BV Capacity for the Schrödinger Operator with an Inverse-Square Potential

Abstract

For $a \ge - ( \frac{d }{2} - 1)^2$ and $2\sigma = {{d - 2}}-( {{{(d - 2)}^2} + 4a})^{1/2}$, let $\mathcal {\widetilde{H}}_{\sigma }= 2( { - \Delta + \frac{{{\sigma ^2}}}{{{{ | x |}^2}}}})$ be a Schrödinger operator with an inverse-square potential on ${{\mathbb {R}}^d}\backslash \{0\}$. In this paper, we introduce and investigate the ${{\mathcal {\widetilde{H}}_{\sigma }}}$-BV capacity, whence discovering some capacitary inequalities on $\Omega \subseteq {{\mathbb {R}}^d}\backslash \{0\}$.

1 Introduction

In this paper, we discuss several basic questions of the geometric measure theory related to the Schrödinger operator with inverse-square potential that is defined as

$$\begin{aligned} \mathcal {\widetilde{H}}_{\sigma }= 2( { - \Delta + \frac{{{\sigma ^2}}}{{{{ | x |}^2}}}}),\ \ 2\sigma = {{d - 2}}-\big ( {{{(d - 2)}^2} + 4a}\big )^{1/2} \end{aligned}$$

on the Euclidean space ${\mathbb {R}}^d\backslash \{0\}$ with $d \ge 2$, which is derived from ${\mathcal {H}}_{a} = - \Delta + \frac{a}{{{{ | x |}^2}}},\ a \ge - {( {\frac{{d - 2}}{2}} )^2}$. The operator ${\mathcal {H}}_{a}$ often appears in the field of mathematics and physics, and is usually used as a scale limit for more complex problems. In [3], the authors have introduced and studied the basic properties of the $\mathcal {\widetilde{H}}_{\sigma }$-BV space ${\mathcal {B}} {{\mathcal {V}} _{{\mathcal {{\widetilde{H}}} _\sigma }}}(\Omega )$. See Sect. 2 for the definition and properties of the space ${\mathcal {B}} {{\mathcal {V}} _{{\mathcal {{\widetilde{H}}} _\sigma }}}(\Omega )$.

The aim of this paper is to investigate the capacity associated with the operator ${\widetilde{{\mathcal {H}}} _\sigma }$. In the study of the pointwise behavior of the bounded variation function, the concept of capacity plays a crucial role. The functional capacities are of fundamental importance in various branches of mathematics such as analysis, geometry, probability theory, partial differential equations, and mathematical physics, see [1, 4, 5, 9] for the details. In recent years, the capacity related to bounded variation functions has attracted the attention of many researchers, and a lot of progress has been made. We refer to [15] for information on the classical BV-capacity on ${\mathbb {R}}^{d}$. In [14], J. Xiao introduced the BV-type capacity on Gaussian spaces ${\mathbb {G}}^{n}$, and, as an application, the Gaussian BV-capacity was used to study the trace theory of Gaussian BV-space. On the generalized Grushin plane ${\mathbb {G}}^{2}_{\alpha }$, Liu obtained some sharp trace and isocapacity inequalities based on the BV capacity in [8]. For further information on this topic, refer to [2, 6, 7, 11, 13] and the references therein.

Let $\Omega \subseteq {{\mathbb {R}}^n}$ be a bounded open domain away from origin. We follow this convention throughout this article. The rest of the paper is structured as follows. In Sect. 2, we introduce the ${{\mathcal {\widetilde{H}}_{\sigma }}}$-BV capacity denoted by $\widetilde{\mathrm {cap}}(E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))$ for a set $E\subseteq \Omega$ and investigate the measure-theoretic nature of $\widetilde{\mathrm {cap}}(\cdot , \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))$. Theorem 2.12 indicates that $\widetilde{\mathrm {cap}}(E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))$ is not only an outer measure (obeying (i), (ii) & (iv)), but also a Choquet capacity (satisfying (i), (ii), (v) & (vi)). Section 3 is devoted to the Poincaré type inequality and ${\mathcal {\widetilde{H}} _\sigma }$-capacitary inequalities. In Theorem 3.1, we obtain some equivalent conditions for a p-poincaré type inequality associated with $\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega )$ functions, which arise from the end-point ${\mathcal {\widetilde{H}} _\sigma }$-Sobolev space $W^{1,1}_{\mathcal {\widetilde{H}} _\sigma }(\Omega )$. Furthermore, we derive an imbedding result for the operator ${\mathcal {\widetilde{H}} _\sigma }$. Let

$$\begin{aligned} {\mathfrak {C}}(f):=\left( \int ^{\infty }_0\big (\widetilde{\mathrm {cap}}(\{x\in \Omega : |f(x)|\ge t\}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\big )^{{\frac{d}{d-1}}}dt^{{\frac{d}{d-1}}}\right) ^{{\frac{d-1}{d}}}. \end{aligned}$$

In Theorem 3.2, we establish the following equivalent relation: for any compactly supported $L^{{d}/{(d-1)}}$-function f, we have the following analytic inequality

$$\begin{aligned} \Vert f\Vert _{L^{\frac{d}{d-1}}(\Omega )}\lesssim \mathfrak {{C}}(f) \Longleftrightarrow |M|^{\frac{d-1}{d}}\lesssim \widetilde{\mathrm {cap}} \big (M, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )\big ), \end{aligned}$$

where M is any compact set in $\Omega$.

Theorem 3.3 derives that for any $f\in {C}^{1}_{c}(\Omega )$,

$$\begin{aligned} {{\mathfrak {C}}}(f)\lesssim {\left\| f \right\| _{{L^1}(\Omega )}} + |\nabla _{\mathcal {\widetilde{H}}_\sigma }f|(\Omega ). \end{aligned}$$

In Theorem 3.4, we establish the following equivalent relation: for any $f\in C^\infty _c(\Omega )$,

$$\begin{aligned} {{\mathfrak {C}}}(f) \lesssim \Vert f\Vert _{\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )} \Longleftrightarrow \widetilde{\mathrm {cap}}\big (M, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )\big )\lesssim |M|+ P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega ), \end{aligned}$$

where M is any connected compact set in $\Omega$ with smooth boundary, and $P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega )$ is the ${\mathcal {\widetilde{H}}_\sigma }$-perimeter of M.

Throughout this article, we will use c and C to denote the positive constants, which are independent of the main parameters and may be different at each occurrence. ${{\mathsf {U}}}\approx {{\mathsf {V}}}$ indicates that there is a constant $c>0$ such that $c^{-1}{\mathsf V}\le {{\mathsf {U}}}\le c{{\mathsf {V}}}$, whose right inequality is also written as ${{\mathsf {U}}}\lesssim {{\mathsf {V}}}$. Similarly, one writes ${\mathsf {V}} \gtrsim {{\mathsf {U}}}$ for ${{\mathsf {V}}}\ge c{{\mathsf {U}}}$. For $k \in {\mathbb {N}}_{0} \cup \{+\infty \}$ $C_{c}^{k}(\Omega )$ denotes the space of $C^{k}$-regular functions with compact support in $\Omega$.

2 ${\mathcal {\widetilde{H}} _\sigma }$-BV capacity

Based on the results for the ${{\mathcal {\widetilde{H}}_{\sigma }}}$-BV space [3], we introduce the ${{\mathcal {\widetilde{H}}_{\sigma }}}$-BV capacity and investigate its properties. As in [3], we recall the definition of ${\mathcal {\widetilde{H}} _\sigma }$-BV space.

The $\mathcal {\widetilde{H}}_{\sigma }$-divergence of a vector valued function

$$\begin{aligned}\Phi = ({\varphi _1}, {\varphi _2}, \ldots ,{\varphi _{2d}}) \in C_c^1(\Omega , {{\mathbb {R}} ^{2d}}) \end{aligned}$$

is defined as

$$\begin{aligned} \mathrm {div}_{\mathcal {\widetilde{H}}_{\sigma }}\Phi ={A_{ - 1,a}} {\varphi _1} + \ldots + {A_{ - d,a}}{\varphi _d} - {A_{ 1,a}}{\varphi _ {d + 1}} - \ldots - {A_{ d,a}}{\varphi _{2d}}, \end{aligned}$$

where

$$\begin{aligned} {A_{i,a}} = \frac{\partial }{{\partial {x_i}}} + \sigma \frac{{{x_i}}}{{{{ | x |}^2}}}, {A_{ - i,a}} = - \frac{\partial }{{\partial {x_i}}} + \sigma \frac{{{x_i}}}{{{{ | x |}^2}}}, 1 \le i \le d \end{aligned}$$

with

$$\begin{aligned} \sigma = \frac{{d - 2}}{2} - \frac{1}{2}\sqrt{{{(d - 2)}^2} + 4a}. \end{aligned}$$

For $u\in C_c^1(\Omega )$, the $\mathcal {\widetilde{H}}_{\sigma }$-gradient of u is defined as

$$\begin{aligned} {\nabla _ {\mathcal {\widetilde{H}}_{\sigma }}}u = ({A_{1,a}}u, \ldots ,{A_{d,a}}u, -{A_{ - 1,a}}u, \ldots ,-{A_{ - d,a}})u. \end{aligned}$$

Let $\Omega \subseteq {\mathbb {R}}^{d}$ be an open set. The ${\mathcal {\widetilde{H}}_{\sigma }}$-variation of $f \in {L^1}(\Omega )$ is defined by

$$\begin{aligned} | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f} |(\Omega ) = \mathop {\sup } \limits _{\Phi \in {\widetilde{{\mathcal {F}}}(\Omega )}} \Big \{ {\int _\Omega {f(x) \mathrm {div}{_{{\mathcal {\widetilde{H}}_{\sigma }}}}\Phi (x)dx} } \Big \}, \end{aligned}$$

where $\widetilde{{{\mathcal {F}}}}(\Omega )$ denotes the class of all functions

$$\begin{aligned} \Phi = ( {{\varphi _1},{\varphi _2}, \ldots ,{\varphi _ {2d}}} ) \in C_c^1(\Omega , {{\mathbb {R}}^{2d}}) \end{aligned}$$

satisfying

$$\begin{aligned} { \Vert \Phi \Vert _\infty } = \mathop {\sup }\limits _ {x \in \Omega } \Big \{( {{{ | {{\varphi _1}(x)} |}^2} + \ldots + {{ | {{\varphi _{2d}}(x)} |}^2}} )^{\frac{1}{2}}\Big \} \le 1. \end{aligned}$$

A function ${f\in L^1}(\Omega )$ is said to have the ${\mathcal {\widetilde{H}}_{\sigma }}$-bounded variation on $\Omega$ if

$$\begin{aligned} | {{\nabla _ {{\mathcal {\widetilde{H}}_{\sigma }}}}f} |(\Omega ) < \infty , \end{aligned}$$

and the collection of all such functions is denoted by $\mathcal{BV}\mathcal{}_{{\mathcal {\widetilde{H}}_{\sigma }} } ( \Omega )$, which is a Banach space with the norm

$$\begin{aligned} { \Vert f \Vert _{{{{\mathcal {B}}}}{{{{\mathcal {V}}}}_{{\mathcal {\widetilde{H}} _{\sigma }}}}(\Omega )}} = { \Vert f \Vert _{{L^{1}(\Omega )}}} + | {{\nabla _ {{\mathcal {\widetilde{H}}_{\sigma }}}}f} |(\Omega ). \end{aligned}$$

In what follows, we will collect some properties of the space $\mathcal{BV}\mathcal{}_ {{{\mathcal {H}}}_a}(\Omega )$ in [3] and we omit the details of their proofs.

Proposition 2.1

Let $u \in {{{\mathcal {B}}}} {{{{\mathcal {V}}}}_{{\mathcal {\widetilde{H}}_{\sigma }}}}(\Omega )$. There exists a unique ${\mathbb {R}}^{2d}$-valued finite Radon measure ${D_{{{\mathcal {\widetilde{H}}}}_\sigma }}u= (D_{{A_{1,a}}}u,\ldots ,D_{{A_{d,a}}}u,D_{{A_{-1,a}}}u,\ldots ,D_{{A_{-d,a}}}u)$ such that

$$\begin{aligned} \int _\Omega u (x)\mathrm {div}{_{\mathcal {\widetilde{H}} _\sigma }}\Phi (x)dx = \int _\Omega \Phi (x) \cdot d{D_{{{\mathcal {\widetilde{H}}}}_\sigma }}u \end{aligned}$$

for every $\Phi \in C^{\infty }_c(\Omega , {\mathbb {R}}^{2d})$ and

$$\begin{aligned} \big | {\nabla _{\mathcal {\widetilde{H}}_{\sigma }}}u\big | (\Omega ) = |{D_{{{\mathcal {\widetilde{H}}}}_\sigma }}u|(\Omega ), \end{aligned}$$

where $|{D_{{{\mathcal {\widetilde{H}}}}_\sigma }}u|$ is the total variation of the measure ${D_{{{\mathcal {\widetilde{H}}}}_\sigma }}u$.

Proposition 2.2

(i) Suppose that $f \in W_{{\mathcal{\widetilde{H}}_\sigma }}^{1,1}(\Omega )$. Then,

$$\begin{aligned} | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f} |(\Omega ) = \int _ \Omega { | {{\nabla _{\mathcal {\widetilde{H}}_{\sigma }}}f(x)} |dx}. \end{aligned}$$

(ii) Suppose that ${f_k} \in {{{\mathcal {B}}}}{{{{\mathcal {V}}}}_ {{\mathcal {\widetilde{H}}_{\sigma }}}}(\Omega ), k\in {\mathbb {N}}$ and ${f_k} \rightarrow f$ in $L^{1}_{loc}(\Omega )$. Then,

$$\begin{aligned} | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f} |(\Omega ) \le \mathop {\lim \inf }\limits _{k \rightarrow \infty } | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}} {f_k}} |(\Omega ). \end{aligned}$$

Proposition 2.3

Let $\Omega \subset {{\mathbb {R}} ^d}$ be an open and bounded domain. Assume that $u \in {{{\mathcal {B}}}} {{{\mathcal V}}_{{\mathcal {\widetilde{H}}_{\sigma }}}}(\Omega )$ satisfies the condition (1). Then, there exists a sequence ${ \{ {{u_h}} \}_{h \in {\mathbb {N}} }} \in {{ {\mathcal {B}}}}{{{\mathcal V}}_{{\mathcal {\widetilde{H}}_{\sigma }}}}(\Omega ) \cap C_c^ \infty (\Omega )$ such that

$$\begin{aligned} \mathop {\lim }\limits _{h \rightarrow \infty } { \Vert {{u_h} - u} \Vert _{{L^1}(\Omega )}} = 0 \end{aligned}$$

and

$$\begin{aligned} \mathop {\lim } \limits _{h \rightarrow \infty } \int _\Omega { | {{\nabla _{{{\mathcal {\widetilde{H}}}}_ \sigma }}{u_h}(x)} |dx} = | {{\nabla _{\mathcal {\widetilde{H}}_{\sigma }}}u} |(\Omega ). \end{aligned}$$

It should be noted that Proposition 2.3 implies that we need to add the additional condition for the function in $\mathcal{BV}\mathcal{}_{{\mathcal {\widetilde{H}}_{\sigma }} } ( \Omega )$ to obtain the approximation result, that is,

$$\begin{aligned} {\int _\Omega { | {f(y)} | {{ | y |}^{ - 2}}dy}}<\infty . \end{aligned}$$

(1)

In order to investigate capacity theory, we let $\Omega \subseteq {{\mathbb {R}}^d}$ be a bounded open domain away from the origin, which can guarantee the function $f \in {{{\mathcal {B}}}} {{{\mathcal V}}_{{\mathcal {\widetilde{H}}_{\sigma }}}}(\Omega )$ satisfying the condition (1) for $d\ge 3$.

Proposition 2.4

Let $\Omega \subset {{\mathbb {R}} ^d}$ be a bounded open domain. Suppose that $u,v\in {L^1}(\Omega )$ satisfy the condition (1). Then,

$$\begin{aligned} | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}\max \{ {u,v} \}} |(\Omega ) + | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}\min \{ {u,v} \}} |(\Omega ) \le | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}u} |(\Omega ) + | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}v} |(\Omega ). \end{aligned}$$

Proposition 2.5

Let $\Omega \subset {{\mathbb {R}} ^d}$ be a bounded open domain. If $f \in {{{\mathcal {B}}}}{{{\mathcal V}}_{{\mathcal {\widetilde{H}}_{\sigma }}}} (\Omega )$ satisfies the condition (1), then the following coarea formula holds:

$$\begin{aligned} | {{ \nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f} |(\Omega ) \approx \int _ { - \infty }^{ + \infty } {{P_{{\mathcal {\widetilde{H}}_{\sigma } }}}({E_t},\Omega ) dt}, \end{aligned}$$

(2)

where ${E_t}=\{x\in \Omega : f(x)>t\}$ for $t\in {\mathbb {R}}$.

The ${\mathcal {\widetilde{H}}_{\sigma }}$-perimeter of $E\subseteq \Omega$ can be defined as follows:

$$\begin{aligned} P_{\mathcal {\widetilde{H}}_{\sigma }}(E,\Omega )=\big | \nabla _ {\mathcal {\widetilde{H}}_{\sigma }} 1_E \big |(\Omega ) =\sup _{\Phi \in \mathcal {\widetilde{F}}(\Omega ) } \Big \{\int _E \mathrm {div}_{{\mathcal {\widetilde{H}}_{\sigma }}}\Phi (x)dx\Big \}. \end{aligned}$$

The following propositions proved in [3] give the properties of the ${\mathcal {\widetilde{H}}_{\sigma }}$-perimeter of $P_{\mathcal {\widetilde{H}}_{\sigma }}(E,\Omega )$.

Proposition 2.6

(Lower semicontinuity of $P_{\mathcal {\widetilde{H}}_{\sigma }}$) Suppose ${1_{{E_k}}} \rightarrow {1_E}$ in $L_{loc}^1(\Omega )$, where E and ${E_k}$, $k \in {\mathbb {N}},$ are subsets of $\Omega$. Then

$$\begin{aligned} P_{\mathcal {\widetilde{H}}_{\sigma }}(E,\Omega )\le \mathop {\lim \inf }\limits _{k \rightarrow \infty } P_{\mathcal {\widetilde{H}}_{\sigma }}({E_k},\Omega ). \end{aligned}$$

Proposition 2.7

For any two compact subsets E, F in $\Omega$, we have

$$\begin{aligned} P_{\mathcal {\widetilde{H}}_{\sigma }}(E\cap F,\Omega ) +P_{\mathcal {\widetilde{H}}_{\sigma }}(E\cup F,\Omega )\le P_{\mathcal {\widetilde{H}}_{\sigma }}(E,\Omega )+P_{\mathcal {\widetilde{H}}_{\sigma }}( F,\Omega ). \end{aligned}$$

(3)

Especially, if $P_{\mathcal {\widetilde{H}}_{\sigma }}(E{\setminus } (E\cap F),\Omega ) \cdot P_{\mathcal {\widetilde{H}}_{\sigma }}(F{\setminus } (F\cap E),\Omega )=0$, the equality of (3) holds true.

In addition, [3] investigates the following Sobolev’s inequality and the isoperimetric inequality for $\mathcal {\widetilde{H}}_{\sigma }$-BV functions.

Proposition 2.8

(i) (Sobolev inequality) Let $\Omega \subset {{\mathbb {R}} ^d}$ be an open and bounded domain. For all $f \in {{{\mathcal {B}}}} {{{{\mathcal {V}}}}_{\mathcal {\widetilde{H}}_{\sigma }}}(\Omega )$ satisfying the condition (1), we have

$$\begin{aligned} { \Vert f \Vert _{L^{{d}/{(d-1)}}(\Omega )}}\lesssim | {{\nabla _{\mathcal {\widetilde{H}}_{\sigma }}}f} |(\Omega ). \end{aligned}$$

(4)

(ii) (Isoperimetric inequality) Let E be a bounded set of finite ${\mathcal {\widetilde{H}}_{\sigma }}$-perimeter in $\Omega$. Then

$$\begin{aligned} | E |^{{1-1/d}} \lesssim {{P_{\mathcal {\widetilde{H}}_{\sigma }}}(E,\Omega )} . \end{aligned}$$

(5)

(iii) The above two statements are equivalent.

Definition 2.9

Let $\Omega \subseteq {{\mathbb {R}}^n}$ be a bounded open domain away from the origin, and for a set $E\subseteq \Omega$, let ${{\mathcal {A}}}(E, {\mathcal{BV}\mathcal{}}_{\mathcal {\widetilde{H}} _\sigma } ({\Omega }))$ be the class of admissible functions on $\Omega$, i.e., functions $f\in \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma } (\Omega )$ satisfying $0\le f\le 1$ and $f=1$ in a neighborhood of E (an open set containing E). The ${\mathcal {\widetilde{H}} _\sigma }$-BV capacity of E is defined by

$$\begin{aligned} \widetilde{\mathrm {cap}}(E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))=\inf \{\parallel f \parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}} _\sigma } f | (\Omega ): f\in {{\mathcal {A}}}(E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))\}. \end{aligned}$$

Via the coarea formula in Proposition 2.5 for ${\mathcal {\widetilde{H}} _\sigma }$-BV function, we can obtain the following basic assertions.

Theorem 2.10

A geometric description of the ${\mathcal {\widetilde{H}} _\sigma }$-BV capacity of a set in $\Omega$ is given as follows:

(i) For any set $K\subseteq \Omega$,

$$\begin{aligned} \widetilde{\mathrm {cap}}(K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))\approx \inf _A \{|A|+P_{\mathcal {\widetilde{H}} _\sigma }(A,\Omega )\}, \end{aligned}$$

where the infimum is taken over all sets $A\subseteq \Omega$ such that $K\subseteq int(A)$.

(ii) For any compact set $K\subseteq \Omega$,

$$\begin{aligned} \widetilde{\mathrm {cap}}(K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))\approx \inf _A \{|A|+P_{\mathcal {\widetilde{H}} _\sigma }(A,\Omega )\}, \end{aligned}$$

where the infimum is taken over all bounded open sets A with smooth boundary in $\Omega$ containing K.

Proof

(i) If $A\subseteq \Omega$ with $K\subseteq int(A)$ and $|A|+P_{\mathcal {\widetilde{H}} _\sigma }(A,\Omega )<\infty$, $1_A\in {{\mathcal {A}}}(K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))$, then

$$\begin{aligned} \widetilde{\mathrm {cap}}(K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))\le |A|+ P_{\mathcal {\widetilde{H}} _\sigma }(A,\Omega ). \end{aligned}$$

By taking the infimum over all such sets A, we have

$$\begin{aligned} \widetilde{\mathrm {cap}}\big (K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega )\big )\le \inf _A \{|A|+ P_{\mathcal {\widetilde{H}} _\sigma }(A,\Omega )\}. \end{aligned}$$

In the following, the inverse inequality is also true. Assume that $\widetilde{\mathrm {cap}}(K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))<\infty$. Let $\varepsilon >0$ and $f\in {{\mathcal {A}}}\big (K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega )\big )$ such that

$$\begin{aligned} \parallel f \parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}} _\sigma } f | (\Omega )< \widetilde{\mathrm {cap}}\big (K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega )\big )+\varepsilon . \end{aligned}$$

Via the coarea formula in Proposition 2.5 and the Cavalieri principle, we have

$$\begin{aligned}&\int _{\Omega }f(x)dx+ | \nabla _{\mathcal {\widetilde{H}} _\sigma } f | (\Omega )\\&\quad \approx \int _0^1 {\left[ {| {\left\{ {x \in {\Omega }:f(x)> t} \right\} } | + {P_{{{{\widetilde{{\mathcal {H}}}}}_\sigma }}}\left( {\left\{ {x \in {\Omega }:f(x) > t} \right\} },\Omega \right) } \right] dt} \\ \end{aligned}$$

Then, there exists a $t_0\in (0,1)$ such that

$$\begin{aligned} {P_{{{{\widetilde{{\mathcal {H}}}}}_\sigma }}}\left( {\left\{ {x \in {\Omega }:f(x) > t} \right\} },\Omega \right) \lesssim \widetilde{\mathrm {cap}}(K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))+\varepsilon \end{aligned}$$

Since $K\subseteq int\{x\in \Omega : f(x)>t_0\}$ for $0<t_0<1,$

$$\begin{aligned} \inf _{A,K\subseteq int(A)} \{|A|+ P_{\mathcal {\widetilde{H}} _\sigma }(A,\Omega )\}\lesssim \widetilde{\mathrm {cap}}(K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))+\varepsilon . \end{aligned}$$

This completes the proof as $\varepsilon \rightarrow 0$.

(ii) Similarly to the proof of (i), via using the coarea formula in Proposition 2.5 and Cavalieri principle again, we can prove that (ii) is also valid, and we omit the details here. $\square$

The following capacitary estimates for balls can be obtained via using Theorem 2.10 and Sobolev inequality in Proposition 2.8.

Corollary 2.11

$$\begin{aligned}|B(x,r)|+|B (x,r)|^{\frac{d-1}{d}}\lesssim \widetilde{\mathrm {cap}}(B(x,r), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))\lesssim |B(x,r)|+ P_{{{\widetilde{{\mathcal {H}}}}}_\sigma } (B(x,r),\Omega ), \end{aligned}$$

where $B(x,r)\subset \Omega$ is an open ball centered at x with radius r.

Proof

Using Theorem 2.10, we easily obtain

$$\begin{aligned}\widetilde{\mathrm {cap}}(B(x,r), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega ))\lesssim |B(x,r)|+ P_{{{\widetilde{{\mathcal {H}}}}}_\sigma } (B(x,r),\Omega ). \end{aligned}$$

On the other hand, for any $f\in \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega )$ with $f=1$ in a neighborhood of B(x, r) and $0\le f\le 1$ on $\Omega$, using the Sobolev inequality in Proposition 2.8, we have

$$\begin{aligned}|B (x,r)|^{\frac{d-1}{d}}\le \Big ( \int _{\Omega } |f(y)|^{\frac{d}{d-1}}dy \Big )^{\frac{d-1}{d}}\lesssim | {{\nabla _{\mathcal {\widetilde{H}}_{\sigma }}}f} |(\Omega ). \end{aligned}$$

A further application of the definition of ${\mathcal {\widetilde{H}} _\sigma }$-BV capacity derives

$$\begin{aligned} |B(x,r)|+|B (x,r)|^{\frac{d-1}{d}}\lesssim \widetilde{\mathrm {cap}}(B(x,r), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}} _\sigma }(\Omega )). \end{aligned}$$

$\square$

In what follows, we obtain some measure-theoretic results of ${\mathcal {\widetilde{H}} _\sigma }$-BV capacity.

Theorem 2.12

Assume that A, B are subsets of $\Omega$.

• $\mathrm{(i)}$

  $$\begin{aligned} \widetilde{\mathrm {cap}}(\emptyset , \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))=0. \end{aligned}$$

• $\mathrm{(ii)}$ If $A\subseteq B$, then

  $$\begin{aligned} \widetilde{\mathrm {cap}}(A, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))\le \widetilde{\mathrm {cap}}(B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )). \end{aligned}$$

• $\mathrm{(iii)}$

  $$\begin{aligned}&\widetilde{\mathrm {cap}}(A\cup B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))+ \widetilde{\mathrm {cap}}(A \cap B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))\\&\quad \le \widetilde{\mathrm {cap}}(A, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))+ \widetilde{\mathrm {cap}}(B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )). \end{aligned}$$

• $\mathrm{(iv)}$ If $A_k, k=1,2,\ldots$, are subsets in $\Omega$, then

  $$\begin{aligned} \widetilde{\mathrm {cap}}(\mathop \cup \limits _{k = 1}^\infty {A_k}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))\le \sum ^\infty _{k=1}\widetilde{\mathrm {cap}}(A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )). \end{aligned}$$

• $\mathrm{(v)}$ For any sequence $\{A_k\}^\infty _{k=1}$ of subsets in $\Omega$ with $A_1\subseteq A_2\subseteq A_3\subseteq \ldots$, we have

  $$\begin{aligned} \lim _{k\rightarrow \infty } \widetilde{\mathrm {cap}}(A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))= \widetilde{\mathrm {cap}}(\mathop \cup \limits _{k = 1}^\infty {A_k}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )). \end{aligned}$$

• $\mathrm{(vi)}$ If $A_k, k=1,2,\ldots$, are compact sets in $\Omega$ and $A_1\supseteq A_2\supseteq A_3\supseteq \ldots$, then

  $$\begin{aligned} \lim _{k\rightarrow \infty }\widetilde{\mathrm {cap}}(A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))=\widetilde{\mathrm {cap}}(\mathop \cap \limits _{k = 1}^\infty {A_k}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )). \end{aligned}$$

Proof

(i-ii). It is obvious that statements (i) and (ii) are valid from Definition 2.9.

(iii) Without loss of generality, we may assume

$$\begin{aligned} \widetilde{\mathrm {cap}}\big (A, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )+ \widetilde{\mathrm {cap}}\big (B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )<\infty . \end{aligned}$$

For any $\varepsilon >0$, there are two functions $\phi \in {{\mathcal {A}}}\big (A,\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )$ and $\psi \in {{\mathcal {A}}}(B,\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \parallel \phi \parallel _{L^1(\Omega )}+ | { \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi } |(\Omega )< \widetilde{\mathrm {cap}}(A, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))+\frac{\varepsilon }{2};\\ \parallel \psi \parallel _{L^1(\Omega )}+| {\nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \psi } | (\Omega )< \widetilde{\mathrm {cap}}(B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))+\frac{\varepsilon }{2}. \end{array}\right. } \end{aligned}$$

Let

$$ \begin{aligned} \varphi _1=\max \{\phi ,\psi \}\ \ \& \ \ {\varphi }_2=\min \{\phi ,\psi \}. \end{aligned}$$

It is easy to see that

$$ \begin{aligned} \varphi _1\in {{\mathcal {A}}}\big (A\cup B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )\ \ \& \ \ \varphi _2\in {{\mathcal {A}}}\big (A\cap B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big ). \end{aligned}$$

Then, by Proposition 2.4,

$$\begin{aligned}&\widetilde{\mathrm {cap}}\big (A\cup B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )+ \widetilde{\mathrm {cap}}\big (A \cap B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )\\&\quad \le \int _{\Omega }\varphi _1(x)dx+\int _{\Omega }\varphi _2(x)dx+ | \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \varphi _1|(\Omega )+ | \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \varphi _2|(\Omega )\\&\quad \le \int _{\Omega }\phi (x)dx+\int _{\Omega }\psi (x)dx+ | \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi |(\Omega )+ | \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \psi |(\Omega )\\&\quad \le \widetilde{\mathrm {cap}}(A, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))+ \widetilde{\mathrm {cap}}(B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))+\varepsilon . \end{aligned}$$

Hence, the assertion (iii) is proved.

(iv) Suppose that

$$\begin{aligned} \sum ^\infty _{k=1}\widetilde{\mathrm {cap}}(A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega ))<\infty . \end{aligned}$$

For any $\varepsilon >0$ and $k=1,2,\ldots$, there is $f_k\in {\mathcal {A}}\big (A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )$ such that

$$\begin{aligned} \parallel f_k\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} f_k| (\Omega )< \widetilde{\mathrm {cap}}\big (A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )+\frac{\varepsilon }{2^k}. \end{aligned}$$

Setting $f = \mathop {\sup }\limits _k {f_k}$, then

$$\begin{aligned} \int _{\Omega }f(x)dx\le \sum ^\infty _{k=1}\int _{\Omega }f_k(x)dx< \sum ^\infty _{k=1}\widetilde{\mathrm {cap}}\big (A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )+\varepsilon <\infty , \end{aligned}$$

which implies $f\in L^1(\Omega )$.

Via the lower semicontinuity in Proposition 2.2, we have

$$\begin{aligned} \int _{\Omega }f(x)dx+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} f| (\Omega )\le & {} \sum ^\infty _{k=1}\int _{\Omega }f_k(x)dx+\mathop {\lim \inf }\limits _{k \rightarrow \infty } | {\nabla _{\mathcal {\widetilde{H}}_{{\sigma }}}}\max \{ {f_1}, \ldots ,{f_k}\} |(\Omega )\\\le & {} \sum ^\infty _{k=1}\int _{\Omega }f_k(x)dx+\sum ^\infty _{k=1}| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} f_k|(\Omega )\\\le & {} \sum ^\infty _{k=1}\widetilde{\mathrm {cap}}\big (A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )+\varepsilon . \end{aligned}$$

Consequently, $f\in {\mathcal {A}}\big (\mathop \cup \limits _{k = 1}^\infty {A_k},\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )$ and this completes the proof of (iv) when $\varepsilon \rightarrow 0$.

(v) It is obvious that

$$\begin{aligned} \lim _{k\rightarrow \infty }\widetilde{\mathrm {cap}}\big (A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )\le \widetilde{\mathrm {cap}}( \mathop \cup \limits _{k = 1}^\infty {A_k}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )). \end{aligned}$$

First, the equality holds if

$$\begin{aligned} \lim _{k\rightarrow \infty }\widetilde{\mathrm {cap}}\big (A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )=\infty . \end{aligned}$$

Second, let $\varepsilon > 0$ and assume that

$$\begin{aligned} \lim _{k\rightarrow \infty }\widetilde{\mathrm {cap}}\big (A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\big )<\infty . \end{aligned}$$

For $k = 1, 2, \ldots$, there is $f_k\in {{\mathcal {A}}}\left( A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right)$ such that

$$\begin{aligned} \parallel f_k\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} f_k|(\Omega )<\widetilde{\mathrm {cap}}\left( A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\frac{\varepsilon }{2^k}. \end{aligned}$$

Set

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi _k=\mathop {\max }\limits _{1 \le i \le k} {f_i}=\max \{\phi _{k-1},\ f_{k}\};\\ \phi _0=0;\\ A_0=\emptyset ;\\ \varphi _k=\min \{\phi _{k-1},\ f_k\}. \end{array}\right. } \end{aligned}$$

Then

$$ \begin{aligned} \phi _k,\varphi _k\in \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\ \ \& \ \ A_{k-1}\subseteq int\{x\in \Omega : \varphi _k(x)=1\}. \end{aligned}$$

Since $\phi _k=\max \{\phi _{k-1}, \phi _k\}$, an application of Proposition 2.4 yields

$$\begin{aligned}&| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \max \{\phi _{k-1}, \phi _k\}|(\Omega )+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \min \{\phi _{k-1}, \phi _k\}|(\Omega )\\&\quad \le | \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k-1}|(\Omega ) +| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k}|(\Omega ) \end{aligned}$$

and then

$$\begin{aligned}&\parallel \phi _{k}\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k}|(\Omega )+\widetilde{\mathrm {cap}}\left( A_{k-1}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \\&\le \parallel \phi _{k}\parallel _{L^1(\Omega )}+ | \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k}|(\Omega ) +\parallel \varphi _{k}\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \varphi _{k}| (\Omega )\\&\le \parallel \phi _{k}\parallel _{L^1(\Omega )}+\parallel \phi _{k-1}\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k}|(\Omega )+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k-1}|(\Omega )\\&\le \parallel \phi _{k-1}\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k-1}|(\Omega )+\widetilde{\mathrm {cap}}\left( A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\frac{\varepsilon }{2^k}, \end{aligned}$$

where we have used the fact that $A_{k-1}\subseteq A_{k}$. Therefore,

$$\begin{aligned}&\parallel \phi _{k}\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k}|(\Omega )-\parallel \phi _{k-1}\parallel _{L^1(\Omega )}- | \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _{k-1}|(\Omega )\\&\le \widetilde{\mathrm {cap}}\left( A_{k}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) -\widetilde{\mathrm {cap}}\left( A_{k-1}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\frac{\varepsilon }{2^k}. \end{aligned}$$

By adding the above inequalities from $k=1$ to $k=j$, we have

$$\begin{aligned} \parallel \phi _{j}\parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _j|(\Omega )\le \widetilde{\mathrm {cap}}\left( A_{j}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\varepsilon . \end{aligned}$$

Let ${\tilde{\phi }} = \mathop {\lim }\limits _{j \rightarrow \infty } {\phi _j}$. Via the monotone convergence theorem, we get

$$\begin{aligned} \int _{\Omega }{\tilde{\phi }}(x)dx=\lim _{j\rightarrow \infty }\int _{\Omega }\phi _j(x)dx\le \lim _{j\rightarrow \infty }\widetilde{\mathrm {cap}}\left( A_{j}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\varepsilon . \end{aligned}$$

Then, via the lower semicontinuity in Proposition 2.2, we obtain

$$\begin{aligned} {\tilde{\phi }}\in {{\mathcal {A}}}\left( \mathop \cup \limits _{j = 1}^\infty {A_j},\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \end{aligned}$$

and

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( \mathop \cup \limits _{j = 1}^\infty {A_j}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right)\le & {} \parallel {\tilde{\phi }} \parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} {\tilde{\phi }}|(\Omega )\\\le & {} \liminf _{j\rightarrow \infty }\big (\int _{\Omega }\phi _j(x)dx+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} \phi _j|(\Omega )\big )\\\le & {} \lim _{j\rightarrow \infty }\widetilde{\mathrm {cap}}\left( A_{j}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\varepsilon . \end{aligned}$$

(vi) Let $A=\mathop \cap \limits _{k = 1}^\infty {A_k}$. By monotonicity, we have

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( \mathop \cap \limits _{k = 1}^\infty {A_k}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \le \lim _{k\rightarrow \infty } \widetilde{\mathrm {cap}}\left( A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) . \end{aligned}$$

Let U be an open set containing A. Then, by the compactness of A, we know that $A_k\subseteq U$ for all sufficiently large k. Thus,

$$\begin{aligned} \lim _{k\rightarrow \infty } \widetilde{\mathrm {cap}}\left( A_k, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \le \widetilde{\mathrm {cap}}\left( U, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) . \end{aligned}$$

Corollary 2.13 implies that $\widetilde{\mathrm {cap}}\left( \cdot , \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right)$ is an outer capacity. Then, we obtain the claim by taking infimum over all open sets U containing A. $\square$

Corollary 2.13

(i) If $E\subseteq \Omega$, then

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) =\inf \left\{ \mathop {\widetilde{\mathrm {cap}}}\limits _{{\mathrm{open}}{} O \supseteq E}\left( O, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \right\} . \end{aligned}$$

(ii) If $E\subseteq \Omega$ is a Borel set, then

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) =\sup \left\{ \mathop {\widetilde{\mathrm{cap}}}\limits _{{\mathrm{compact}}{} K \subseteq E}\left( K, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \right\} . \end{aligned}$$

Proof

(i) By the statement (ii) of Theorem 2.12, we have

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \le \inf \{\mathop {\widetilde{\mathrm{cap}}}\limits _{{\mathrm{open}}{} O \supseteq E}\left( O, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \}. \end{aligned}$$

To prove the reverse inequality, we may assume

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) <\infty . \end{aligned}$$

By Definition 2.9, for any $\varepsilon >0$, there is $f\in {{\mathcal {A}}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right)$ such that

$$\begin{aligned} \parallel f \parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} f| (\Omega )< \widetilde{\mathrm {cap}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\varepsilon . \end{aligned}$$

Therefore, there is an open set $O\supseteq E$ such that $f=1$ on O and

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( O, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \le \parallel f \parallel _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_{{\sigma }}} f|(\Omega )< \widetilde{\mathrm {cap}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) +\varepsilon . \end{aligned}$$

Moreover,

$$\begin{aligned} \widetilde{\mathrm {cap}}\left( E, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \ge \inf \{\mathop {\widetilde{\mathrm{cap}}}\limits _{{\mathrm{open}}{} O \supseteq E} \left( O, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_{{\sigma }}}(\Omega )\right) \}. \end{aligned}$$

(ii) This follows from (v) and (vi) in Theorem 2.12. $\square$

3 Poincaré Type and ${\mathcal {\widetilde{H}} _\sigma }$-Capacitary Inequalities

In this section, similarly to Xiao’s result in [14, Theorem 2], we investigate the relation between the nonnegative Radon measure $\mu$ and the Poincaré type inequality:

$$\begin{aligned} \mathop {\sup }\limits _{0 \ne f \in \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )} \frac{{{{\left\| f \right\| }_{L_\mu ^q({\Omega })}}}}{{{{\left\| f \right\| }_{\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }({\Omega })}}}} < \infty . \end{aligned}$$

(6)

As shown in the next theorem for ${\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }({\Omega })}$, the validity of (6) is closely linked with the domination of $\mu (B)$ via $P_{\mathcal {\widetilde{H}}_\sigma }(B,\Omega )$ or $\widetilde{\mathrm {cap}}\left( B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )\right)$ for any Borel set $B \subseteq {\Omega }$.

Theorem 3.1

Given $1\le p\le \frac{d}{d-1}$ and a nonnegative Radon measure $\mu$ on $\Omega$. The following three statements are equivalent:

(i)

$$\begin{aligned} {\left( {\int _{{\Omega }} {{{| f |}^p}d\mu } } \right) ^{ \frac{1}{p}}} \lesssim \Vert f\Vert _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_\sigma } f|({\Omega }) \end{aligned}$$

for all functions $f \in {{{\mathcal {B}}}} {{{\mathcal V}}_{{\mathcal {\widetilde{H}}_{\sigma }}}}(\Omega )$ which are defined $\mu$-a.e..

(ii)

$$\begin{aligned} \mu {(B)^{\frac{1}{p}}}\lesssim |B|+ P_{\mathcal {\widetilde{H}}_\sigma }(B,\Omega ) \end{aligned}$$

for all Borel sets $B\subseteq \Omega$.

(iii)

$$\begin{aligned} \mu {(B)^{ \frac{1}{p}}} \lesssim \, \widetilde{\mathrm {cap}}(B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )) \end{aligned}$$

for all Borel sets $B\subseteq \Omega$.

Proof

(i)$\Rightarrow$(ii). By taking $f=1_B$ and considering the definition of $P_{\mathcal {\widetilde{H}}_\sigma }(\cdot ,\Omega )$, we can deduce that (ii) is valid.

(ii)$\Rightarrow$(iii). For all bounded open sets $O\subseteq \Omega$ with smooth boundary containing B which is a compact subset, by the above assumption, we obtain

$$\begin{aligned} {(\mu (O))^{\frac{1}{p}}} \le {(\mu ({\bar{O}}))^{\frac{1}{p}}} \lesssim |\bar{O}|+P_{\mathcal {\widetilde{H}}_\sigma }(\bar{O},\Omega )=|{O}|+ P_{\mathcal {\widetilde{H}}_\sigma }(O,\Omega ). \end{aligned}$$

Then, by Theorem 2.10, we have

$$\begin{aligned} {(\mu (B))^{\frac{1}{p}}} \lesssim \widetilde{\mathrm {cap}}(B, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\approx \inf _{O\supseteq B} \{P_{\mathcal {\widetilde{H}}_\sigma }(O,\Omega )+|O| \}. \end{aligned}$$

Hence, using (ii) of Corollary 2.13 and the inner regularity of $\mu$, we conclude that (iii) is true.

(iii)$\Rightarrow$(i). Suppose (iii) holds. First, we claim that f is finite almost everywhere with respect to the measure $\mu$ for $f\in \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )$. Indeed, we can assume $f\in C^\infty _0(\Omega ) \bigcap \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )$. For $t>0$, let $E_t=\{x\in \Omega : |f(x)|>t\}$. By the coarea formula in Proposition 2.5, we know that $E_t$ has finite ${\mathcal {\widetilde{H}}_\sigma }$-perimeter for a.e. t and

$$\begin{aligned} \int ^\infty _0 P_{\mathcal {\widetilde{H}}_\sigma }(E_t,\Omega )dt\approx | \nabla _{\mathcal {\widetilde{H}}_\sigma } |f||(\Omega )<\infty . \end{aligned}$$

From this, we conclude that $\mathop {\lim \inf }\limits _{t \rightarrow \infty } P_{\mathcal {\widetilde{H}}_\sigma }(E_t,\Omega )=0.$ By Theorem 2.10, we have

$$\begin{aligned} \widetilde{\mathrm {cap}}(\{x\in \Omega : |f(x)|=\infty \}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )) \lesssim \mathop {\lim \inf }\limits _{t \rightarrow \infty } \{|E_t|+P_{\mathcal {\widetilde{H}}_\sigma }(E_t,\Omega )\}=0. \end{aligned}$$

By assumption, we know that $\mu (\{x\in \Omega : |f(x)|=\infty \})=0.$ This completes the proof of the claim.

For $f\in C^\infty _c(\Omega )\bigcap \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )$, combining the layer-cake formula, Theorem 2.10, and the coarea formula in Proposition 2.5, we have

$$\begin{aligned} \left( \int _{\Omega } |f|^pd\mu \right) ^{\frac{1}{p}}\le & {} \left( \int ^\infty _0 \mu (\{x\in \Omega : |f(x)|>t\})dt^p\right) ^{\frac{1}{p}}\\\le & {} \int ^\infty _0 \frac{d}{dt}\left( \int ^t_0 \mu (\{x\in \Omega : |f(x)|>s\})ds^p\right) ^{\frac{1}{p}}dt\\= & {} \int ^\infty _0 \left( \int ^t_0 \mu (\{x\in \Omega : |f(x)|>s\})ds^p\right) ^{\frac{1}{p}-1}\\&\times \mu (\{x\in \Omega : |f(x)|>t\})t^{p-1}dt\\\le & {} \int ^\infty _0 \left( \mu (\{x\in \Omega : |f(x)|>t\}) \right) ^{\frac{1}{p}}dt\\\lesssim & {} \int ^\infty _0 \widetilde{\mathrm {cap}}(\{x\in \Omega : |f(x)|>t\}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))dt\\\lesssim & {} \int _{0}^\infty \Big [|\{x\in \Omega :\ |f(x)|>s\}|+ P_{\mathcal {\widetilde{H}}_\sigma }\big (\{x\in \Omega :\ |f(x)|>s\},\Omega \big )\Big ]\,ds\\\lesssim & {} \Vert f\Vert _{L^1(\Omega )}+| \nabla _{\mathcal {\widetilde{H}}_\sigma } |f||(\Omega )\\\lesssim & {} \Vert f\Vert _{L^1(\Omega )}+ | \nabla _{\mathcal {\widetilde{H}}_\sigma } f |(\Omega ). \end{aligned}$$

Therefore, combining the approximation results in Proposition 2.3 with the above proofs, we know that (i) is true. $\square$

We can obtain the following imbedding result for the ${\mathcal {\widetilde{H}}_\sigma }$ case if $\mu$ in the above theorem is taken as the Lebesgue measure.

Theorem 3.2

The analytic inequality

$$\begin{aligned} \parallel f\parallel _{ L^{ \frac{d}{d-1}}(\Omega )}\lesssim \left( \int ^{\infty }_0 \big (\widetilde{\mathrm {cap}}(\{x\in \Omega : |f(x)|\ge t\}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\big )^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\right) ^{\frac{d-1}{d}} \end{aligned}$$

(7)

holds for any $f \in L^{\frac{d}{d-1}}(\Omega )$ with compact support if and only if the geometric inequality

$$\begin{aligned} |M|^{\frac{d-1}{d}}\lesssim \widetilde{\mathrm {cap}}(M, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )) \end{aligned}$$

(8)

holds for any compact set M in $\Omega$. Moreover, inequalities (7) and (8) are true.

Proof

We adopt Xiao’s method in [12] for the proof. In what follows, we assume:

$$\begin{aligned} \Omega _t(f)=\{x\in \Omega : |f(x)|\ge t\} \end{aligned}$$

and

$$\begin{aligned} \partial \Omega _t(f)=\{x\in \Omega : |f(x)|=t\} \end{aligned}$$

for a function f defined on $\Omega$ and a number $t>0$.

Given a compact set $M\subseteq \Omega ,$ let $f=1_M$. Then,

$$\begin{aligned} \Vert f\Vert _{L^{\frac{d}{d-1}}(\Omega )}=|M|^{1- \frac{1}{d}} \end{aligned}$$

and

$$\begin{aligned} \Omega _t(f)=\left\{ \begin{array}{cc} M, \hbox {if} t\in (0,1],\\ \ \ \emptyset ,\ \hbox {if} t\in (1,\infty ). \end{array}\right. \end{aligned}$$

Therefore,

$$\begin{aligned}&\int ^\infty _0 \left( \widetilde{\mathrm {cap}}\left( \Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )\right) \right) ^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\quad =\int ^1_0 \left( \widetilde{\mathrm {cap}}\left( \Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )\right) \right) ^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\qquad +\int ^\infty _1( \widetilde{\mathrm {cap}} (\Omega _t(f)), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\quad =\left( \widetilde{\mathrm {cap}}\left( M, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )\right) \right) ^{\frac{d}{d-1}}, \end{aligned}$$

which implies that (7) implies (8).

Conversely, we show that (8) implies (7). Suppose (8) holds for any compact set in $\Omega$. For $t>0$ and f, an $L^{\frac{d}{d-1}}$ integrable function with compact support in $\Omega$, we get

$$\begin{aligned} \parallel f\parallel ^{\frac{d}{d-1}}_{L^{\frac{d}{d-1}}(\Omega )}=\int ^{\infty }_0|\Omega _t(f)|dt^{\frac{d}{d-1}}\lesssim \int ^{\infty }_0\big ( \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\big )^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}. \end{aligned}$$

Moreover, since (7) is equivalent to (8), it suffices to prove that (8) is valid. Actually, for any bounded set B with smooth boundary containing M, using (ii) of Proposition 2.8, we have

$$\begin{aligned} |M|^{\frac{d-1}{d}}\le |B|^{\frac{d-1}{d}} \lesssim P_{\mathcal {\widetilde{H}}_\sigma }(B,\Omega )\lesssim |B|+P_{\mathcal {\widetilde{H}}_\sigma }(B,\Omega ). \end{aligned}$$

Theorem 2.10 implies that (8) holds true for $\widetilde{\mathrm {cap}}(\cdot , \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))$. $\square$

Theorem 3.3

For any $f\in {C}^{1}_{c}(\Omega )$, the following inequality

$$\begin{aligned}&\Big (\int ^{\infty }_0 (\widetilde{\mathrm {cap}}(\{x\in \Omega : |f(x)|\ge t\},\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\Big )^{\frac{d-1}{d}}\\&\quad \lesssim {\left\| f \right\| _{{L^1}(\Omega )}} + |\nabla _{\mathcal {\widetilde{H}}_\sigma }f|(\Omega ) \end{aligned}$$

holds true.

Proof

Let $f\in {C}^{1}_{c}(\Omega )$. Using (ii) of Theorem 2.12, we obtain

$$\begin{aligned}&\int _0^\infty (\widetilde{\mathrm {cap}}(\Omega _t(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\quad =\sum _{k=-\infty }^{+\infty }\int _{2^k}^{2^{k+1}} (\widetilde{\mathrm {cap}}(\Omega _t(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\quad \le \sum _{k=-\infty }^{+\infty }(2^{\frac{(k+1)d}{d-1}}-2^{\frac{kd}{d-1}})(\widetilde{\mathrm {cap}}(\Omega _{2^k}(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^\frac{d}{d-1}\\&\quad =(2^{\frac{d}{d-1}}-1)\sum _{k=-\infty }^{+\infty }2^{\frac{kd}{d-1}}(\widetilde{\mathrm {cap}} (\Omega _{2^k}(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^{\frac{d}{d-1}}\\&\quad \le (2^{\frac{d}{d-1}}-1)\Big (\sum _{k=-\infty }^{+\infty }2^k\widetilde{\mathrm {cap}}(\Omega _{2^k}(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\Big )^{\frac{d}{d-1}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Big (\int _0^\infty (\widetilde{\mathrm {cap}}(\Omega _t(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\Big )^{\frac{d-1}{d}} \lesssim \sum _{k=-\infty }^{+\infty }2^{k}\widetilde{\mathrm {cap}}(\Omega _{2^k}(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )). \end{aligned}$$

In the following, we estimate the series $\sum \limits _{k=-\infty }^{+\infty }2^{k}\widetilde{\mathrm {cap}}(\Omega _{2^k}(f),\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))$. By [10, page 155, Remark 1], we can find an even function $\tau : {\mathbb {R}}\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} 0\le \tau (t)\le 1,\ t\ge 0;\\ \tau (t)=0,\ t\in [0, {1}/{2}];\\ \tau (t)=1,\ t\in [1, \infty );\\ 0\le \tau '(t)\le 3,\ t\ge 0. \end{array}\right. } \end{aligned}$$

For any $k\in {\mathbb {Z}}$ and $x\in \Omega$, let

$$\begin{aligned} f_k(x)=\tau \Big (\frac{f(x)}{2^k}\Big )= {\left\{ \begin{array}{ll} 0,\quad x\notin \Omega _{2^{k-1}};\\ 1,\quad x\in \Omega _{2^k}, \end{array}\right. } \end{aligned}$$

and $0\le f_k(x)\le 1$ for all $x\in \Omega _{2^{k-1}}\setminus \Omega _{2^k}$. Then,

$$\begin{aligned} \int _{\Omega }|f_k(x)|dx =\int _{\Omega _{2^{k-1}}}|f_k(x)|dx\le \int _{\Omega _{2^{k-1}}}dx. \end{aligned}$$

On the other hand, since $|\nabla f|+\frac{{| \sigma |}}{{| x |}}| f | \lesssim |\nabla _{\mathcal {\widetilde{H}}_\sigma }f|,$ which has been proved in [3, Lemma 3.11]. Consequently, $f_k\in \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )$. Noting that $f_k\equiv 1$ on $\Omega _{2^k}$, we get

$$\begin{aligned} {\left\{ \begin{array}{ll} \Omega _{2^k}\subset \Big \{x\in \Omega :\, f_k(x)\ge 1\Big \}^\circ ;\\ f_k\in {\mathcal {A}}(\Omega _{2^k},\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )). \end{array}\right. } \end{aligned}$$

Then, we obtain

$$\begin{aligned} \widetilde{\mathrm {cap}}(\Omega _{2^k},\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\le & {} \int _{\Omega }|f_k(x)|dx+|\nabla _{\mathcal {\widetilde{H}}_\sigma }f_k(x)|(\Omega )\\\le & {} \int _{\Omega _{2^{k-1}}}dx+|\nabla _{\mathcal {\widetilde{H}}_\sigma }f_k(x)|(\Omega )\\:= & {} I_{1}+I_{2}. \end{aligned}$$

As for $I_{1}$, we have

$$\begin{aligned} \sum ^{+\infty }_{k=-\infty }2^{k}\int _{\Omega _{2^{k-1}}}dx\lesssim \sum ^{+\infty }_{k=-\infty }2^{k}\int _{\Omega _{2^{k-1}}}\frac{|f(x)|}{2^{k}}dx \lesssim \int _{\Omega }|f(x)|dx. \end{aligned}$$

As for $I_{2}$, for all $\Phi \in C_c^1(\Omega , {{\mathbb {R}}^{2d}})$ and ${ \Vert \Phi \Vert _\infty }\le 1$, by (i) of Proposition 2.2, we have

$$\begin{aligned} \int _\Omega {{f_k}{\mathrm {div}_{{\mathcal {\widetilde{H}}_\sigma }}} \Phi dx}= & {} \int _\Omega { \Phi \cdot {\nabla _{\mathcal {\widetilde{H}}_\sigma }}{f_k}dx}\\= & {} 2^{-k}\int _{{\Omega _{{2^{k - 1}}}}\backslash {\Omega _{{2^k}}}} { \Phi \cdot {\nabla _{\mathcal {\widetilde{H}}_\sigma }}fdx}\\= & {} 2^{-k}\int _{{\Omega _{{2^{k - 1}}}}\backslash {\Omega _{{2^k}}}} { {f}{\mathrm {div}_{{\mathcal {\widetilde{H}}_\sigma }}} \Phi dx}\\\le & {} 2^{-k}|{{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f}|({{\Omega _{{2^{k - 1}}}}\backslash {\Omega _{{2^k}}}}), \end{aligned}$$

which implies that

$$\begin{aligned} | {{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f_k} |(\Omega ) \le 2^{-k}|{{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f}|({{\Omega _{{2^{k - 1}}}}\backslash {\Omega _{{2^k}}}}).\end{aligned}$$

Therefore,

$$\begin{aligned} \sum ^{+\infty }_{k=-\infty }2^{k}\widetilde{\mathrm {cap}}(\Omega _{2^k},\mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\lesssim & {} \sum ^{+\infty }_{k=-\infty }2^{k}\Big (\int _{\Omega _{2^{k-1}}}dx+|\nabla _{\mathcal {\widetilde{H}}_\sigma }f_k(x)|(\Omega )\Big )\\\lesssim & {} \int _{\Omega }|f(x)|dx+\sum ^{+\infty }_{k=-\infty }|{{\nabla _{{\mathcal {\widetilde{H}}_{\sigma }}}}f}|({{\Omega _{{2^{k - 1}}}}\backslash {\Omega _{{2^k}}}})\\\lesssim & {} {\left\| f \right\| _{{L^1}(\Omega )}} + |\nabla _{\mathcal {\widetilde{H}}_\sigma }f|(\Omega ). \end{aligned}$$

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

Theorem 3.4

The analytic inequality

$$\begin{aligned}&\left( \int ^{\infty }_0\big (\widetilde{\mathrm {cap}}(\{x\in \Omega : |f(x)|\ge t\}, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\big )^{{\frac{d}{d-1}}}dt^{{\frac{d}{d-1}}}\right) ^{{\frac{d-1}{d}}}\nonumber \nonumber \\&\quad \lesssim \int _{\Omega }| f(x)|dx+ |\nabla _{\mathcal {\widetilde{H}}_\sigma }f(x)|(\Omega ) \end{aligned}$$

(9)

holds for any $f\in C^{1}_c(\Omega )$ if and only if the geometric inequality

$$\begin{aligned} \widetilde{\mathrm {cap}}(M, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\lesssim |M|+ P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega ) \end{aligned}$$

(10)

holds for any connected compact set M in $\Omega$ with smooth boundary. Moreover, inequalities (9) and (10) are true.

Proof

We adopt Xiao’s method in [12] for the proof. For a connected compact set $M\subseteq \Omega$ with smooth boundary, choose $\delta >0$ such that $2\delta <{\mathrm {dist}_{{\mathbb {R}}^{d}}({M},\partial \Omega })$, where ${\mathrm {dist}_{{\mathbb {R}}^{d}}({M},\Omega })$ represents the Euclidean distance from M to $\partial \Omega$.

Define the Lipschitz function

$$\begin{aligned} f_\delta (x)=\left\{ \begin{array}{cc}1 -{\delta ^{ -1}}{\mathrm {dist}_{{{\mathbb {R}}^{d}}}}(x,M), \hbox {if} \mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})<\delta ,\\ 0,\ \hbox {if} \mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})\ge \delta . \end{array}\right. \end{aligned}$$

If (9) holds, then due to $M\subseteq \Omega _t(f_\delta )$ for $t\in [0,1]$,

$$\begin{aligned} \widetilde{\mathrm {cap}}(M, \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\le & {} \left( \int ^{1}_0 \big ( \widetilde{\mathrm {cap}}(\Omega _t(f_\delta ), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }( \Omega ))\big )^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\right) ^{\frac{d-1}{d}}\\\le & {} \parallel f_\delta \parallel _{L^1(\Omega )}+ |\nabla _{\mathcal {\widetilde{H}}_\sigma }f_\delta |(\Omega ). \end{aligned}$$

Using the coarea formula for the ${\mathcal {\widetilde{H}}_\sigma }$-BV function and (i) in Proposition 2.2, we have

$$\begin{aligned} |\nabla _{\mathcal {\widetilde{H}}_\sigma }f_\delta |(\Omega )\approx & {} \int ^1_0 P_{\mathcal {\widetilde{H}}_\sigma }(\{x\in \Omega : f_\delta (x)>t \},\Omega )dt\\= & {} \int ^1_0 P_{\mathcal {\widetilde{H}}_\sigma }(\{x\in \Omega : \mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})<\delta (1-t) \},\Omega )dt\\= & {} \int ^1_0 \big (P_{\mathcal {\widetilde{H}}_\sigma }(\{x\in \Omega : \mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})<\delta (1-t) \},\Omega ) -P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega )\big )dt\\&+P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega ). \end{aligned}$$

Next, we deal with the following integral

$$\begin{aligned} \int ^1_0 \big (P_{\mathcal {\widetilde{H}}_\sigma }(\{x\in \Omega : \mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})<\delta (1-t) \},\Omega )-P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega )\big )dt. \end{aligned}$$

By Proposition 2.7, we have

$$\begin{aligned}&\int ^1_0 \big (P_{\mathcal {\widetilde{H}}_\sigma }(\{x\in \Omega : \mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})<\delta (1-t) \},\Omega )-P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega )\big )dt\\&\quad \le \int ^1_0 P_{\mathcal {\widetilde{H}}_\sigma }(\{x\in \Omega : 0<\mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})<\delta (1-t) \},\Omega )dt. \end{aligned}$$

Denoted by

$$\begin{aligned} E_\delta =\{x\in \Omega : 0<\mathrm {dist}_{{\mathbb {R}}^{d}}(x,{M})<\delta (1-t) \}. \end{aligned}$$

Then, from the definition of ${\mathcal {\widetilde{H}}_\sigma }$-perimeter, we have

$$\begin{aligned} P_{\mathcal {\widetilde{H}}_\sigma }(E_\delta ,\Omega )= & {} |\nabla _{\mathcal {\widetilde{H}}_\sigma } 1_{E_\delta }|(\Omega )\\= & {} \sup _{\Phi \in \mathcal {\widetilde{F}}(\Omega ) } \Big \{\int _{E_\delta } \mathrm {div}_{{\mathcal {\widetilde{H}}_{\sigma }}}\Phi (x)dx\Big \}\\\lesssim & {} \sup _{\Phi \in \mathcal {\widetilde{F}}(\Omega ) } \left\{ {\int _{{E_\delta }} \big [-\sum ^{d}_{i=1}\frac{\partial \varphi _i}{\partial x_i}-\sum ^{d}_{i=1}\frac{\partial \varphi _{d+i}}{\partial x_{i}}\big ]dx} \right\} \\&+ \sup _{\Phi \in \mathcal {\widetilde{F}}(\Omega ) } \left\{ {\int _{{E_\delta }} \sigma \sum ^d_{i=1} \frac{x_i(\varphi _i(x)-\varphi _{d+i}(x)) }{|x|^2} dx} \right\} \\\lesssim & {} (P(E_\delta ,\Omega )+{ \big \Vert \frac{1}{|x|} \big \Vert _{{L^\infty }(\Omega )}}|{E_\delta }|)\rightarrow 0 \end{aligned}$$

when $\delta \rightarrow 0$, where $P(E_\delta ,\Omega )$ is the classical perimeter of $E_\delta$ and by the fact on page 125 in [10]. Therefore, we know that $P_{\mathcal {\widetilde{H}}_\sigma }(E_\delta ,\Omega )\rightarrow 0$ when $\delta \rightarrow 0$. Hence, $|\nabla _{\mathcal {\widetilde{H}}_\sigma }f_\delta |(\Omega )\rightarrow P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega )$ when $\delta \rightarrow 0.$

We also have

$$\begin{aligned} \parallel f_\delta \parallel _{L^1(\Omega )}= & {} \frac{1}{\delta }(\int _0^\delta \mid \{ x \in {\Omega }:{\mathrm {dis}}{{\mathrm {t}}_{{{\mathbb {R}}^d}}}(x,M) < s\} \mid ds). \end{aligned}$$

Then, we conclude that

$$\begin{aligned} \parallel f_\delta \parallel _{L^1(\Omega )}+ |\nabla _{\mathcal {\widetilde{H}}_\sigma }f_\delta |(\Omega )\rightarrow |M|+P_{\mathcal {\widetilde{H}}_\sigma }(M,\Omega ) \end{aligned}$$

when $\delta \rightarrow 0$. Hence, we show that (9) implies (10).

Conversely, suppose (10) is true for any connected compact set M in $\Omega$ with smooth boundary. By the monotonicity of $\widetilde{\mathrm {cap}}(\cdot , \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))$, we conclude that $t\rightarrow \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))$ is a decreasing function on $[0,\infty )$, and we get

$$\begin{aligned}&t^{\frac{1}{d-1}}( \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))^{\frac{d}{d-1}}\\&\quad = \Big [t \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\Big ]^{\frac{1}{d-1}} \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\\&\quad \le \left( \int ^t_0 \widetilde{\mathrm {cap}}(\Omega _r(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))dr\right) ^{\frac{1}{d-1}} \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))\\&\quad = \left( \frac{d-1}{d}\right) \frac{d}{dt}\left( \int ^t_0 \widetilde{\mathrm {cap}}(\Omega _r(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))dr\right) ^{{\frac{d}{d-1}}}. \end{aligned}$$

Via (10) and the estimate, we have

$$\begin{aligned}&\int ^\infty _0 ( \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\quad = \frac{d}{d-1}\int ^\infty _0 ( \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )))^{\frac{d}{d-1}}t^{\frac{1}{d-1}}dt\\&\quad \le \int ^\infty _0 \left[ \frac{d}{dt}\left( \int ^t_0 \widetilde{\mathrm {cap}}(\Omega _r(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega ))dr\right) ^{\frac{1}{d-1}}\right] dt\\&\quad =\left( \int ^\infty _0 \widetilde{\mathrm {cap}}(\Omega _t(f), \mathcal{BV}\mathcal{}_{\mathcal {\widetilde{H}}_\sigma }(\Omega )) dt\right) ^{\frac{d}{d-1}}\\&\quad \le \left( \int ^\infty _0 (|\Omega _t(f)|+P_{\mathcal {\widetilde{H}}_\sigma }(\Omega _t(f),\Omega )) dt\right) ^{\frac{d}{d-1}}\\&\quad \approx \left( \int _{\Omega }|f(x)|dx+|\nabla _{\mathcal {\widetilde{H}}_\sigma }f|(\Omega )\right) ^{\frac{d}{d-1}}, \end{aligned}$$

where we used the coarea formula for ${\mathcal {\widetilde{H}} _\sigma }$-BV function in the last step.

Finally, since (9) is equivalent to (10), it suffices to check that (9) is valid for any connected compact set M in $\Omega$ with smooth boundary, while Theorem 3.3 implies that (9) is valid. $\square$