let $z'_1$ be the intersection point of the ray starting from $p_1$ and passing through $z_1$ with $\partial B_r$, and let $z'_2$ be the intersection point of the ray starting from $p_1$ and passing through $z_2$ with $\partial B_r$, then we have

We use $U_r$ to denote the disk determined by $z'_1$, $z'_2$ and $p_1$ in $\overline{B_r}$ with center $p_1$ and radius $r$. Then, $z'_1$ and $z'_2$ divide $\partial U_r$ into two subarcs. Let $\alpha$ denote the subarc with the shorter arclength (if they have the same arclength, then we choose one of them to be $\alpha$). We set

where $[z_i, z'_i]$ denotes the line segments in metric space of $z_i$ and $z'_i$, $i = 1, 2$.

If $z\in \alpha\subseteq \partial U_r$, it is clear that

Together with (2.9) and (2.10), it follows that

If $z\in [z_1, z'_1]$ or $z\in [z_2, z'_2]$, by symmetry, it is sufficient to show that $z\in [z_1, z'_1]$. For any $z\in [z_1, z'_1]$, we have the desired estimate

Hence, by combing (2.11) with (2.12), for any $z\in \beta$, we deduce that

Case 3. $z_1\in D \backslash \overline{B_r} \quad {\text {and}}\quad z_2\in B_r \backslash \{p_1\}$.

Since $D$ is a $\varphi$-John domain, there must exist a curve $\gamma$ joining $z_1$ and $z_2$ such that

for all $z\in \gamma$.

Let $z'_1$ to be the first intersection point of $\gamma$ from $z_1$ to $z_2$ with $\partial B_r$, and let $z'_2$ be the intersection point of the ray starting from $p_1$ and passing through $z_2$ with $\partial B_r$. We use $U_r$ to denote the disk determined by $z'_1$, $z'_2$ and $p_1$ in $\overline{B_r}$ with center $p_1$ and radius $r$. Then, $z'_1$ and $z'_2$ divide $\partial U_r$ into two subarcs. Let $\alpha$ denote the subarc with the shorter arclength (if they have the same arclength, then we choose one of them to be $\alpha$). Then, according to the description above, we have

Set

where $[z_2, z'_2]$ represents a straight line segment joining $z_2$ to $z'_2$.

We now consider three subcases:

Subcase 3.1. $z\in \alpha\subseteq \partial U_r$.

Using a similar argument as in Case 2, we have $\delta_G(z) = r.$ According to (2.13), from the definition of $\varphi$-John domain, we get that

Subcase 3.2. $z\in [z_2, z'_2]$.

From the definition of $z'_2$, it follows from $z\in [z_2, z'_2]$ that

Subcase 3.3. $z\in \gamma[z_1, z'_1]$.

If $l(\gamma[z_1, z])\leq l(\gamma[z_2, z])$. Since $D$ is a $\varphi$-John domain, by using the conclusion of Claim 1, it follows that

If $l(\gamma[z_1, z]) > l(\gamma[z_2, z])$, by the conclusion of Subcase 3.1, we deduce that

Now, for any $z\in \gamma[z_1, z'_1]$ with $l(\gamma[z'_1, z]) < r/2$, then we have

Together with (2.14) and (2.15), it follows that

If $l(\gamma[z'_1, z])\geq r/2$, for any $z\in \gamma[z_1, z_1']\subseteq D\backslash \overline{B_r}$, by Claim 1, we know that $\delta_D(z)\leq 3\delta_G(z)$. According to inequality (2.14) and the definition of the $\varphi$-John domain, we get

Therefore, as discussed above, it follows that $G = D\backslash P$ is a $\varphi_1$-John domain. Here,

Hence, this completes the proof of Lemma 2.1. □

Lemma 2.2. Under the assumptions of Theorem 1.7. If $G = D\backslash P$ is a $\varphi$-John domain, then $D$ is also a $\varphi_2$-John domain, where $\varphi_2$ depends only on $\varphi$ and $c$.

Proof. We always assume that $D\subseteq X$ is a $c$-annular quasiconvex, and that $G = D\backslash P$ is a domain, where

Let $\varphi$ be a continuous, increasing function with $\varphi(0) = 0$ and $\varphi(t)\geq t$ for all $t > 0$. For any pair of points $z_1, z_2 \in D$, we divide the discussions into three cases:

Case 1. $z_1, z_2 \in G$.

According to the assumptions of Theorem 1.7, we know that $G = D\backslash P$ is a $\varphi$-John domain, that is, there is a rectifiable curve $\gamma \subset G$ connecting $z_1$ and $z_2$ such that

for all $z \in \gamma$, where $C \geq 1$ is a constant. Since $G = D\backslash P \subseteq D$, we have

According to $\varphi$ being a continuous, increasing function, it follows that

for all $z \in \gamma$.

Case 2. $z_1, z_2 \in D \backslash G$.

Let $z_1, z_2 \in P = D \backslash G$, and $P = \{p_1, p_2, \cdots, p_m\}$, $p_i\in D$ $(i = 1, 2, \cdots, m)$. Set

We choose

where

According to the definition of the $\varphi$-John domain, there must exist a curve $\gamma \subseteq G$ joining $z_1'$ with $z_2'$ such that

for all $z\in \gamma$.

Since $D\subseteq X$ is a $c$-annular quasiconvex, by Remark 1, we have the $D$ is $9c$-quasiconvex, that is, there exists a rectifiable curve $\gamma_1$ in $D$ joining $z_1$ to $z'_1$ with $l(\gamma_1) \leq 9c|z_1-z'_1|$, and there exists a rectifiable curve $\gamma_2$ in $D$ joining $z_2$ to $z'_2$ with $l(\gamma_2) \leq 9c|z_2-z'_2|$, where $c\geq 1$. Hence, we have

Let

For any $z \in \beta$, if $z\in\gamma_1$ or $z\in\gamma_2$, by symmetry, we assume that $z\in\gamma_1$. From the definition of quasiconvexity, it follows that

If $z\in \gamma$ and $\min\{l(\gamma[z'_1, z]), l(\gamma[z'_2, z])\} \leq cr$, we deduce that

Hence, by inequality (2.18) and the definition of function $\varphi$, it is clear that

If $z \in \gamma$ and $\min\{l(\gamma[z'_1, z]), l(\gamma[z'_2, z])\} > cr$, it follows from the definition of the $\varphi$-John domain that

Together with (2.17), (2.19) and (2.20), which shows that in this subcase, the lemma holds with

Case 3. $z_1\in G$ and $z_2\in D \backslash G$.

Using a similar argument as in Case 2, we can show that there is a rectifiable curve $\gamma \subseteq D$ connecting $z_1$ and $z_2$ such that for any $z\in \gamma$,

By combining (2.16), (2.21) and (2.22), we get that $D$ is a $\varphi_2$-John domain, that is

where $C_2\geq 1$ is a constant, and

Hence, Lemma 2.2 is proved. □

The proof of Theorem 1.7. Under the assumptions of Theorem 1.7, Theorem 1.7 follows from Lemmas 2.1 and 2.2. □

3.   The union of generalized John domains

The proof of Theorem 1.8. The assumption implies that $D_1, D_2\subseteq X$ are two $c$-quasiconvex and $\varphi$-John domains. Furthermore, we suppose that $z_0\in D_1\cap D_2$ and $r > 0$ with

Let $D = D_1\cup D_2$. Under these assumptions, in order to prove Theorem 1.8, we need only to show that there exist constant $C''\geq 1$ and function $\varphi''$ with the property that each pair of points $a, b$ in $D$ can be joined by a rectifiable arc $\gamma$ in $D$ satisfying

for all $z\in \gamma$. Here, $\varphi''$ is a continuous, increasing function with $\varphi''(0) = 0$ and $\varphi''(t)\geq t$ for all $t > 0$.

Without loss of generality, we assume that $\mathrm{diam}(D_1)\leq \mathrm{diam}(D_2)$. Let $a\in D_1$ and $b\in D_2$, and we can choose $\varphi$-John curves $\alpha:a\curvearrowright z_0$ and $\beta:b\curvearrowright z_0$ in $D_1$ and $D_2$, respectively. The continuum $\alpha\cup\beta$ contains a curve $\gamma:a\curvearrowright b$. It suffices to show that $\gamma$ is a $\varphi''$-John curve in $D = D_1\cup D_2$.

We choose two points $a_1\in \alpha$ and $b_1\in \beta$ dividing $\alpha$ and $\beta$ to subarcs of equal length, respectively. Let

and

In what follows, we will divide the proof into two steps.

Step 1. For all $x\in \alpha$, we prove that

with constant $C_3\geq 1$.

Since $D_1$ is $\varphi$-John domain and $\alpha:a\curvearrowright z_0$ is $\varphi$-John curve in $D_1$, thus, we have

Let $x\in \alpha$. Now, to prove inequality (3.1), we divide the discussions into three cases:

Case 1. $x\in \alpha[a, a_1]$.

Since $D = D_1\cup D_2$, it is clear that

From the definitions of $a_1$ and the $\varphi$-John domain, for any $x\in \alpha[a, a_1]$, by (3.2), it follows that

Case 2. $x\in \alpha[a_2, z_0]$.

Now that $\mathrm{diam}(D_1)\leq c_0 r$, according to the definition of $a_2$, it is clear that

Since $D_1$ is a $c$-quasiconvex domain, we have

where $c\geq 1$ is a constant. Therefore, according to (3.3), (3.4) and the definition of function $\varphi$, it follows that

Case 3. $x\in \alpha[a_1, a_2]$ and $a_2\in \alpha[a_1, z_0]$.

This case may be empty. From the construction of $a_2$, it is obvious that

From the definition of $c$-quasiconvex domain and inequality $\mathrm{diam}(D_1)\leq c_0 r$, we obtain that

Hence, Combing (3.2), (3.5) and (3.6), it follows that

Therefore, for all $x\in \alpha$, we have

where

and $C_3\geq 1$ is a constant.

Step 2. For all $y\in \beta$, we prove that

with constant $C_4\geq 1$.

Let $y\in \beta$. To prove inequality (3.7), our proof consists of three parts. For the first part, if $y\in \beta[b, b_1]$, Since $D_2$ is $\varphi$-John domain and $\beta:b\curvearrowright z_0$ is a $\varphi$-John curve in $D_2$, thus, we get that

By the inequality (3.8), we deduce that

For the second part, if $y\in\beta[b_2, z_0]$, according to the definition of $b_2$, we obtain that

Since $D_2$ is a $c$-quasiconvex domain, and $\beta$ is a $\varphi$-John curve, from the definitions of $a_2$ and $b_2$, it is clear that

Combing (3.4), (3.9) and (3.10), it follows that

For the final part, if $b_2\in\beta[b_1, z_0]$ and $y\in\beta[b_1, b_2]$, this case may again be empty. Since $D_1$ is a $c$-quasiconvex domain, and $\mathrm{diam}(D_1)\leq c_0 r$, we get that

and since $D_2\subseteq D$ is a $\varphi$-John domain, we have

In addition, from the definition of $b_2$, we deduce that

According to (3.11)–(3.13), we get

Now, it follows immediately from the inequality (3.14) that

Therefore, for all $y\in \beta$, we have

where

and $C_4\geq 1$ is a constant.

Hence, we verified all the cases and our conclusion holds, that is, $D_1\cup D_2$ is a $\varphi''$-John domain with

where $C''\geq 1$ is a constant. □

4.   Conclusions

In summary, we investigated the removability and union of generalized John domain, that is, the main result of this paper showed that $D$ is a $\varphi$-John domain if, and only if, $D\backslash P$ is a $\varphi'$-John domain, where $P$ is a subset of $D$ containing finitely many points of $D$, $\varphi$ and $\varphi'$ depend on each other, and finally we prove the union of $\varphi$-John domains is $\varphi''$-John domain.

Given the Theorem 1.7 of the paper, it is natural to ask the following question:

Question 4.1. Let $X$ be a rectifiably connected, locally compact and $c$-annular quasiconvex metric space, and let $P$ be a countable subset of $X$. Is $X$ $\varphi$-John metric space if and only if $X\backslash P$ $\varphi'$-John metric space?

Acknowledgments

The authors thank the referee for their careful reading and valuable comments that led to the improvement of the paper.

This research work is supported by the National Natural Science Foundation of China (Grant No. 11671057), and the Guizhou Province Science and Technology Foundation (Grant No. QianKeHeJiChu[2020]1Y003, QianKeHeJiChu-ZK[2021] general 001).

Use of AI tools declaration

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

Conflict of interest

All authors declare no conflicts of interest in this paper.