BV Capacity and Sobolev Capacity for the Laguerre Operator

Abstract

This paper considers some topics of geometric measure theory related to the Laguerre operator. Firstly, this paper is devoted to introducing and investigating the so-called Laguerre bounded variation capacity, thereby discovering some Poincaré type inequalities and BV-isocapacity inequalities. Secondly, we define the Laguerre p-capacity and discuss properties of capacity for the space $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$. Moreover, we study the relation between the $\mathcal {L}^{\alpha }$-BV capacity and the Laguerre 1-capacity. Finally, we prove the Laguerre p-capacitary-strong-type inequality and the trace inequality for $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$.

1 Introduction

In the last few years, there has been an increasing interest in the function of bounded variation, simply the BV-function in various function spaces. On the Euclidean space, the BV-function spaces are by now a classical setting where several problems, mainly (but not exclusively) of variational nature, find their natural framework. For instance, the reflexivity or the weak compactness property of the function space $W^{1,p}(\mathbb R^{d})$ for $p>1$ usually performs an important role when dealing with minimization problems. However, the importance of extending the classical notion of variation has been pointed out in several occasions by Giorgi [5]. Recently, Huang et al. [14] investigate the space of functions with $\alpha$-Hermite bounded variation and its functional capacity and geometrical perimeter. Alonso-Ruiz et al. [1] introduce the class of bounded variation functions in a general framework of strictly local Dirichlet spaces with doubling measure, and prove the Sobolev inequality under the Bakry-$\acute{\text {E}}$mery curvature type condition. For fruitful explorations on this topic, we refer the reader to [4, 11, 13, 19, 26, 27] and the references therein.

The modern theory of capacity was introduced to mathematics by Wiener around 1923–1925 and nowadays is widely used in studying various problems arising from partial differential equations, potential theory, geometric analysis and mathematical physics (cf. [3, 16, 20, 24]). For the information of the classical BV-capacity in $\mathbb R^{d}$, we refer to [35]. In 2010, Hakkarainen and Kinnunen [9] studied basic properties of the BV-capacity and the Sobolev capacity in a complete metric space equipped with a doubling measure and supporting a weak Poincaré inequality. They further investigated the relation between the variational Sobolev 1-capacity and versions of the variational BV-capacity in a complete metric space in [10]. In [33], Xiao introduced the BV-type capacity on Gaussian spaces $\mathbb G^{d}$, and as an application, the Gaussian BV-capacity was used to investigate the trace theory of Gaussian BV-space. On the generalized Grushin plane, Liu [21] obtained some sharp trace and isocapacity inequalities via the BV-capacity. We refer the reader to [19, 23, 29, 31] for more information on this topic.

In the theory of calculus of variations, partial differential equations and potential theory, many interesting features of Sobolev functions are investigated in terms of capacity, see for instance the monographs by Maz’ya [25], Evans and Gariepy [6], and Heinonen et al. [12]. In 1996, Kinnunen and Martio investigated the Sobolev p-capacity in the metric measure space, and they developed a set of capacity theory using Sobolev functions on metric spaces with Borel regular outer measure. By adding the regularity assumption to the measure, they proved the relationship between the Sobolev 1-capacity and the Hausdorff measure, see [17]. In [18], Kinnunen and Martio proved that the Sobolev p-capacity is a Choquet capacity for $1<p<\infty$. Up to now, there are still some problems to be studied about the Sobolev p-capacity in metric spaces. Recently, Liu et al. [23] established the theory of the Gaussian Sobolev p-capacity in Gaussian Sobolev p-spaces, and obtained the Gaussian p-capacitary-strong-type inequality and the trace inequality for $W^{1,p}(\mathbb {G}^d)$. One aim of this paper is to generalize the Sobolev p-capacity to the Laguerre case, and, more importantly, show that this generalization make sense. Also, another aim of this paper is intended to discuss several basic questions of geometric measure theory associated with the Laguerre operator in the Laguerre BV-space.

At first, we will present a very short introduction to the Laguerre operator. Given a multiindex $\alpha =(\alpha _1,\ldots ,\alpha _d)$ with $\alpha \in (-1,\infty )^d$, the Laguerre differential operator is defined as:

$$\begin{aligned} \mathcal {L}^{\alpha }=-\sum _{i=1}^{d}\Big [x_i\frac{\partial ^2}{\partial {x_i}^2}+(\alpha _i+1-x_i)\frac{\partial }{\partial {x_i}}\Big ]. \end{aligned}$$

Consider the probabilistic gamma measure $\mu _{\alpha }$ in $\mathbb {R}_{+}^{d}=(0,\infty )^{d}$ given by

$$\begin{aligned} d\mu _{\alpha }(x)=\prod _{i=1}^{d}\frac{{x_i}^{\alpha _i}e^{-x_i}}{\Gamma (\alpha _i+1)}dx:=\omega (x)dx. \end{aligned}$$

(1)

It is well-known that $\mathcal {L}^{\alpha }$ is positive and symmetric in $L^2(\mathbb {R}_{+}^{d},d\mu _{\alpha })$. Moreover, $\mathcal {L}^{\alpha }$ has a closure which is selfadjoint in $L^2(\mathbb {R}_{+}^{d},d\mu _{\alpha })$ and will also be denoted by $\mathcal {L}^{\alpha }$. For $i=1,2,\ldots , d$, we define the i-th partial derivative associated with $\mathcal {L}^{\alpha }$ by

$$\begin{aligned} \delta _{i}=\sqrt{x_i}\frac{\partial }{\partial {x_i}}, \end{aligned}$$

see [7] or [8]. One of the motivations of such definition is that

$$\begin{aligned} \mathcal {L}^{\alpha }=\sum _{i=1}^{d}\delta _{i}^{*}\delta _{i}, \end{aligned}$$

where

$$\begin{aligned} \delta _{i}^{*}=-\sqrt{x_i}\Big (\partial {x_i}+\frac{\alpha _i+\frac{1}{2}-x_i}{x_i}\Big ) \end{aligned}$$

is the formal adjoint of $\delta _i$ in $L^2(\mathbb {R}_{+}^{d},\textrm{d}\mu _{\alpha })$. Throughout this paper, we always assume that $\Omega \subseteq \mathbb R_{+}^{d}$ be an open set. For $u \in C^1(\mathbb R_{+}^d)$ and $\varphi =(\varphi _1, \varphi _2, \ldots , \varphi _d) \in C^1(\mathbb R_{+}^d,\mathbb {R}^{d})$, we introduce the ${\mathcal L}^{\alpha }$-gradient operator and the ${\mathcal L}^{\alpha }$-divergence operator associated to $\mathcal {L}^{\alpha }$:

$$\begin{aligned} \left\{ \begin{array}{ll} \nabla _{\mathcal L^{\alpha }}u:=(\delta _{1}u, \delta _{2}u,\ldots , \delta _{d}u),\\ \textrm{div}_{\mathcal {L}^{\alpha }}\varphi :=\delta ^{*}_{1}{\varphi _1} + \delta ^{*}_{2}{\varphi _2}+\cdots +\delta ^{*}_{d}{\varphi _d}, \end{array} \right. \end{aligned}$$

which also give $\mathcal {L}^{\alpha }u=\mathrm {div_{\mathcal {L}^{\alpha }}}(\nabla _{\mathcal {L}^{\alpha }}u).$

Naturally, we use ${\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega )$ to represent the class of all functions with the Laguerre bounded variation ($\mathcal {L}^{\alpha }$-BV in short) on $\Omega$, as a continuation of [14], the goal of this paper is to consider some related topics for the Laguerre setting, and contents of this paper are given as follows.

In Sect. 2.1, we introduce the $\mathcal {L}^{\alpha }$-BV capacity denoted by $\textrm{cap}(E,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))$ for a set $E\subseteq \Omega$. In Sect. 2.2, we investigate the measure-theoretic nature of $\textrm{cap}(\cdot , \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))$. Theorem 2.2 indicates that $\textrm{cap}(\cdot ,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))$ is not only an outer measure (obeying (i), (ii) & (iv)), but also a Choquet capacity (satisfying (i), (ii), (v) & (vi)). Section 2.3 is devoted to the Poincaré type inequality and the $\mathcal {L}^{\alpha }$-BV isocapacity inequality in $\Omega$. In Theorem 2.3, we obtain some equivalent conditions for the p-Poincaré inequality associated with $\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )$ functions arising from the end-point Laguerre Sobolev space $W^{1,1}_{\mathcal {L}^{\alpha }}(\Omega )$. Furthermore, we derive an imbedding result for the Laguerre operator $\mathcal {L}^{\alpha }$. Let $\Omega _1$ be an open set defined in (3) from [22], that is,

$$\begin{aligned} \Omega _{1}=\Omega \setminus \{x\in \mathbb {R}_{+}^d:\exists i\in 1,\ldots ,d\ \textrm{such}\ \textrm{that}\ \sqrt{x_i}<1\},\end{aligned}$$

(2)

and let

$$\begin{aligned} \mathfrak C(f):=\Big \{\int ^{\infty }_0 \Big [\textrm{cap}\big (\{x\in \Omega _1: |f(x)|\ge t\}, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _1)\big )\Big ]^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\Big \}^{\frac{d-1}{d}}. \end{aligned}$$

In Theorem 2.4, we establish the following equivalent relation:

The analytic inequality

$$\begin{aligned} \Vert f\Vert _{L^{\frac{d}{d-1}}(\Omega _1,d\mu _{\alpha })}\lesssim \mathfrak C(f) \end{aligned}$$

holds for all compactly supported $L^{\frac{d-1}{d}}(\Omega _1,d\mu _{\alpha })$-functions f if and only if

$$\begin{aligned} \mu _{\alpha }(M)^{\frac{d-1}{d}}\lesssim \textrm{cap}(M,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _1)) \end{aligned}$$

holds for all compact sets M in $\Omega _1$.

In Sect. 3, we introduce the Laguerre p-capacity and investigate related topics. In Sect. 3.1, we define the Sobolev 1-capacity $\textrm{Cap}_1^{\mathcal {L}^{\alpha }}(\cdot )$ in the Laguerre case and compare it with the $\mathcal {L}^{\alpha }$-BV capacity $\textrm{cap}(\cdot ,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))$ introduced previously (see Proposition 3.1). More generally, we study the Laguerre p-capacity for $1<p<\infty$ and investigate the measure-theoretic nature of $\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\cdot ,\mathbb R_{+}^d)$ in Propositions 3.2 and 3.4. Section 3.2 is devoted to a brief discussion of the alternative of the Laguerre p-capacity for $1\le p<\infty$. As applications, Sect. 3.3 presents the Laguerre p-capacitary-strong-type inequality as follows:

Let $1\le p<\infty$ and $f\in C^0(\mathbb R_{+}^d)\bigcap W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$. For any $t\in (0,\infty )$ set

$$\begin{aligned} E_t(f):=\{x\in \mathbb R_{+}^d:\ |f(x)|>t\}. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t(f),\mathbb R_{+}^d)dt^p\lesssim \Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p, \end{aligned}$$

see Lemma 3.4. Section 3.4 addresses the trace inequality for $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ under $1\le p\le q<\infty$. More explicitly, in Theorem 3.2 we obtain the following two equivalent relations:

• (i) There exists a positive constant $C_1$ such that for all compact sets $K\subseteq \mathbb R_{+}^d$,

  $$\begin{aligned} \mu (K)\le C_1(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d))^{\frac{q}{p}}. \end{aligned}$$

• (ii) There exists a positive constant $C_2$ such that

  $$\begin{aligned} \bigg (\int _{\mathbb R_{+}^d}|f|^q d\mu \bigg )^{\frac{1}{q}}\le C_2\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\quad \forall \ f\in C^0(\mathbb R_{+}^d)\cap W^{1,p}_{\mathcal {L}^{\alpha }} (\mathbb R_{+}^d). \end{aligned}$$

  Moreover, $C_1\approx C_2^q$ with the implicit constants depending only on p and q.

Finally, as the other half of Theorem 3.2, in Sect. 3.5 we derive the following two equivalent relations (see Theorem 3.3):

• (i) The function

  $$\begin{aligned} (0,\infty )\ni t\mapsto h_{\mu ,p}(t):=\inf \{\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d):\ \mathbb R_{+}^d\supset K\ \textrm{is}\ \textrm{compact}\ \textrm{with}\ \mu (K)\ge t\} \end{aligned}$$

  satisfies

  $$\begin{aligned} \Vert h_{\mu ,p}\Vert :=\bigg (\int _{0}^{\infty }\frac{ds^{\frac{p}{p-q}}}{(h_{\mu ,p}(s))^{ \frac{q}{p-q}}}\bigg )^{\frac{p-q}{p}}<\infty . \end{aligned}$$

• (ii) There exists a positive constant C such that

  $$\begin{aligned} \bigg (\int _{\mathbb R_{+}^d}|f|^q d\mu \bigg )^{\frac{1}{q}}\le C\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)} \quad \forall \ f\in C^0(\mathbb R_{+}^d)\cap W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

  Moreover,

  $$\begin{aligned} \Vert h_{\mu ,p}\Vert \approx C^q \end{aligned}$$

  whose implicit constants depend only on p and q.

Throughout this article, we will use C to denote positive constants, which are independent of 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 $k \in \mathbb {N}_{0} \cup \{+\infty \}$ and $\Omega \subseteq \mathbb R_{+}^d$, $C^{k}(\Omega )$ denotes the space of $C^{k}$-regular functions in $\Omega$; $C^k(\Omega ,\mathbb {R}^{d})$ denotes the space of the vector-valued $C^{k}$-regular functions in $\Omega$ and $C^{0}(\Omega )$ denotes the space of continuous functions in $\Omega$.

2 $\mathcal {L}^{\alpha }$-BV Capacity

2.1 Fundamentals of $\mathcal {L}^{\alpha }$-BV Capacity

In this section we describe the definition and fundamental characteristics of the Laguerre bounded variation, which are established by referring to the following articles on the BV capacity (see [33, 35]). At first, we recall the definition of the $\mathcal {L}^{\alpha }$-BV space in [22], i.e. the class of all functions with the Laguerre bounded variation. The Laguerre variation ($\mathcal {L}^{\alpha }$-variation in short) of $f \in {L^1}(\Omega ,d\mu _{\alpha })$ is defined by

$$\begin{aligned} |\nabla _{\mathcal {L}^{\alpha }}f|(\Omega )= \mathop {\sup }\limits _{\varphi \in \mathcal F(\Omega )}\left\{ \int _\Omega f(x)\mathrm {{div}_{\mathcal {L}^{\alpha }}}\varphi (x)d\mu _\alpha (x)\right\} , \end{aligned}$$

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

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

satisfying

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

An function $f \in {L^1}(\Omega ,d\mu _{\alpha })$ is said to have the $\mathcal {L}^{\alpha }$-bounded variation on $\Omega$ if

$$\begin{aligned} |\nabla _{\mathcal {L}^{\alpha }}f|(\Omega )<\infty , \end{aligned}$$

and the collection of all such functions is denoted by ${\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega )$, which is a Banach spaces with the norm

$$\begin{aligned}\Vert f\Vert _{{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega )} = \Vert f\Vert _{L^{1}(\Omega ,d\mu _{\alpha })}+|\nabla _{\mathcal {L}^{\alpha }}f| (\Omega ).\end{aligned}$$

Definition 2.1

For a set $E\subseteq \Omega$, let $\mathcal {A}(E, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))$ be the class of admissible functions on $\Omega$, i.e., functions $f\in \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )$ satisfying $0\le f\le 1$ and $f=1$ in a neighborhood of E (an open set containing E). The $\mathcal {L}^{\alpha }$-BV capacity of E is defined by

$$\begin{aligned} \textrm{cap}(E,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )):=\inf _{f\in \mathcal {A}(E, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))}\Big \{\Vert f \Vert _{L^1(\Omega ,d\mu _{\alpha })}+ |\nabla _{\mathcal {L}^{\alpha }} f|(\Omega )\Big \}. \end{aligned}$$

Via the coarea formula for $\mathcal {L}^{\alpha }$-BV functions in [22, Theorem 2.11] and the layer-cake formula, we obtain the following basic assertions.

Theorem 2.1

A geometric description of the $\mathcal {L}^{\alpha }$-BV capacity of a set in $\Omega$ is given as follows:

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

  $$\begin{aligned} \textrm{cap}(K,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\approx \inf _A \Big \{\mu _{\alpha }(A)+P_{\mathcal {L}^{\alpha }}(A)\Big \}, \end{aligned}$$

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

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

  $$\begin{aligned} \textrm{cap}(K,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\approx \inf _A \Big \{\mu _{\alpha }(A)+P_{\mathcal {L}^{\alpha }}(A)\Big \}, \end{aligned}$$

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

2.2 Measure-Theoretic Nature of $\mathcal {L}^{\alpha }$-BV Capacity

Similarly to [22, 23], we can obtain the following theorem to give the measure-theoretic properties of $\mathcal {L}^{\alpha }$-BV capacity. We only give the proof of converse inequality of (3) under (4).

Theorem 2.2

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

• (i)
  $$\begin{aligned} \textrm{cap}(\emptyset , \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=0. \end{aligned}$$

• (ii) If $A\subseteq B$, then

  $$\begin{aligned} \textrm{cap}(A, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\le \textrm{cap}(B, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )). \end{aligned}$$
• (iii)
  $$\begin{aligned}{} & {} \textrm{cap}(A\cup B, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))+\textrm{cap}(A \cap B, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\nonumber \\{} & {} \quad \le \textrm{cap}(A,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )) +\textrm{cap}(B,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )), \end{aligned}$$

  (3)

  whose equality can be assured by the subadditivity above and the constraint below

  $$\begin{aligned} \textrm{cap}(A\backslash (A \cap B), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\cdot \textrm{cap}(B\backslash (B \cap A), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=0. \end{aligned}$$

  (4)

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

  $$\begin{aligned} \textrm{cap}(\cup ^\infty _{k=1}A_k, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\le \sum ^\infty _{k=1}\textrm{cap}(A_k,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )). \end{aligned}$$

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

  $$\begin{aligned} \lim _{k\rightarrow \infty }\textrm{cap}(A_k, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=\textrm{cap}(\cup ^\infty _{k=1}A_k, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )). \end{aligned}$$

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

  $$\begin{aligned} \lim _{k\rightarrow \infty }\textrm{cap}(A_k,\mathcal {BV_{\mathcal {L}^{\alpha }}}(\Omega ))=\textrm{cap}(\cap ^\infty _{k=1}A_k,\mathcal {BV_{\mathcal {L}^{\alpha }}}(\Omega )). \end{aligned}$$

Proof

(iii) We adapt the method of the proof in Xiao and Zhang’s paper [32, Section 1.1 (iii)] to prove the equality condition of (3). Since (3) is valid, we only need to show that its converse inequality holds true under the above condition (4). Obviously, the condition

$$\begin{aligned} \textrm{cap}(A\backslash (A \cap B), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\cdot \textrm{cap}(B\backslash (B \cap A), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=0 \end{aligned}$$

implies that $\textrm{cap}(A\backslash (A \cap B), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=0$ or $\textrm{cap}(B\backslash (B \cap A), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=0$. Suppose

$$\begin{aligned} \textrm{cap}(A\backslash (A \cap B), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=0. \end{aligned}$$

Via (3), we have

$$\begin{aligned} \textrm{cap}(A,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega ))&=\textrm{cap}((A \setminus (A\cap B))\cup (A\cap B),{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega )) \nonumber \\&\le \textrm{cap}(A\setminus (A\cap B),{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega ))+\textrm{cap}(A\cap B,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega ))\nonumber \\&=\textrm{cap}(A\cap B,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega )). \end{aligned}$$

(5)

Using (ii), we obtain

$$\begin{aligned} \begin{aligned} \textrm{cap}(B,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega ))&\le \textrm{cap}(A\cup B,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega )). \end{aligned} \end{aligned}$$

(6)

Combining (5) with (6) deduces that

$$\begin{aligned}{} & {} \textrm{cap}(A,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega ))+\textrm{cap}(B,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega ))\\{} & {} \qquad \le \textrm{cap}(A\cap B,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega ))+\textrm{cap}(A\cup B,{\mathcal B}{\mathcal V}_{\mathcal L^{\alpha }}(\Omega )), \end{aligned}$$

which derives the desired result. Another case can be similarly proved, we omit the details. The assertion (iii) is proved. $\square$

Using Theorem 2.2, we have the following capacitability.

Corollary 2.1

• (i) If $E\subseteq \Omega$, then

  $$\begin{aligned} \textrm{cap}(E,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))=\inf _{\textrm{open}\,O\supseteq E}\Big \{\textrm{cap}(O,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\Big \}. \end{aligned}$$

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

  $$\begin{aligned} \textrm{cap}(E,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )) =\sup _{\textrm{compact}\, K\subseteq E}\Big \{\textrm{cap}(K,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\Big \}. \end{aligned}$$

2.3 Poincaré Type Inequality and $\mathcal {L}^{\alpha }$-BV Isocapacity Inequality

Similarly to Xiao’s result in [33, Theorem 10], in this section, we investigate the relation between the nonnegative Radon measure $\mu$ and the Poincaré type inequality:

$$\begin{aligned} \sup _{0\ne f\in \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )}\frac{\Vert f\Vert _{L^q_\mu (\Omega )}}{\Vert f\Vert _{\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )}}<\infty . \end{aligned}$$

(7)

As shown in the next theorem for $\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )$, the validity of (7) is closely linked with the domination of $\mu (B)$ via $P_{\mathcal {L}^{\alpha }}(B)$ or $\textrm{cap}(B, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))$ for any Borel set $B\subseteq \Omega$.

Theorem 2.3

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

• (i)
  $$\begin{aligned} \Big (\int _{\Omega }|f|^p d\mu \Big )^{\frac{1}{p}}\lesssim \Vert f\Vert _{L^1(\Omega ,d\mu _{\alpha })}+|\nabla _{\mathcal {L}^{\alpha }} f |(\Omega ) \end{aligned}$$

  for all $f\in \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )$ which are defined $\mu$-a.e..

• (ii)
  $$\begin{aligned} \mu (B)^{\frac{1}{p}}\lesssim \mu _{\alpha }(B)+ P_{\mathcal {L}^{\alpha }}(B) \end{aligned}$$

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

• (iii)
  $$\begin{aligned} \mu (B)^{\frac{1}{p}}\lesssim \textrm{cap}(B,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )) \end{aligned}$$

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

Proof

(i)$\Rightarrow$(ii). By taking $f=1_B$ and the definition of $P_{\mathcal {L}^{\alpha }}(\cdot )$, we can deduce that (ii) is valid.

(ii)$\Rightarrow$(iii). For all bounded open sets $O\subseteq \Omega$ with smooth boundary containing K which is a compact subset, using the regularity of $\mu$ and the assumption we obtain

$$\begin{aligned} \mu (K)^{\frac{1}{p}}&=\Big (\inf \{\mu (O):\ O\supseteq K\} \Big )^{\frac{1}{p}}\\&\lesssim \Big (\inf \{(\mu _{\alpha }(O)+P_{\mathcal {L}^{\alpha }}(O))^p:\ O\supseteq K\}\Big )^{\frac{1}{p}}\\&\lesssim \inf \{\mu _{\alpha }(O)+P_{\mathcal {L}^{\alpha }}(O):\ O\supseteq K\}. \end{aligned}$$

The above arguments and Theorem 2.1 imply that

$$\begin{aligned} \mu (K)^{\frac{1}{p}}\lesssim \textrm{cap}(K, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )). \end{aligned}$$

Then following from (ii) of Corollary 2.1 and the inner regularity of $\mu$, we conclude that (iii) is true.

(iii)$\Rightarrow$(i). Suppose (iii) holds. If $f\in C^\infty _c(\Omega )\cap \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )$, we first claim that f is finite almost everywhere with respect to the measure $\mu$. Indeed, for $t>0$, let $E_t=\{x\in \Omega : |f(x)|>t\}$. By the coarea formula (Theorem 2.4 in [22]), we know $E_t$ has finite $\mathcal {L}^{\alpha }$-perimeter for a.e. t and

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

From this, we conclude that $\lim \inf _{t\rightarrow \infty } P_{\mathcal {L}^{\alpha }}(E_t)=0.$ Via Theorem 2.1, we have

$$\begin{aligned} \textrm{cap}(\{x\in \Omega :\ |f(x)|=\infty \}, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\lesssim \mathop {\lim \inf }_{t\rightarrow \infty } \{\mu _{\alpha }(E_t)+P_{\mathcal {L}^{\alpha }}(E_t)\}=0. \end{aligned}$$

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

Using the layer-cake formula, Theorem 2.1 and the coarea formula in [22, Theorem 2.11], we conclude that

$$\begin{aligned}&\Big (\int _{\Omega } |f|^pd\mu \Big )^{1/p}\le \Big (\int ^\infty _0 \mu (\{x\in \Omega :\ |f(x)|>t\})dt^p\Big )^{\frac{1}{p}}\\&\quad =\int ^\infty _0 \frac{d}{dt}\Big (\int ^t_0 \mu (\{x\in \Omega :\ |f(x)|>s\})ds^p\Big )^{\frac{1}{p}}dt\\&\quad =\int ^\infty _0 \Big (\int ^t_0 \mu (\{x\in \Omega :\ |f(x)|>s\})ds^p\Big )^{\frac{1}{p}-1}\mu (\{x\in \Omega :\ |f(x)|>t\})t^{p-1}dt\\&\quad \le \int ^\infty _0\Big (\mu (\{x\in \Omega :\ |f(x)|>t\}) \Big )^{\frac{1}{p}}dt\\&\quad \lesssim \int ^\infty _0 \textrm{cap}(\{x\in \Omega :\ |f(x)|>t\}, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))dt\\&\quad \lesssim \int _{0}^\infty \Big [\mu _{\alpha }(\{x\in \Omega :\ |f(x)|>s\})+P_{\mathcal {L}^{\alpha }}(\{x\in \Omega :\ |f(x)|>s\})\Big ]ds\\&\quad \lesssim \Vert f\Vert _{L^1(\Omega ,d\mu _{\alpha })}+| \nabla _{\mathcal {L}^{\alpha }} |f||(\Omega ) \\&\quad \lesssim \Vert f\Vert _{L^1(\Omega ,d\mu _{\alpha })}+| \nabla _{\mathcal {L}^{\alpha }} f|(\Omega ) . \end{aligned}$$

Hence, we use Theorem 2.4 in [22] and the above proofs to deduce that (i) is true. $\square$

If $\mu$ in Theorem 2.3 is taken as the measure given by (1), we can obtain the isocapacity inequalities associated with $\mathcal {L}^{\alpha }$-BV functions.

Theorem 2.4

Let $\Omega _1$ be an open set defined in (2). The analytic inequality

$$\begin{aligned} \Vert f\Vert _{L^{\frac{d}{d-1}}(\Omega _{1},d\mu _{\alpha })} \lesssim \bigg (\int ^{\infty }_0 \Big (\textrm{cap}(\{x\in \Omega _{1}: |f(x)|\ge t\}, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _{1}))\Big )^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\bigg )^{\frac{d-1}{d}}\nonumber \\ \end{aligned}$$

(8)

holds for all $f\in L^{\frac{d}{d-1}}(\Omega _1,d\mu _{\alpha })$ with compact support if and only if the geometric inequality

$$\begin{aligned} \mu _{\alpha }(M)^{\frac{d-1}{d}}\lesssim \textrm{cap}(M, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _1)) \end{aligned}$$

(9)

holds for all compact sets M in $\Omega _{1}$. Moreover, the inequalities (8) and (9) are true.

Proof

We adopt the method in [30] to give the proof. In what follows, we always adopt two short notations:

$$\begin{aligned} \Omega _t(f)=\{x\in \Omega _{1}: |f(x)|\ge t\} \end{aligned}$$

and

$$\begin{aligned} \partial \Omega _t(f)=\{x\in \Omega _{1}: |f(x)|=t\} \end{aligned}$$

for a function f defined on $\Omega _{1}$ and a number $t>0$.

Given a compact set $M\subseteq \Omega _{1}$, let $f=1_M$. Then $\Vert f\Vert _{{L^{\frac{d}{d-1}}}(\Omega _1,d\mu _{\alpha })}=\mu _{\alpha }(M)^{1-\frac{1}{d}}$ and

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

Hence,

$$\begin{aligned}&\int ^\infty _0 \Big [\textrm{cap}(\Omega _t(f), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _1))\Big ]^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\quad =\int ^1_0 \Big [\textrm{cap}(\Omega _t(f), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _{1}))\Big ]^{\frac{d}{d-1}}dt^{\frac{d}{d-1} }\\&\qquad \quad +\int ^\infty _1\Big [ \textrm{cap}(\Omega _t(f), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _1))\Big ]^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}\\&\quad =\Big [\textrm{cap}(M, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _1))\Big ]^{\frac{d}{d-1}}, \end{aligned}$$

which derives that (8) implies (9).

Conversely, we show that (9) implies (8). Suppose (9) holds for all compact sets in $\Omega _1$. For $t>0$ and f, an $L^{\frac{d}{d-1}}(\Omega _1,d\mu _{\alpha })$ integrable function with compact support in $\Omega _1$, we use the inequality (9) to get

$$\begin{aligned} \Vert f\Vert ^{\frac{d}{d-1} }_{L^{\frac{d}{d-1}}(\Omega _1,d\mu _{\alpha }) }=\int ^{\infty }_0|\Omega _t(f)|dt^{\frac{d}{d-1}}\lesssim \int ^{\infty }_0\Big [\textrm{cap}(\Omega _t(f), \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega _1))\Big ]^{\frac{d}{d-1}}dt^{\frac{d}{d-1}}. \end{aligned}$$

Since (8) is equivalent to (9), it suffices to prove that (9) is valid. In fact, for all bounded sets B with smooth boundaries containing M, using (ii) of Theorem 2.12 in [22], we have

$$\begin{aligned} \mu _{\alpha }(M)^{1-\frac{1}{d}}\le \mu _{\alpha }(B)^{1-\frac{1}{d}}\lesssim P_{\mathcal {L}^{\alpha }}(B, \Omega _1)\le \mu _{\alpha }(B)+P_{\mathcal {L}^{\alpha }}(B,\Omega _1). \end{aligned}$$

Theorem 2.1 implies that (9) holds true. $\square$

3 Laguerre p-Capacity

3.1 Basic Properties of Laguerre p-Capacity

In [14], the authors introduced the Sobolev type capacity associated with the $\alpha$-Hermite operator $\mathcal {H}_{\alpha }$ and investigated the related topics. Following from [14], we similarly give the definition of the Sobolev 1-capacity associated with the Laguerre operator $\mathcal {L}^{\alpha }$. Before giving the definition of the Laguerre p-capacity, let us review the definition of the Laguerre Sobolev space $W^{1,p} _{\mathcal {L}^{\alpha }}(\Omega )$.

Definition 3.1

Suppose $\Omega$ is an open set in $\mathbb {R}_{+}^{d}$. Let $1 \le p \le \infty$. The Sobolev space $W_{\mathcal {L}^{\alpha }}^{k,p}(\Omega )$ associated with $\mathcal {L}^{\alpha }$ is defined as the set of all functions $f \in {L^p}(\Omega ,d\mu _{\alpha })$ such that

$$\begin{aligned}\delta _{j_1} \ldots \delta _{j_m}f\in {L^p}(\Omega ,d\mu _\alpha ),\ 1 \le {j_1}, \ldots ,{j_m}\le d,\ 1\le m \le k.\end{aligned}$$

The norm of $f\in W_{\mathcal {L}^{\alpha }}^{k,p}(\Omega )$ is given by

$$\begin{aligned}\Vert f\Vert _{W_{\mathcal {L}^{\alpha }}^{k,p}}:=\sum \limits _{1\le {j_1} \ldots {j_m} \le d,\ 1\le m \le k} {\Vert {\delta _{j_1}} \ldots \delta _{j_m}f\Vert }_{L^p(\Omega ,d\mu _{\alpha })} + \Vert f\Vert _{L^p(\Omega ,d\mu _{\alpha })}.\end{aligned}$$

Definition 3.2

Let $E\subseteq \Omega$ and

$$\begin{aligned} \mathcal {A}_1(E)=\Big \{f\in W^{1,1}_{\mathcal {L}^{\alpha }}(\Omega ):\ E\subseteq \textrm{int}\{x\in \Omega :\ f(x)\ge 1\}\Big \}. \end{aligned}$$

The Sobolev 1-capacity of E is defined by

$$\begin{aligned} \textrm{Cap}^{\mathcal {L}^{\alpha }}_1(E,\Omega )=\inf _{f\in \mathcal {A}_1(E)}\Big \{\Vert f\Vert _{W^{1,1}_{\mathcal {L}^{\alpha }}(\Omega )}\Big \}. \end{aligned}$$

Proposition 3.1

For any set $E\subseteq \Omega$, then

$$\begin{aligned} \textrm{cap}(E,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\lesssim \textrm{Cap}^{\mathcal {L}^{\alpha }}_1(E,\Omega ). \end{aligned}$$

Proof

For any $f\in \mathcal {A}_1(E)$, via (i) of Lemma 2.1 in [22], we have

$$\begin{aligned} \Vert f\Vert _{W^{1,1}_{\mathcal {L}^{\alpha }}(\Omega )}&=\int _{\Omega } |\nabla _{\mathcal {L}^{\alpha }}f(x)|d\mu _\alpha (x)+\int _{\Omega } |f(x)|d\mu _\alpha (x)\\&\gtrsim \int ^1_0\mu _{\alpha }(\{x\in \Omega :\ f(x)>t\})+ P_{\mathcal {L}^{\alpha }}(\{x\in \Omega :\ f(x)>t\})dt\\&\gtrsim \textrm{cap}(E, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega )), \end{aligned}$$

where we have used Theorem 2.1 in the last step. Hence, Definition 3.2 implies

$$\begin{aligned} \textrm{cap}(E, \mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))\lesssim \textrm{Cap}^{\mathcal {L}^{\alpha }}_1(E,\Omega ). \end{aligned}$$

$\square$

Remark 3.1

By Theorem 4.3 of [9], the Sobolev 1-capacity is equivalent to the $\textrm{BV}$-capacity in a complete metric space equipped with a doubling measure and supporting a weak Poincaré inequality. Since the Laguerre measure doesn’t satisfy doubling condition and 1-Poincaré inequality, the arguments of Theorem 4.3 in [9] can’t be extended to the Laguerre setting.

Definition 3.3

For $p\in [1,\infty )$ and $E\subseteq \mathbb R_{+}^d$ let

$$\begin{aligned} \mathcal {A}_{p}(E):=\Big \{f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d):\ E\subseteq \textrm{int}\{x\in \mathbb R_{+}^d:\ f(x)\ge 1\}\Big \}. \end{aligned}$$

Define that Laguerre p-capacity of E as:

$$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d):=\inf \Big \{\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p:\ f\in \mathcal {A}_{p}(E) \Big \}. \end{aligned}$$

(10)

Remark 3.2

Similarly to Xiao’s results or Huang, Li and Liu’s paper (cf. [15, 30]), we also introduce the the corresponding relative capacity, which is given as follows: for $p\in [1,\infty )$ and for any compact set $E\subseteq \mathbb R_{+}^d$ the relative Laguerre p-capacity of E is

$$\begin{aligned} \textrm{Cap}^{\mathcal {L}^{\alpha }}_{p,r}(E,\mathbb R_{+}^d):=\inf \Big \{\sum \limits _{1\le {j} \le d} {\Vert {\delta _{j}} f\Vert }_{L^p(\Omega ,d\mu _{\alpha })}:\ f\in C^\infty _0(\mathbb R_{+}^d), f \ge 1_E\Big \}, \end{aligned}$$

where $1_E$ denotes the characteristic function of the set E. As for some properties and related results of the relative Laguerre p-capacity, we investigate them in the future.

Lemma 3.1

Given $p\in [1,\infty )$ and $i=1,2,\ldots ,d$. For a sequence of functions

$$\begin{aligned} {\{f_k\}_{k\in \mathbb {N}}}\subseteq W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d), \end{aligned}$$

let

$$\begin{aligned} g= {\sup _{k\in \mathbb {N}}}f_k\ \textrm{and}\ h= {\sup _{k\in \mathbb {N}}}|\delta _i f_k|. \end{aligned}$$

If both g and h are in $\mathcal{B}\mathcal{V}(\mathbb R_{+}^d)$, then

$$\begin{aligned} |\delta _i g(x)|\le h(x) \end{aligned}$$

holds for almost all $x\in \mathbb R_{+}^d$.

Proof

Since

$$\begin{aligned} g\in \mathcal{B}\mathcal{V}(\mathbb R_{+}^d)\subseteq L^1(\mathbb R_{+}^d,d\mu _{\alpha })\subseteq L^1_{\textrm{loc}}(\mathbb R^d,dx), \end{aligned}$$

therefore, it makes sense to consider $\delta _i g$ in the sense of distribution. According to integration by part, we have

$$\begin{aligned} \int _{\mathbb R_{+}^d}(\delta _{i}g)\Phi d\mu _{\alpha }=-\int _{\mathbb R_{+}^d}(\delta _{i}^{*}\Phi )g d\mu _{\alpha }\ \forall \ \Phi \in C_c^{\infty }(\mathbb R_{+}^d,\mathbb R). \end{aligned}$$

For any $l\in \mathbb {N}$, define

$$\begin{aligned} g_l=\sup _{1\le k\le l}f_k. \end{aligned}$$

Due to

$$\begin{aligned} f_k\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d), \end{aligned}$$

one has

$$\begin{aligned} g_l\in L^p(\mathbb R_{+}^d,d\mu _{\alpha })\subseteq L^1_{\textrm{loc}}(\mathbb R^d,dx). \end{aligned}$$

An application of [6, p. 148, Lemma  2(iii)] yields

$$\begin{aligned} |\delta _{i}g_l|=\Big |\sqrt{x_i}\frac{\partial }{\partial x_i}\big (\sup _{1\le k\le l}f_k\big )\Big |\le \sqrt{x_i}\sup _{1\le k\le l} \Big |\frac{\partial f_k}{\partial x_i}\Big |=\sup _{1\le k\le l}\Big |\sqrt{x_i}\frac{\partial f_k}{\partial x_i}\Big |= \sup _{1\le k\le l}|\delta _{i}f_k|, \end{aligned}$$

a.e. on $\mathbb R_{+}^d$. Of course, this follows by induction and by verifying the case $l=2$:

$$\begin{aligned} \delta _{i}\max \{f_1,f_2\}(x)&=2^{-1}\big (\delta _{i}f_1(x)+\delta _{i}f_2(x)+\delta _{i}|f_1(x)-f_2(x)|\big )\\&={\left\{ \begin{array}{ll} \delta _{i}f_1(x), \ \ \textrm{for}\ \textrm{almost}\ \textrm{all}\ x\in \mathbb R_{+}^d\ \textrm{with}\ f_1(x)\ge f_2(x),\\ \delta _{i}f_2(x),\ \ \textrm{for}\ \textrm{almost}\ \textrm{all}\ x\in \mathbb R_{+}^d\ \textrm{with}\ f_1(x)\le f_2(x). \end{array}\right. } \end{aligned}$$

Using integration by part and the Lebesgue dominated convergence theorem, for any

$$\begin{aligned} \Phi \in C_{c}^{\infty }(\mathbb R_{+}^d, \mathbb R), \end{aligned}$$

we have

$$\begin{aligned} \Big |\int _{\mathbb R_{+}^d}(\delta _{i}^{*}\Phi )g d\mu _{\alpha }\Big |&=\Big |\lim _{l\rightarrow \infty }\int _{\mathbb R_{+}^d}(\delta _{i}^{*}\Phi )g_l d\mu _{\alpha }\Big |\\ {}&=\Big |\lim _{l\rightarrow \infty }\int _{\mathbb R_{+}^d}(\delta _{i}g_l)\Phi d\mu _{\alpha }\Big |\\ {}&\le \int _{\mathbb R_{+}^d}|\Phi |h d\mu _{\alpha }. \end{aligned}$$

Combining the above result with [28, p. 58, Theorem 3.3], the linear functional L defined by

$$\begin{aligned} L(\Phi ):=\int _{\mathbb R_{+}^d}(\delta _{i}^{*}\Phi )g d\mu _{\alpha } \ \forall \ \Phi \in C_{c}^{\infty }(\mathbb R_{+}^d,\mathbb R) \end{aligned}$$

extends to a linear functional $\bar{L}$ on $C_{c}(\mathbb R_{+}^d,\mathbb R)$ such that

$$\begin{aligned} |\bar{L}(\Phi )|\le \int _{\mathbb R_{+}^d}|\Phi |hd\mu _{\alpha }\ \forall \ \Phi \in C_{c}(\mathbb R_{+}^d,\mathbb R). \end{aligned}$$

Now, Lemma 2.3 in [22] deduces that there exists a Radon measure $\mu _{\mathcal {L}^{\alpha }}$ on $\mathbb R_{+}^d$ such that

$$\begin{aligned} \int _{\mathbb R_{+}^d}(\delta _{i}^{*}\Phi )g d\mu _{\alpha }=\int _{\mathbb R_{+}^d}\Phi \cdot d\mu _{\mathcal {L}^{\alpha }} \ \forall \ \Phi \in C_c^{\infty }(\mathbb R_{+}^d, \mathbb R). \end{aligned}$$

Moreover, the outer regularity of $\mu _{\mathcal {L}^{\alpha }}$ implies that for any Lebesgue measurable set $A\subseteq \mathbb R_{+}^d$,

$$\begin{aligned} \mu _{\mathcal {L}^{\alpha }}(A)&=\inf \{\mu _{\mathcal {L}^{\alpha }}(O):\ O\supset A\}\\&=\inf _{\textrm{open}\, O\supset A}\sup \{\bar{L}(\Phi ):\ \Phi \in C_c (\mathbb R_{+}^d,\mathbb R),\ |\Phi |\le 1,\ \textrm{supp}\ \Phi \subseteq O\}\\&\le \inf _{\textrm{open}\, O\supset A}\int _{O}hd\mu _{\alpha }\\&=\int _{A}h d\mu _{\alpha }. \end{aligned}$$

Therefore, $\mu _{\mathcal {L}^{\alpha }}$ is absolutely continuous with respect to the Lebesgue measure, so that

$$\begin{aligned} d\mu _{\mathcal {L}^{\alpha }}=ud\mu _{\alpha } \end{aligned}$$

for some function u satisfying that

$$\begin{aligned} |u(x)|\le h(x) \end{aligned}$$

for almost all $x\in \mathbb R_{+}^d$. Hence, for all $\Phi \in C_c^{\infty }(\mathbb R_{+}^d, \mathbb R)$, we have

$$\begin{aligned} \int _{\mathbb R_{+}^d}(\delta _{i}^{*}\Phi )gd\mu _{\alpha }=L(\Phi )=\bar{L}(\Phi )=\int _{\mathbb R_{+}^d}\Phi \cdot d\mu _{\mathcal {L}^{\alpha }}=\int _{\mathbb R_{+}^d}\Phi \cdot ud\mu _{\alpha }, \end{aligned}$$

which implies

$$\begin{aligned} \delta _{i}g=u \end{aligned}$$

and

$$\begin{aligned} |\delta _{i}g|=|u|\le h\ \textrm{for}\ \textrm{almost}\ \textrm{all}\ x\in \mathbb R_{+}^d. \end{aligned}$$

This completes the argument. $\square$

Similarly to [23], we can obtain the following proposition giving the measure-theoretic properties of the Laguerre p-capacity. We only give the proof of (iii) and (iv).

Proposition 3.2

Let $p\in [1,\infty )$. The set-function $\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\cdot ,\mathbb R_{+}^d)$ enjoys the following properties:

• (i)
  $$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\emptyset ,\mathbb R_{+}^d)=0\ \textrm{and}\ \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d,\mathbb R_{+}^d)\le 1. \end{aligned}$$

• (ii) If $E_1\subseteq E_2\subseteq \mathbb R_{+}^d$, then

  $$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_1,\mathbb R_{+}^d)\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_2,\mathbb R_{+}^d). \end{aligned}$$

• (iii) For any $\{E_j\}_{j=1}^{\infty }$ of subsets of $\mathbb R_{+}^d$,

  $$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\cup _{j=1}^{\infty } E_j,\mathbb R_{+}^d)\le \sum _{j=1}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_j,\mathbb R_{+}^d). \end{aligned}$$

• (iv) For any $1\le p< q<\infty$ and any $E\subseteq \mathbb R_{+}^d$,

  $$\begin{aligned} 2^{-\frac{d}{p}}(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d))^{\frac{1}{p}}\le 2^{-\frac{d}{q}}(\textrm{Cap}_{q}^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d))^{\frac{1}{q}}. \end{aligned}$$

• (v) For any sequence $\{K_j\}_{j=1}^{\infty }$ of compact subsets of $\mathbb R_{+}^d$ with $K_1\supseteq K_2\supseteq \ldots$,

  $$\begin{aligned} \lim _{j\rightarrow \infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K_j,\mathbb R_{+}^d)=\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\cap _{j=1}^{\infty }K_{j},\mathbb R_{+}^d). \end{aligned}$$

Proof

   

• (iii) We can apply Lemma 3.1 to deduce this part.

• (iv) When $1\le p<q<\infty$, using the Hölder inequality and the elementary inequality

  $$\begin{aligned} \sum _{i=1}^{d}a_i+b\le 2^{d(1-\frac{1}{k})}\bigg (\sum _{i=1}^d a_i^k+b^k\bigg )^{\frac{1}{k}}\quad \forall \ (k,a_i,b)\in [1,\infty )\times (0,\infty )\times (0,\infty ), \end{aligned}$$

  we deduce that if $k=\frac{q}{p}$, then

  $$\begin{aligned} \begin{aligned} 2^{-\frac{d}{p}}\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}&=2^{-\frac{d}{p}} \bigg ( {\sum _{1\le i\le d}}\Vert \delta _{i}f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p +\Vert f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\bigg )^{\frac{1}{p}}\\&\le 2^{-\frac{d}{p}} \bigg ( {\sum _{1\le i\le d}}\Vert \delta _{i}f\Vert _{L^q(\mathbb R_{+}^d,d\mu _{\alpha })}^p +\Vert f\Vert _{L^q(\mathbb R_{+}^d,d\mu _{\alpha })}^p\bigg )^{\frac{1}{p}}\\&\le 2^{-\frac{d}{q}}\Vert f\Vert _{W^{1,q}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}. \end{aligned} \end{aligned}$$

  (11)

  For any set $E\subseteq \mathbb R_{+}^d$, notice that

  $$\begin{aligned} { {\mathcal {A}_q(E)\subseteq \mathcal {A}_p(E)}.} \end{aligned}$$

  So, we use (11) and (10) to obtain

  $$\begin{aligned} 2^{-\frac{d}{p}}(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d))^{\frac{1}{p}}&=\inf _{f\in \mathcal {A}_{p}(E)} 2^{-\frac{d}{p}}\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\\&\le \inf _{f\in \mathcal {A}_{p}(E)}2^{-\frac{d}{q}}\Vert f\Vert _{W^{1,q}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\\&\le \inf _{f\in \mathcal {A}_{q}(E)}2^{-\frac{d}{q}}\Vert f\Vert _{W^{1,q}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\\&=2^{-\frac{d}{q}}(\textrm{Cap}_{q}^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d))^{\frac{1}{q}}. \end{aligned}$$

$\square$

Before giving the proof of Proposition 3.4, we need some preparations. For any two functions f and g, define their Laguerre inner product:

$$\begin{aligned} \langle f,g\rangle _{\mu _{\alpha }}=\int _{\mathbb R_{+}^d}fg d\mu _{\alpha }. \end{aligned}$$

Proposition 3.3

Let $p\in (1,\infty )$. Assume that the sequence $\{f_k\}_{k\in \mathbb {N}}$ satisfies

$$\begin{aligned} {\sup _{k\in \mathbb {N}}\Vert f_k\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}<\infty .} \end{aligned}$$

(12)

Then there exists a subsequence

$$\begin{aligned} {\{f_{k_l}\}_{l\in \mathbb {N}}\subseteq \{f_k\}_{k\in \mathbb {N}}} \end{aligned}$$

and a function

$$\begin{aligned} f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d) \end{aligned}$$

such that for any $i=1,2,\ldots ,d$

$$\begin{aligned} \{(f_{k_l},\delta _i f_{k_l})\}_{l\in \mathbb {N}} \end{aligned}$$

converges to $(f,\delta _i f)$ weakly in

$$\begin{aligned} L^p(\mathbb R_{+}^d)\times L^p(\mathbb R_{+}^d,\mathbb R^d), \end{aligned}$$

this is,

$$\begin{aligned} \lim _{l\rightarrow \infty }\langle f_{k_l},\phi \rangle _{\mu _{\alpha }}=\langle f,\phi \rangle _{\mu _{\alpha }} \quad \phi \in L^{p^{\prime }}(\mathbb R_{+}^d) \end{aligned}$$

(13)

and

$$\begin{aligned} \lim _{l\rightarrow \infty }\langle \delta _{i}f_{k_l},\Phi \rangle _{\mu _{\alpha }}=\langle \delta _{i}f,\Phi \rangle _{\mu _{\alpha }} \quad \Phi \in L^{p^{\prime }}(\mathbb R_{+}^d,\mathbb R^d), \end{aligned}$$

(14)

where $p^{\prime }=\frac{p}{p-1}$. Moreover,

$$\begin{aligned} \lim _{l\rightarrow \infty }\int _{\mathbb R_{+}^d}f_{k_l}d\mu _{\alpha }=\int _{\mathbb R_{+}^d}f d\mu _{\alpha }. \end{aligned}$$

(15)

Proof

The condition (12) implies that we can seek a subsequence $\{f_{k_l}\}_{l\in \mathbb {N}}$ such that

$$\begin{aligned} \{(f_{k_l},\delta _i f_{k_l})\}_{l\in \mathbb {N}} \end{aligned}$$

tends to some (f, F) weakly in

$$\begin{aligned} L^p(\mathbb R_{+}^d)\times L^p(\mathbb R_{+}^d,\mathbb R^d) \end{aligned}$$

for any $i=1,2,\ldots ,d$. This indicates that not only (13) holds, but also (14) holds with $\delta _i f$ there replaced by F.

To get (14) fully, we need to verify

$$\begin{aligned} F=\delta _i f,\ \textrm{for} \ \textrm{any}\ \textrm{given}\ i=1,2,\ldots ,d. \end{aligned}$$

Using Mazur’s Theorem in [34, p. 120, Theorem 2], we obtain a convex combination

$$\begin{aligned} \sum _{l=1}^{m}\lambda _{m,l}(f_{k_l},\delta _{i}f_{k_l}) \end{aligned}$$

converging to (f, F) strongly in

$$\begin{aligned} L^p(\mathbb R_{+}^d)\times L^p(\mathbb R_{+}^d,\mathbb R^d), \end{aligned}$$

where

$$ \begin{aligned} \lambda _{m,l}\in (0,1]\quad \& \quad \sum _{l=1}^{m}\lambda _{m,l}=1. \end{aligned}$$

In particular, we have

$$\begin{aligned} \lim _{m\rightarrow \infty }\sum _{l=1}^{m}\lambda _{m,l}f_{k_{l}}=f\quad \textrm{in} \quad L^p(\mathbb R_{+}^d) \end{aligned}$$

and for any $i=1,2,\ldots ,d$

$$\begin{aligned} \lim _{m\rightarrow \infty }\sum _{l=1}^{m}\lambda _{m,l}\delta _{i}f_{k_{l}}=F\quad \textrm{in} \quad L^p(\mathbb R_{+}^d,\mathbb R^d), \end{aligned}$$

thereby getting

$$\begin{aligned} F=\delta _{i}f\ \mathrm {a.e.}\ \textrm{on}\ \mathbb R_{+}^d. \end{aligned}$$

This completes the proof of (14). Finally, we take $\phi =1$ in (13) and then obtain (15). $\square$

Proposition 3.4

Let $p\in (1,\infty )$. Then for any sequence $\{E_l\}_{l=1}^{\infty }$ with

$$\begin{aligned} E_l\subseteq E_{l+1}\subseteq \mathbb R_{+}^d, \ \forall \ l\in \mathbb {N}, \end{aligned}$$

one has

$$\begin{aligned} {\left\{ \begin{array}{ll} {\lim _{l\rightarrow \infty }}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_l,\mathbb R_{+}^d)\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\cup _{l=1}^{\infty }E_l,\mathbb R_{+}^d);\\ \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(\cup _{l=1}^{\infty }E_l,\mathbb R_{+}^d)\lesssim \lim _{l\rightarrow \infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_l,\mathbb R_{+}^d). \end{array}\right. } \end{aligned}$$

Proof

We adopt the idea used in Costea [2, Theorem 3.1(iv)]. Let

$$\begin{aligned} E=\cup _{l=1}^{\infty }E_{l}. \end{aligned}$$

By Proposition 3.2 (ii), we obtain

$$\begin{aligned} \lim _{l\rightarrow \infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_l,\mathbb R_{+}^d)\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d). \end{aligned}$$

Thus, we only need to prove the second inequality. Without loss of generality, we may assume

$$\begin{aligned} \lim _{l\rightarrow \infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_l,\mathbb R_{+}^d)<\infty . \end{aligned}$$

Fix $\epsilon \in (0,1)$. For any $l\in \mathbb {N}$, choose

$$\begin{aligned} u_l\in \mathcal {A}_p(E_l) \end{aligned}$$

such that

$$\begin{aligned} \Vert u_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_l,\mathbb R_{+}^d)+\epsilon . \end{aligned}$$

Since $\{E_l \}_{l=1}^{\infty }$ increases and

$$\begin{aligned} \lim _{l\rightarrow \infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_l,\mathbb R_{+}^d)<\infty , \end{aligned}$$

it follows that

$$\begin{aligned} \sup _{l\in \mathbb {N}}\Vert u_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}<\infty . \end{aligned}$$

Applying Proposition 3.3, we find a subsequence, which we denote again by $\{u_l \}_{l\in \mathbb {N}}$, and a function

$$\begin{aligned} u\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d) \end{aligned}$$

such that $\{(u_l,\delta _{i}u_l)\}_{l\in \mathbb {N}}$ converges to $(u,\delta _i u)$ (for any given $i=1,2,\ldots ,d$) weakly in

$$\begin{aligned} L^p(\mathbb R_{+}^d)\times L^p(\mathbb R_{+}^d,\mathbb R^d). \end{aligned}$$

Upon fixing $l_0\in \mathbb {N}$ and using Mazur’s Theorem for the sequence $\{u_l\}_{l\ge l_0}$, we find a finite convex combination of $\{u_l\}_{l\ge l_0}$, denote by $v_{l_0}$, such that

$$\begin{aligned} \Vert v_{l_0}-u\Vert _{W_{\mathcal {L}^{\alpha }}^{1,p}(\mathbb R_{+}^d)}\le 2^{-l_0}. \end{aligned}$$

(16)

Since every $u_l$ with $l\ge l_0$ satisfies

$$\begin{aligned} E_{l_0}\subseteq E_l\subseteq \textrm{int}\{x\in \mathbb R_{+}^d:\ u_l(x)\ge 1\}, \end{aligned}$$

it follows that

$$\begin{aligned} E_{l_0}\subseteq \textrm{int}\{x\in \mathbb R_{+}^d:\ v_{l_0}(x)\ge 1\}. \end{aligned}$$

In this way, we obtain a sequence $v_l$ being a finite convex combination of $\{u_k\}_{k\ge l}$ such that

$$\begin{aligned} v_l\rightarrow u\ \textrm{strongly} \ \textrm{in}\ W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d) \ \textrm{as}\ l\rightarrow \infty , \end{aligned}$$

and so that

$$\begin{aligned} v_l\in \mathcal {A}_p(E_l). \end{aligned}$$

Passing to a subsequence if necessary, we may even assume that for any $l\in \mathbb {N}$,

$$\begin{aligned} \Vert v_{l+1}-v_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}<2^{-l}. \end{aligned}$$

(17)

Next, define

$$\begin{aligned} w_{l}=\sup _{j\ge l}v_j \end{aligned}$$

for any $l\in \mathbb {N}$. It is easy to verify that for all $l\in \mathbb {N}$ and $x\in \mathbb R_{+}^d$,

$$\begin{aligned} |w_l(x)-v_l(x)|\le \sum _{j=l}^{\infty }|v_{j+1}(x)-v_{j}(x)|, \end{aligned}$$

(18)

and for any $i=1,2,\ldots ,d$,

$$\begin{aligned} \bigg |\sup _{j\ge l}|\delta _i v_{j}(x)|-|\delta _{i}v_l(x)|\bigg |\le \sum _{j=l}^{\infty }|\delta _{i}v_{j+1}(x)-\delta _{i}v_{j}(x)|. \end{aligned}$$

(19)

By (17) and (18), we have

$$\begin{aligned} \begin{aligned} \Vert w_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}&\le \Vert v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+\sum _{j=l}^{\infty }\Vert v_{j+1}(x)-v_{j}(x)\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\\&\le \Vert v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+\sum _{j=l}^{\infty }\Vert v_{j+1}(x)-v_{j}(x)\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\\&\le \Vert v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+\sum _{j=l}^{\infty }2^{-j}\\&\le \Vert v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+2^{1-l}. \end{aligned} \end{aligned}$$

(20)

Similarly, by (17) and (19), we obtain

$$\begin{aligned} \begin{aligned} \Big \Vert \sup _{j\ge l}|\delta _i{ v_j}|\Big \Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}&\le \Vert \delta _i v_l \Vert _{L^{p}(\mathbb R_{+}^d,d\mu _{\alpha })}+ \sum _{j=l}^{\infty }\Vert \delta _{i}v_{j+1}-\delta _{i}v_{j} \Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\\&\le \Vert \delta _i v_l \Vert _{L^{p}(\mathbb R_{+}^d,d\mu _{\alpha })}+2^{1-l} \end{aligned} \end{aligned}$$

(21)

for all $i=1,2,\ldots ,d$. Noticing that (20)–(21) and Lemma 3.1 implies that $\delta _{i}w_{l}$, $i=1,2,\ldots ,d$ exists a.e. on $\mathbb R_{+}^d$ and

$$\begin{aligned} |\delta _i w_l(x)|\le \sup _{j\ge l}|\delta _{i}{ v_j}(x)|\ \mathrm {a.e.}\ x\in \mathbb R_{+}^d. \end{aligned}$$

(22)

By (20)–(22), we see

$$\begin{aligned} w_{l}\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

Now, we calculate the $\Vert \cdot \Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}$ norm of $w_{l}$. To this end, we observe that the mean value theorem implies the following inequality:

$$\begin{aligned} (a+b)^{p}\le a^p+p(M+1)^{p-1}b\ \ \textrm{for} \ (M,a,b)\in (0,\infty )\times [0,M]\times [0,1]. \end{aligned}$$

(23)

Notice that (16) implies

$$\begin{aligned} \max \{\Vert v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })},\Vert \delta _{i}v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\} \le \Vert u\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}+1 \end{aligned}$$

for any $i=1,2,\ldots ,d$. Below we will apply (23) with

$$\begin{aligned} M=\Vert u\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}+1. \end{aligned}$$

Consequently, we deduce from (22), (20) and (21) that

$$\begin{aligned} \begin{aligned} \Vert w_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p&= {\sum _{i=1}^d}\Vert \delta _i w_{l}\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p + \Vert w_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\\&\le {\sum _{i=1}^{d}}\Big \Vert \sup _{j\ge l}|\delta _i v_{j}|\Big \Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p+\Vert w_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\\&\le {\sum _{i=1}^d}\Big (\Vert \delta _i v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+2^{1-l}\Big )^p +\Big (\Vert v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+2^{1-l}\Big )^p\\&\le \Vert v_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p+C_{u,p}2^{-l}, \end{aligned} \end{aligned}$$

(24)

where

$$\begin{aligned} C_{u,p}:=4p(\Vert u\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}+2)^{p-1}. \end{aligned}$$

Recall that

$$\begin{aligned} {v_l}\in \mathcal {A}_p(E_l) \end{aligned}$$

and $\{E_l\}_{l}$ increases to E, we have

$$\begin{aligned} E\subseteq \textrm{int}\{x\in \mathbb R_{+}^d:\ w_{l}(x)\ge 1\}. \end{aligned}$$

Thus,

$$\begin{aligned} w_l\in \mathcal {A}_p(E). \end{aligned}$$

Moreover, (24) implies

$$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d)\le \Vert w_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p\le \Vert v_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p+C_{u,p}2^{-l}, \ \forall \ l\in \mathbb {N}. \end{aligned}$$

According to the construction of $v_l$, we may assume

$$\begin{aligned} v_l=\sum _{k=l}^{N_l}\lambda _{l,k}u_k, \end{aligned}$$

where

$$\begin{aligned} {\left\{ \begin{array}{ll} l\le N_l\in \mathbb {N};\\ \lambda _{l,k}\in [0,1];\\ \sum _{k=l}^{N_l}\lambda _{l,k}=1. \end{array}\right. } \end{aligned}$$

Consequently, by the Minkowski inequality, the Hölder inequality, and Proposition 3.2 (ii), we have

$$\begin{aligned} \Vert v_l\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p&=\Vert v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p+\sum _{i=1}^{d}\Vert \delta _i v_l\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\\&\le \bigg (\sum _{k=l}^{N_l}\lambda _{l,k}\Vert u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })} \bigg )^p+\bigg (\sum _{i=1}^{d}\sum _{k=l}^{N_l}\lambda _{l,k}\Vert \delta _i u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\bigg )^p\\&\le \bigg (\sum _{k=l}^{N_l}\lambda _{l,k}\Vert u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })} \bigg )^p+\bigg (\sum _{k=l}^{N_l}\sum _{i=1}^{d}\lambda _{l,k}\Vert \delta _i u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\bigg )^p\\&\le \sum _{k=l}^{N_l}\lambda _{l,k}\Vert u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p+ \bigg (\sum _{k=l}^{N_l}\Big (\sum _{i=1}^{d}\lambda _{l,k}\Vert \delta _i u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\Big )^{\frac{1}{p}}\bigg )^p\\&\lesssim \sum _{k=l}^{N_l}\lambda _{l,k}\Vert u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p+\sum _{k=1}^{N_l}\lambda _{l,k} \bigg (\sum _{i=1}^d\Vert \delta _i u_k\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\bigg )\\&\lesssim \sum _{k=l}^{N_l}\lambda _{l,k}\Vert u_k\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p\\&\lesssim \sum _{k=l}^{N_l}\lambda _{l,k}\big (\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_k,\mathbb R_{+}^d)+\epsilon \big )\\&\lesssim \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{N_l},\mathbb R_{+}^d)+\epsilon . \end{aligned}$$

We then deduce that for any $l\in \mathbb {N}$,

$$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d)\lesssim \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{N_l},\mathbb R_{+}^d)+\epsilon +C_{u,p}2^{-l}. \end{aligned}$$

Letting $l\rightarrow \infty$ and $\epsilon \rightarrow 0$ yields

$$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d)\lesssim \lim _{l\rightarrow \infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{N_l},\mathbb R_{+}^d)=\lim _{l\rightarrow \infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_l,\mathbb R_{+}^d), \end{aligned}$$

thereby completing the desired result. $\square$

3.2 Alternative of Laguerre p-Capacity for $1\le p<\infty$

Definition 3.4

Let $p\in [1,\infty )$ and $K\subseteq \mathbb R_{+}^d$ be a compact set. Define

$$\begin{aligned} \mathcal {A}(K)=\{f\in C_c^1(\mathbb R_{+}^d):\ f\ge 1\ \textrm{on}\ K\} \end{aligned}$$

and

$$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d):=\inf \{\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p:\ f\in \mathcal {A}(K)\}. \end{aligned}$$

(25)

If $O\subseteq \mathbb R_{+}^d$ is an open set, then

$$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d):=\sup \{\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d):\ \textrm{compact}\ K\subseteq O\}. \end{aligned}$$

(26)

Lemma 3.2

Let $p\in [1,\infty )$. Then the following properties are valid.

• (i) For compact sets $K_1$ and $K_2$ satisfying that $K_1\subseteq K_2\subseteq \mathbb R_{+}^d$,

  $$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_1,\mathbb R_{+}^d)\le \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_2,\mathbb R_{+}^d). \end{aligned}$$

• (ii) For compact set K and open set O satisfying $O\subseteq K\subseteq \mathbb R_{+}^d$,

  $$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d)\le \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d). \end{aligned}$$

• (iii) For open sets $O_1$ and $O_2$ satisfying $O_1\subseteq O_2\subseteq \mathbb R_{+}^d$,

  $$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O_1,\mathbb R_{+}^d)\le \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O_2,\mathbb R_{+}^d). \end{aligned}$$

Proof

First of all, if $K_1$ and $K_2$ are compact subsets of $\mathbb R_{+}^d$ satisfying $K_1\subseteq K_2$, then

$$\begin{aligned} \mathcal {A}(K_2)\subseteq \mathcal {A}(K_1), \end{aligned}$$

and hence

$$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_1,\mathbb R_{+}^d)\le \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_2,\mathbb R_{+}^d), \end{aligned}$$

which proves (i).

Next, (ii) follows from (i) and (26).

Finally, (iii) follows directly from (ii) and (26). $\square$

Lemma 3.3

Let $p\in [1,\infty )$ and K be a compact subset of $\mathbb R_{+}^d$. Then

$$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d)=\inf \{\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d):\ \textrm{open}\ O\supset K\}. \end{aligned}$$

Proof

Using Definition 3.4 and Lemma 3.2 (ii) we can conclude that Lemma 3.3 is valid. $\square$

Let $p\in (1,\infty )$. Define the Laguerre Sobolev p-space $W^{1,p}_{0,\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ to be the completion of all $C_{c}^1(\mathbb R_{+}^d)$-functions in

$$\begin{aligned} \big (W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d),\Vert \cdot \Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\big ). \end{aligned}$$

The following assertion just indicates that $W^{1,p}_{0,\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ is nothing but $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ and consequently all polynomials are dense in each Laguerre Sobolev p-space, see Theorem 6.17 in [8].

Proposition 3.5

Let $p\in (1,\infty )$. Then

$$\begin{aligned} W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)=W^{1,p}_{0,\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

More precisely, for any $f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$, there exists a sequence of functions

$$\begin{aligned} \{f_{j}\}_{j\in \mathbb {N}}\subseteq C_c^1(\mathbb R_{+}^d) \end{aligned}$$

such that

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert f_j-f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}=0. \end{aligned}$$

(27)

Moreover, if K is a nonempty compact subset of $\mathbb R_{+}^d$ such that

$$\begin{aligned} K\subseteq \textrm{int}\{x\in \mathbb R_{+}^d:\ f(x)\ge 1\}, \end{aligned}$$

then the above functions $f_j$ enjoy that

$$\begin{aligned} f_j|_{K}\ge 1\quad \forall \ j\in \mathbb {N}. \end{aligned}$$

(28)

Proof

For any $i=1,2,\ldots ,d$, we can choose specific function $\eta \in C_{c}^1(\mathbb R_{+}^d)$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \eta =1 &{}\textrm{on}\ B(0,1)\cap \mathbb R_{+}^d;\\ \eta =0 &{}\textrm{on}\ B(0,2)^{c}\cap \mathbb R_{+}^d;\\ 0\le \eta \le 1\ {} &{}\textrm{on}\ \mathbb R_{+}^d;\\ |\delta _i\eta |\le C &{}\textrm{on}\ \mathbb R_{+}^d. \end{array}\right. } \end{aligned}$$

For

$$ \begin{aligned} k\in \mathbb {N}\ \& \ f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d), \end{aligned}$$

we obtain

$$\begin{aligned} \Vert \eta (2^{-k}\cdot )f-f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}&= \bigg (\int _{\mathbb R_{+}^d}|(\eta (2^{-k}x)-1)f(x)|^p d\mu _{\alpha }(x)\bigg )^{\frac{1}{p}}\\&\le \bigg (\int _{B(0,2^k)^c\cap \mathbb R_{+}^d}|f(x)|^p d\mu _{\alpha }(x)\bigg )^{\frac{1}{p}}, \end{aligned}$$

which tends to 0 as $k\rightarrow \infty$ since $f\in L^p(\mathbb R_{+}^d,d\mu _{\alpha })$.

Meanwhile, for any $k\in \mathbb {N}$,

$$\begin{aligned}&\Vert \delta _{i}(\eta (2^{-k}\cdot )f)-\delta _{i} f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\\&\quad =\bigg (\int _{\mathbb R_{+}^d}|2^{-k}\delta _i\eta (2^{-k}x)f(x)+\eta (2^{-k}x)\delta _i f(x)-\delta _i f(x)|^p d\mu _{\alpha }(x)\bigg )^{\frac{1}{p}}\\&\quad \le C 2^{-k}\Vert f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+\bigg (\int _{\mathbb R_{+}^d} |(\eta (2^{-k}x)-1)\delta _i f(x)|^p d\mu _{\alpha }\bigg )^{\frac{1}{p}}\\&\quad \le C 2^{-k}\Vert f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+\bigg (\int _{B(0,2^k)^c\cap \mathbb R_{+}^d} |\delta _i f(x)|^p d\mu _{\alpha }(x)\bigg )^{\frac{1}{p}}, \end{aligned}$$

which also tends to 0 as $k\rightarrow \infty$. Combining the last two inequalities we conclude that

$$\begin{aligned} {\left\{ \begin{array}{ll} \eta (2^{-k}\cdot )\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d);\\ \lim _{k\rightarrow \infty }\Vert \eta (2^{-k}\cdot )f-f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}=0. \end{array}\right. } \end{aligned}$$

Therefore, each $f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ can be approximated by a sequence of functions in $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ with compact support, and thus we can assume

$$\begin{aligned} f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d) \end{aligned}$$

has a compact support to reduce the proof.

Without loss of generality, we may suppose

$$\begin{aligned} \textrm{supp}\ f\subseteq B(0,R)\cap \mathbb R_{+}^d \end{aligned}$$

for some fixed $R>0$. Choose a nonnegative function $\phi \in C_c^{\infty }(\mathbb R_{+}^d)$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{supp}\ \phi \subseteq B(0,1)\bigcap \mathbb R_{+}^d;\\ \int _{\mathbb R_{+}^d}\phi dx=1. \end{array}\right. } \end{aligned}$$

Let

$$\begin{aligned} \phi _{j}(x):=2^{jd}\phi (2^j x)\quad \forall \ x\in \mathbb R_{+}^d,\ \forall \ j\in \mathbb {N}. \end{aligned}$$

Obviously, for all $j\in \mathbb {N}$ we observe

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{supp}\ \phi _{j}\subseteq B(0,2^{-j})\bigcap \mathbb R_{+}^d;\\ \int _{\mathbb R_{+}^d}\phi _{j}dx=1. \end{array}\right. } \end{aligned}$$

By the Hölder inequality,

$$\begin{aligned} |\phi _{j}*f(x)|=\bigg |\int _{\mathbb R_{+}^d}\big (\phi _j(x-y)\big )^{1-\frac{1}{p}} \big (\phi _{j}(x-y)\big )^{\frac{1}{p}}f(y)dy\bigg |\\ \le \bigg (\int _{\mathbb R_{+}^d}\phi _{j}(x-y)|f(y)|^p dy\bigg )^{\frac{1}{p}}, \end{aligned}$$

whence

$$\begin{aligned}&\int _{\mathbb R_{+}^d}|\phi _{j}*f(x)|^p d\mu _{\alpha }(x)\\&\quad \le \int _{\mathbb R_{+}^d}\int _{\mathbb R_{+}^d}\phi _{j}(x-y)|f(y)|^p dyd\mu _{\alpha }(x)\\&\quad =\int _{y\in B(0,R)\cap \mathbb R_{+}^d}\bigg (\int _{x\in B(y,2^{-j})\cap \mathbb R_{+}^d}\phi _{j}(x-y) \omega (y)^{-1}\omega (x)dx\bigg )|f(y)|^p d\mu _{\alpha }(y). \end{aligned}$$

Upon using

$$\begin{aligned} (x,y)\in \{B(y,2^{-j})\cap \mathbb R_{+}^d\}\times \{B(0,R)\cap \mathbb R_{+}^d\}, \end{aligned}$$

we see that

$$\begin{aligned} \omega (y)^{-1}\omega (x)&=\bigg (\prod _{i=1}^d\frac{y_i^{\alpha _i}e^{-y_i}}{\Gamma ( \alpha _i+1)}\bigg )^{-1}\prod _{i=1}^d\frac{x_i^{\alpha _i}e^{-x_i}}{\Gamma (\alpha _i+1)}\\&=\prod _{i=1}^d \bigg (\frac{x_i}{y_i}\bigg )^{\alpha _i}e^{y_i-x_i}\\&\le \prod _{i=1}^d \bigg (\frac{|x_i|}{|y_i|}\bigg )^{\alpha _i}e^{|y_i-x_i|}\\&\le \prod _{i=1}^d \bigg (\frac{|x_i|}{|y_i|}\bigg )^{\alpha _i}e^{2^{-j}}\\&\le 1, \end{aligned}$$

where we have used the fact that $|x_i|\approx |y_i|$ for sufficiently large j in the last inequality. Furthermore, we have

$$\begin{aligned} \begin{aligned}&\int _{\mathbb R_{+}^d}|\phi _{j}*f(x)|^p d\mu _{\alpha }(x)\\&\quad \le \int _{y\in B(0,R)\cap \mathbb R_{+}^d}\bigg (\int _{x\in B(y,2^{-j})\cap \mathbb R_{+}^d}\phi _{j}(x-y) dx\bigg )|f(y)|^p d\mu _{\alpha }(y)\\&\quad =\int _{B(0,R)\cap \mathbb R_{+}^d}|f(y)|^p d\mu _{\alpha }(y)\quad \forall \ j\in \mathbb {N}. \end{aligned} \end{aligned}$$

(29)

Now fix $\delta >0$. Due to

$$\begin{aligned} f\in L^p(\mathbb R_{+}^d,d\mu _{\alpha })\ \textrm{and}\ p\in (1,\infty ), \end{aligned}$$

there exists a function $g\in C_c(\mathbb R_{+}^d)$ such that

$$\begin{aligned} \Vert g-f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}<\delta . \end{aligned}$$

According to (29), this implies that for all $j\in \mathbb {N}$ one has

$$\begin{aligned} \Vert \phi _{j}*g-\phi _{j}*f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p&= \Vert \phi _{j}*(g-f)\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\\&\le \Vert g-f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\\&<\delta ^p. \end{aligned}$$

Without loss of generality, we may assume

$$\begin{aligned} \textrm{supp}\ g\subseteq B(0,R_1)\cap \mathbb R_{+}^d \end{aligned}$$

for some fixed $R_1>0$.

Similarity, we can proceed as in the proof of (29) to derive

$$\begin{aligned}&\int _{\mathbb R_{+}^d}|\phi _{j}*g(x)-g(x)|^p d\mu _{\alpha }\\&\quad \le \int _{y\in B(0,R_1)\cap \mathbb R_{+}^d}\bigg (\int _{x\in B(y,2^{-j})\cap \mathbb R_{+}^d}\phi _{j}(x-y)|g(y)-g(x)|^p dx\bigg )d\mu _{\alpha }(y). \end{aligned}$$

Thanks to

$$\begin{aligned} g\in C_c(\mathbb R_{+}^d), \end{aligned}$$

there exists $N\in \mathbb {N}$ such that when $j>N$, we obtain

$$\begin{aligned} |g(y)-g(x)|<\delta \quad \textrm{as}\quad |x-y|<2^{-j}, \end{aligned}$$

which implies

$$\begin{aligned} \Vert \phi _{j}*g-g\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\le \delta \quad \forall \ j>N. \end{aligned}$$

Consequently, when $j>N$ we have

$$\begin{aligned}&\Vert \phi _{j}*f-f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\\&\quad \le \Vert \phi _{j}*f-\phi _{j}*g\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+ \Vert \phi _{j}*g-g\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}+\Vert g-f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}\\&\quad \le 3\delta . \end{aligned}$$

Letting first $j\rightarrow \infty$ and then $\delta \rightarrow 0$ in the above formulae yields

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert \phi _{j}*f-f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}=0. \end{aligned}$$

(30)

Upon noticing that

$$\begin{aligned} \delta _{i}(\phi _{j}*f)(x)=\phi _{j}*(\delta _{i}f)(x)\quad \forall \ (x,j,i)\in \mathbb R_{+}^d\times \mathbb {N}\times \{1,2,\ldots ,d\}, \end{aligned}$$

then we apply (30) to deduce that

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert \delta _{i}(\phi _{j}*f)- \delta _{i}f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}=\lim _{j\rightarrow \infty }\Vert \phi _{j}*(\delta _i f) -\delta _i f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}=0.\nonumber \\ \end{aligned}$$

(31)

Via combining (30) and (31), we achieve that (27) holds for

$$\begin{aligned} f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d) \end{aligned}$$

with compact support.

Finally, we show (28). By the above proof, we know that the following functions

$$\begin{aligned} f_{k,j}(x):=\phi _{j}*\big (\eta (2^{-k}\cdot )f\big )(x)\quad \forall \ x\in \mathbb R_{+}^d \end{aligned}$$

satisfy

$$\begin{aligned} \lim _{k\rightarrow \infty }\bigg (\lim _{j\rightarrow \infty }\Vert f_{k,j}-f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\bigg )=0. \end{aligned}$$

Suppose further that K is a nonempty compact subset of $\mathbb R_{+}^d$ and

$$\begin{aligned} K\subseteq \textrm{int}\{x\in \mathbb R_{+}^d:\ f(x)\ge 1\}. \end{aligned}$$

One can choose $R>0$ and $k_{0}\in \mathbb {N}$ such that

$$ \begin{aligned} K\subseteq B(0,2^{-1}R)\cap \mathbb R_{+}^d\quad \& \quad 2^{k_{0}}>R+1. \end{aligned}$$

Choose $j_0\in \mathbb {N}$ such that

$$\begin{aligned} 2^{-j_0}<\textrm{dist}\big (\partial K,\partial (\textrm{int}\{ x\in \mathbb R_{+}^d:\ f(x) \ge 1\})\big ). \end{aligned}$$

Then, for any

$$\begin{aligned} (x,y)\in K\times \{B(x,2^{-j_0})\cap \mathbb R_{+}^d\}, \end{aligned}$$

we observe

$$ \begin{aligned} f(y)\ge 1\quad \& \quad \eta (2^{-k_0}y)=1. \end{aligned}$$

Therefore, when $x\in K$ we have

$$\begin{aligned} f_{k_0,j_0}(x)=\int _{\mathbb R_{+}^d}\phi _{j_0}(x-y)\eta (2^{-k_0}y)f(y)dy\ge \int _{\mathbb R_{+}^d}\phi _{j_0}(x-y)dy=1. \end{aligned}$$

This completes (28), thereby proving the argument for Proposition 3.5. $\square$

Applying Lemma 3.3, we can extend the definition of $\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(\cdot ,\mathbb R_{+}^d)$ from a compact set to any set.

Definition 3.5

Let $p\in [1,\infty )$ and E be an arbitrary subset of $\mathbb R_{+}^d$. Define

$$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d):=\inf \{\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d):\ \textrm{open}\ O\supset E\}. \end{aligned}$$

(32)

Due to Proposition 3.5, we give the following equivalent characterization of the Laguerre p-capacity.

Theorem 3.1

Assume that $p\in (1,\infty )$.

• (i) If K is a compact set, then

  $$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d)=\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d). \end{aligned}$$

  (33)

• (ii) If O is an open set, then

  $$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d)\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d);\\ \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d)\lesssim \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(O,\mathbb R_{+}^d). \end{array}\right. } \end{aligned}$$

  (34)

• (iii) If E is an arbitrary set, then

  $$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d)\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d);\\ \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d)\lesssim \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d). \end{array}\right. } \end{aligned}$$

  (35)

Also, if K is a compact subset of $\mathbb R_{+}^d$, then

$$\begin{aligned} \textrm{Cap}_{0,1}^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d)=\textrm{Cap}_{1}^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d). \end{aligned}$$

(36)

Proof

As in the proof of in [23, Theorem 3.2.6], we can prove (33), (34), (35) and (36) by three steps via (25), (26), Propositions 3.4 and 3.5. $\square$

3.3 Laguerre p-Capacitary-Strong-Type Inequality

In this subsection we investigate the restriction/trace question for the Laguerre Sobolev space $W^{1,p} _{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$. In Lemma 3.4, we adopt the method similar to the proof of [23, Lemma 4.1.1].

Lemma 3.4

Let $1\le p<\infty$ and $f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ be continuous. For any $t\in (0,\infty )$ set

$$\begin{aligned} E_t(f):=\{x\in \mathbb R_{+}^d:\ |f(x)|>t\}, \end{aligned}$$

then

$$\begin{aligned} \int _{0}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t(f),\mathbb R_{+}^d)dt^p\lesssim \Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p. \end{aligned}$$

(37)

Proof

Assume $f\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ be a nonzero continuous function. In what follows, for any $t\in (0,\infty )$, we simply write $E_t(f)$ as $E_t$. By Proposition 3.2 (ii),

$$\begin{aligned} \begin{aligned} \int _{0}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p&=\sum _{k\in \mathbb {Z}}\int _{2^k}^{2^{k+1}} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p\\&\le (2^p-1)\sum _{k\in \mathbb {Z}}2^{kp}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^k},\mathbb R_{+}^d). \end{aligned} \end{aligned}$$

(38)

Similarly to [25, p. 155, Remark 1], we define a function $\tau : \mathbb {R}\rightarrow \mathbb {R}$ as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} \tau \in C^1(\mathbb {R}) &{}\textrm{is}\ \textrm{even};\\ 0\le \tau (t)\le 1 &{}\forall \ t\ge 0;\\ \tau (t)=0 &{}\forall \ t\in [0,2^{-1}];\\ \tau (t)=1 &{}\forall \ t\ge 1;\\ 0\le \tau ^{\prime }(t)\le 3 &{}\forall \ t\ge 0. \end{array}\right. } \end{aligned}$$

In fact, on the interval $[2^{-1},1]$, we can define $\tau$ by smoothing the line passing over the points $(2^{-1},0)$ and (1, 1) so that

$$\begin{aligned} 0\le \tau ^{\prime }\le 2+\epsilon \end{aligned}$$

for some small $\epsilon \in (0,1)$. Due to $f\in C^{0}(\mathbb R_{+}^d),$ for all $k\in \mathbb {Z}$ and $x\in \mathbb R_{+}^d$, we define

$$\begin{aligned} f_{k}(x):=\tau \bigg (\frac{f(x)}{2^{k}}\bigg ) \end{aligned}$$

which is also a continuous function. For all $k\in \mathbb {Z}$, observe that

$$\begin{aligned} f_k(x)=\tau \bigg (\frac{f(x)}{2^k}\bigg )=\tau \bigg (\frac{|f(x)|}{2^k}\bigg ) ={\left\{ \begin{array}{ll} 0&{} \forall \ x\notin E_{2^{k-1}};\\ 1&{} \forall \ x\in E_{2^k} \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} 0\le f_{k}(x)\le 1\quad x\in E_{2^{k-1}}\setminus E_{2^k}. \end{aligned}$$

Hence,

$$\begin{aligned} \int _{\mathbb R_{+}^d}|f_k|^pd\mu _{\alpha }\le \mu _{\alpha }(\mathbb R_{+}^d)=1 \end{aligned}$$

and for any $i=1,2,\ldots ,d$

$$\begin{aligned} \int _{\mathbb R_{+}^d}|\delta _i f_k|^pd\mu _{\alpha }= & {} \int _{\mathbb R_{+}^d}\bigg |\sqrt{x_i}\tau ^{\prime }\Big ( \frac{f(x)}{2^k}\Big )\frac{\partial f}{\partial x_i}2^{-k}\bigg |^p d\mu _{\alpha }\\\le & {} 3^p\int _{E_{2^{k-1}}\setminus E_{2^k}}|\delta _i f(x)|^p2^{-kp}d\mu _{\alpha }, \end{aligned}$$

which implies

$$\begin{aligned} f_k\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

Obviously,

$$\begin{aligned} E_{2^k}\subseteq \textrm{int}\{x\in \mathbb R_{+}^d:\ f_k(x)\ge 1\}, \end{aligned}$$

since $f_k=1$ on the open set $E_{2^k}$. Thus,

$$\begin{aligned} f_k\in \mathcal {A}_p(E_{2^k}). \end{aligned}$$

Moreover,

$$\begin{aligned} \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^k},\mathbb R_{+}^d)&\le \Vert f_k\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p\\&\le \sum _{i=1}^d\int _{\mathbb R_{+}^d}|\delta _i f_k|^pd\mu _{\alpha }+\int _{\mathbb R_{+}^d}|f_k|^pd\mu _{\alpha }\\&\le 3^p\sum _{i=1}^d\int _{E_{2^{k-1}}\setminus E_{2^k}}|\delta _i f(x)|^p2^{-kp}d\mu _{\alpha } +\mu _{\alpha }(E_{2^{k-1}}). \end{aligned}$$

From this and (38), we obtain

$$\begin{aligned} \begin{aligned}&\int _{0}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p\\&\quad \lesssim \sum _{k\in \mathbb {Z}}\sum _{i=1}^d\int _{E_{2^{k-1}}\setminus E_{2^k}} |\delta _i f(x)|^pd\mu _{\alpha }+\sum _{k\in \mathbb {Z}}2^{kp}\mu _{\alpha }(E_{2^{k-1}})\\&\quad \lesssim \sum _{i=1}^d\Vert \delta _i f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p+\sum _{k\in \mathbb {Z}}2^{kp} \mu _{\alpha }(E_{2^{k-1}}). \end{aligned} \end{aligned}$$

(39)

Using the integral formula and exchanging sum order, we have

$$\begin{aligned} \sum _{k\in \mathbb {Z}}2^{kp}\mu _{\alpha }(E_{2^{k-1}})&=\sum _{k\in \mathbb {Z}}2^{kp}\sum _{j=k-1 }^{\infty }\mu _{\alpha }(E_{2^j}\setminus E_{2^{j+1}})\\&=\frac{2^p}{1-2^{-p}}\sum _{j\in \mathbb {Z}}2^{jp}\mu _{\alpha }(E_{2^j}\setminus E_{2^{j+1}})\\&\lesssim \Vert f\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p. \end{aligned}$$

Combining this with (39), we deduce that (37) holds. $\square$

Remark 3.3

It should be noted that the capacitary inequality in Lemma 3.4 is also valid for $\textrm{cap}(E,\mathcal{B}\mathcal{V}_{\mathcal {L}^{\alpha }}(\Omega ))$ by adopting the similar method as Lemma 3.4 and smooth approximation. Here we omit the details of the proof.

3.4 Trace Inequality for $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ Under $1\le p\le q<\infty$

In the paper [23], Liu, Xiao, Yang and Yuan proved the following trace inequality for the Gaussian Sobolev p-space $W^{1,p}(\mathbb {G}^n)$ under $1\le p\le q<\infty$. In the same way, we use Lemma 3.4 to establish the first restriction/trace result for $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$.

Theorem 3.2

Let $1\le p\le q<\infty$ and $\mu$ be a nonnegative Radon measure on $\mathbb R_{+}^d$. Then the following two assertions are equivalent.

• (i) There exists a positive constant $C_1$ such that for all compact sets $K\subseteq \mathbb R_{+}^d$,

  $$\begin{aligned} \mu (K)\le C_1(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d))^{\frac{q}{p}}. \end{aligned}$$

• (ii) There exists a positive constant $C_2$ such that

  $$\begin{aligned} \bigg (\int _{\mathbb R_{+}^d}|f|^q d\mu \bigg )^{\frac{1}{q}}\le C_2\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\quad \forall \ f\in C^0(\mathbb R_{+}^d)\cap W^{1,p}_{\mathcal {L}^{\alpha }} (\mathbb R_{+}^d). \end{aligned}$$

  (40)

  Moreover, $C_1\approx C_2^q$ with the implicit constants depending only on p and q.

Proof

(i) $\Rightarrow$ (ii). Fix

$$\begin{aligned} f\in C^0(\mathbb R_{+}^d)\cap W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

For $t\in (0,\infty )$, define a open set as follows:

$$\begin{aligned} E_t:=\{x\in \mathbb R_{+}^d:\ |f(x)|>t\}. \end{aligned}$$

Consider the set $E_{2^k}$ with $k\in \mathbb {Z}$. Since $\mu$ is a nonnegative Radon measure on $\mathbb R_{+}^d$, we can choose a compact set $K_k\subseteq E_{2^k}$ such that

$$\begin{aligned} \mu (E_{2^k})\le 2\mu (K_k). \end{aligned}$$

Via the above inequality and Proposition 3.2 (ii), we obtain

$$\begin{aligned} \begin{aligned} \int _{\mathbb R_{+}^d}|f|^q d\mu&\le \int _{\mathbb R^d}|f|^q d\mu \\&=\sum _{k\in \mathbb {Z}}\int _{2^k}^{2^{k+1}}\mu (E_t)dt^q\\&\le (2^q-1)\sum _{k\in \mathbb {Z}}2^{kq}\mu (E_{2^k})\\&\le 2^{q+1}\sum _{k\in \mathbb {Z}}2^{kq}\mu (K_k)\\&\le C_1 2^{q+1}\sum _{k\in \mathbb {Z}}2^{kq}(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K_k,\mathbb R_{+}^d))^{\frac{q}{p}}\\&\le C_1 2^{q+1}\sum _{k\in \mathbb {Z}}2^{kq}(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^k},\mathbb R_{+}^d))^{\frac{q}{p}}. \end{aligned} \end{aligned}$$

(41)

Next, we will use the following inequality: for any nonnegative sequence $\{a_j\}_{j\in \mathbb {Z}}$,

$$\begin{aligned} \bigg (\sum _{j\in \mathbb {Z}}a_j\bigg )^k\le \sum _{j\in \mathbb {Z}}a_j^k\quad \forall \ k\in (0,1]. \end{aligned}$$

The above estimation, along with the fact $p\le q$, further implies

$$\begin{aligned} \sum _{k\in \mathbb {Z}}2^{kq}(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^k},\mathbb R_{+}^d))^{\frac{q}{p}} \le \bigg (\sum _{k\in \mathbb {Z}}2^{kp}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^k},\mathbb R_{+}^d)\bigg )^{\frac{q}{p}}. \end{aligned}$$

(42)

Via Proposition 3.2 (ii) again, we have

$$\begin{aligned} \begin{aligned} \sum _{k\in \mathbb {Z}}2^{kp}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^k},\mathbb R_{+}^d)&=(1-2^{-p})^{-1}\sum _{k\in \mathbb {Z}} \int _{2^{k-1}}^{2^k}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^k},\mathbb R_{+}^d)dt^p\\&\le (1-2^{-p})^{-1}\sum _{k\in \mathbb {Z}}\int _{2^{k-1}}^{2^k}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p\\&=(1-2^{-p})^{-1}\int _{0}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p. \end{aligned} \end{aligned}$$

(43)

Combining (41), (42) with (43) implies

$$\begin{aligned} \int _{\mathbb R_{+}^d}|f|^q d\mu \le C_1 2^{q+1}(1-2^{-p})^{-\frac{q}{p}}\bigg ( \int _{0}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p\bigg )^{\frac{q}{p}}, \end{aligned}$$

which, together with Lemma 3.4, further gives (40). Thus, (ii) is valid and

$$\begin{aligned} C_2^q\le 2^{q+1}(1-2^{-p})^{-\frac{q}{p}}\tilde{C} C_1, \end{aligned}$$

where $\tilde{C}$ is the positive constant determined in Lemma 3.4.

(ii) $\Rightarrow$ (i). By Theorem 3.1(i), we only need to prove that (i) holds for $\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(\cdot ,\mathbb R_{+}^d)$. Let K be a compact subset of $\mathbb R_{+}^d$. For any $f\in \mathcal {A}(K)$ we have

$$ \begin{aligned} f\in C_c^1(\mathbb R_{+}^d)\quad \& \quad f|_{K}\ge 1. \end{aligned}$$

It is easy to verify that (40) holds for such f. From this and the fact that $\mu$ is nonnegative, it follows that

$$\begin{aligned} (\mu (K))^{\frac{1}{q}}\le \bigg (\int _{\mathbb R_{+}^d}|f|^q d\mu \bigg )^{\frac{1}{q}}\le C_2\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}. \end{aligned}$$

By taking infimum over all such $f\in \mathcal {A}(K)$, we obtain

$$\begin{aligned} (\mu (K))^{\frac{1}{q}}\le C_2(\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(E,\mathbb R_{+}^d))^{\frac{1}{p}}. \end{aligned}$$

Therefore, (i) holds with $C_1\le C_2^q$. $\square$

3.5 Trace Inequality for $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$ Under $0<q<p<\infty$

In this subsection, we give the second restriction/trace result for $W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)$. We refer the readers to [23, Theorem 4.3.1] for the case of the Gaussian space.

Theorem 3.3

Let $p\in [1,\infty )$, $0<q<p<\infty$, and $\mu$ be a nonnegative Radon measure. Then the following two conditions are equivalent.

• (i) The function

  $$\begin{aligned} (0,\infty )\ni t\mapsto h_{\mu ,p}(t):=\inf \{\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K,\mathbb R_{+}^d):\ \mathbb R_{+}^d\supset K\ \textrm{is}\ \textrm{compact}\ \textrm{with}\ \mu (K)\ge t\} \end{aligned}$$

  satisfies

  $$\begin{aligned} \Vert h_{\mu ,p}\Vert :=\bigg (\int _{0}^{\infty }\frac{ds^{\frac{p}{p-q}}}{(h_{\mu ,p}(s))^{ \frac{q}{p-q}}}\bigg )^{\frac{p-q}{p}}<\infty . \end{aligned}$$

  (44)

• (ii) There exists a positive constant C such that

  $$\begin{aligned} \bigg (\int _{\mathbb R_{+}^d}|f|^q d\mu \bigg )^{\frac{1}{q}}\le C\Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)} \quad \forall \ f\in C^0(\mathbb R_{+}^d)\cap W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

  Moreover,

  $$\begin{aligned} \Vert h_{\mu ,p}\Vert \approx C^q \end{aligned}$$

  whose implicit constants depend only on p and q.

Proof

Assume that (i) holds. At this time, we prove (ii). Fix

$$\begin{aligned} f\in C^0(\mathbb R_{+}^d)\cap W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

For $t\in (0,\infty )$, define the set

$$\begin{aligned} E_t:=\{x\in \mathbb R_{+}^d:\ |f(x)|>t\}. \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} \int _{\mathbb R_{+}^d}|f|^q d\mu&\le \sum _{k\in \mathbb {Z}}\int _{2^k}^{2^{k+1}}\mu (E_t)dt^q\\&\le (2^q-1)\sum _{k\in \mathbb {Z}}2^{kq}\mu (E_{2^k})\\&=(2^q-1)\sum _{k\in \mathbb {Z}}\sum _{j=k}^{\infty }2^{kq}\mu (E_{2^j}\setminus E_{2^{j+1}})\\&\lesssim \sum _{j\in \mathbb {Z}}2^{jq}(\mu (E_{2^j})-\mu (E_{2^{j+1}})). \end{aligned} \end{aligned}$$

(45)

Since $q<p$, by the Hölder inequality, we have

$$\begin{aligned} \begin{aligned}&{ {\sum _{j\in \mathbb {Z}}2^{jq}(\mu (E_{2^j})-\mu (E_{2^{j+1}}))}}\\&\quad =\sum _{j\in \mathbb {Z}}\bigg (2^{jp}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j},\mathbb R_{+}^d)\bigg )^{\frac{q}{p}} \frac{\mu (E_{2^j})-\mu (E_{2^{j+1}})}{(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j},\mathbb R_{+}^d))^{\frac{q}{p}}}\\&\quad \le \bigg (\sum _{j\in \mathbb {Z}}2^{jp}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j},\mathbb R_{+}^d)\bigg )^{\frac{q}{p}} \bigg (\sum _{j\in \mathbb {Z}}\bigg (\frac{\mu (E_{2^j})-\mu (E_{2^{j+1}})}{(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j}, \mathbb R_{+}^d))^{\frac{q}{p}}}\bigg )^{\big (\frac{p}{q}\big )^{\prime }}\bigg )^{\frac{p-q}{p}}. \end{aligned} \end{aligned}$$

(46)

Combining Proposition 3.2 (ii) with Lemma 3.4, we obtain

$$\begin{aligned} \sum _{j\in \mathbb {Z}}2^{jp}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j},\mathbb R_{+}^d)&=(1-2^{-p})^{-1}\sum _{j\in \mathbb {Z}} \int _{2^{j-1}}^{2^j}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j},\mathbb R_{+}^d)dt^p\\&\le (1-2^{-p})^{-1}\sum _{j\in \mathbb {Z}}\int _{2^{j-1}}^{2^j}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p\\&=(1-2^{-p})^{-1}\int _{0}^{\infty }\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_t,\mathbb R_{+}^d)dt^p\\&\lesssim \Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p, \end{aligned}$$

which derives

$$\begin{aligned} \bigg (\sum _{j\in \mathbb {Z}}2^{jp}\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j},\mathbb R_{+}^d)\bigg )^{\frac{q}{p}}\lesssim \Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^q. \end{aligned}$$

(47)

As for the other part

$$\begin{aligned} \bigg (\sum _{j\in \mathbb {Z}}\bigg (\frac{\mu (E_{2^j})-\mu (E_{2^{j+1}})}{(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j}, \mathbb R_{+}^d))^{\frac{q}{p}}}\bigg )^{\big (\frac{p}{q}\big )^{\prime }}\bigg )^{\frac{p-q}{p}} \end{aligned}$$

in (46). Since $f\in C^{0}(\mathbb R_{+}^d)$, it follows that every set $E_{2^j}$ is open. Since $\mu$ is a Radon measure, there exists a compact set $K_{j}\subseteq E_{2^j}$ such that

$$\begin{aligned} 2^{-1}\mu (E_{2^j})\le \mu (K_{j})\le \mu (E_{2^j}). \end{aligned}$$

Furthermore, according to the definition of the function $h_{\mu ,p}$, we see that

$$\begin{aligned} h_{\mu ,p}(2^{-1}\mu (E_{2^j}))\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(K_{j},\mathbb R_{+}^d)\le \textrm{Cap}_p^{\mathcal {L}^{\alpha }}(E_{2^j},\mathbb R_{+}^d), \end{aligned}$$

where the second inequality holds due to Proposition 3.2 (ii). It follows from this,

$$\begin{aligned} \bigg (\frac{p}{q}\bigg )^{\prime }=\frac{p}{p-q}>1, \end{aligned}$$

the monotone increasing property of $h_{\mu ,p}$, and (44) that

$$\begin{aligned} \begin{aligned} \sum _{j\in \mathbb {Z}}\bigg (\frac{\mu (E_{2^j})-\mu (E_{2^{j+1}})}{(\textrm{Cap}_p^{\mathcal {L}^{\alpha }}( E_{2^j},\mathbb R_{+}^d))^{\frac{q}{p}}}\bigg )^{\big (\frac{p}{q}\big )^{\prime }}&\le \sum _{j\in \mathbb {Z}}\frac{(\mu (E_{2^j})-\mu (E_{2^{j+1}}))^{\frac{p}{p-q}}}{ (h_{\mu ,p}(2^{-1}\mu (E_{2^j})))^{\frac{q}{p-q}}}\\&\le \sum _{j\in \mathbb {Z}}\frac{(\mu (E_{2^j}))^{\frac{p}{p-q}} -(\mu (E_{2^{j+1}}))^{\frac{p}{p-q}}}{(h_{\mu ,p}(2^{-1}\mu (E_{2^j})))^{\frac{q}{p-q}}}\\&=2^{\frac{p}{p-q}}\sum _{j\in \mathbb {Z}}\int _{2^{-1}\mu (E_{2^{j+1}})}^{2^{-1}\mu (E_{2^j})}\frac{ds^{\frac{p}{p-q}}}{(h_{\mu ,p}(2^{-1}\mu (E_{2^j})) )^{\frac{q}{p-q}}}\\&\le 2^{\frac{p}{p-q}}\int _{0}^{\infty }\frac{ds^{\frac{p}{p-q}}}{(h_{\mu ,p}(s))^{ \frac{q}{p-q}}}\\&=2^{\frac{p}{p-q}}\Vert h_{\mu ,p}\Vert ^{\frac{p}{p-q}}. \end{aligned} \end{aligned}$$

(48)

Inserting the estimates of (47) and (48) into (46) leads to

$$\begin{aligned} { {\sum _{j\in \mathbb {Z}}2^{jq}(\mu (E_{2^j})-\mu (E_{2^{j+1}}))}} \lesssim \Vert h_{\mu ,p}\Vert \Vert f\Vert _{ W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^q, \end{aligned}$$

which combined with (45) further implies that

$$\begin{aligned} \int _{\mathbb R_{+}^d}|f|^q d\mu \lesssim \Vert h_{\mu ,p}\Vert \Vert f\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^q. \end{aligned}$$

Thus (ii) is valid with

$$\begin{aligned} C\lesssim \Vert h_{\mu ,p}\Vert ^{\frac{1}{q}}. \end{aligned}$$

Now we show that (ii) implies (i). Via Theorem 3.1(i), $\textrm{Cap}_p^{\mathcal {L}^{\alpha }}$ in the definition of $h_{\mu ,p}$ can be replaced by $\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}$. Observe that if $\mu (\mathbb R_{+}^d)\le t$, then we tacitly approve that $h_{\mu ,p}(t)=\infty$. Thus

$$\begin{aligned} \Vert h_{\mu ,p}\Vert =\bigg (\int _{0}^{\mu (\mathbb R_{+}^d)}\frac{ds^{\frac{p}{p-q}}}{(h_{\mu ,p}(s) )^{\frac{q}{p-q}}}\bigg )^{\frac{p-q}{p}}. \end{aligned}$$

If $\mu (\mathbb R_{+}^d)<\infty$, there exists a unique $J_0\in \mathbb {Z}$ such that

$$\begin{aligned} 2^{J_0}<\mu (\mathbb R_{+}^d)\le 2^{J_0+1}. \end{aligned}$$

If $\mu (\mathbb R_{+}^d)=\infty$, we let $J_0=\infty$. Since $h_{\mu ,p}$ is an increasing function, we have

$$\begin{aligned} \begin{aligned} \Vert h_{\mu ,p}\Vert&=\bigg (\sum _{j=-\infty }^{J_0}\int _{2^j}^{2^{j+1}}\frac{ds^{\frac{p}{p-q}}}{(h_{\mu ,p}(s))^{\frac{q}{p-q}}}\bigg )^{\frac{p-q}{p}}\\&\le (2^{\frac{p}{p-q}}-1)^{\frac{p-q}{p}}\bigg (\sum _{j=-\infty }^{J_0}\frac{2^{j\frac{p}{p-q}}}{(h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}\bigg )^{\frac{p-q}{p}}. \end{aligned} \end{aligned}$$

(49)

For each $j\in \mathbb {Z}$ such that $j<J_0+1$, by the definition of $h_{\mu ,p}$ and Theorem 3.1(i), there exists a compact set $K_j\subseteq \mathbb R_{+}^d$ such that

$$\begin{aligned} \mu (K_j)\ge 2^j \end{aligned}$$

and

$$\begin{aligned} \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_j,\mathbb R_{+}^d)-\epsilon h_{\mu ,p}(2^j)\le h_{\mu ,p}(2^j)\le \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_j,\mathbb R_{+}^d) \end{aligned}$$

(50)

for any $\epsilon \in (0,\infty )$. Moreover, via the definition of $\textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_j,\mathbb R_{+}^d)$, there exists a function $f_j\in \mathcal {A}(K_j)$ such that

$$\begin{aligned} \Vert f_j\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p-2^{-j}\epsilon \le \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_j,\mathbb R_{+}^d)\le \Vert f_j\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p. \end{aligned}$$

(51)

Define

$$\begin{aligned} F_{l,m}:=\max _{l\le j\le m}\gamma _{j}f_j\quad \textrm{with}\quad \gamma _{j}:=\bigg (\frac{2^j}{h_{\mu ,p}(2^j)}\bigg )^{\frac{1}{p-q}}. \end{aligned}$$

Since

$$\begin{aligned} f_j\in C^0(\mathbb R_{+}^d)\Rightarrow F_{l,m}\in C^0(\mathbb R_{+}^d), \end{aligned}$$

we apply

$$\begin{aligned} f_j\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d) \end{aligned}$$

and Lemma 3.1 to deduce that for any $i=1,2,\ldots ,d$, $\delta _{i}$ exists a.e. on $\mathbb R_{+}^d$ and that

$$\begin{aligned} |\delta _i F_{l,m}|\le \max _{l\le j\le m}\gamma _{j}|\delta _i f_j| \quad \mathrm {a.e.} \ \textrm{on}\ \mathbb R_{+}^d. \end{aligned}$$

Then it is obvious that

$$\begin{aligned} \Vert F_{l,m}\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}\le \sum _{j=l}^{m}\gamma _{j}\Vert f_j\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }} (\mathbb R_{+}^d)}<\infty . \end{aligned}$$

Hence,

$$\begin{aligned} F_{l,m}\in W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d). \end{aligned}$$

It follows from (ii) that

$$\begin{aligned} \int _{\mathbb R_{+}^d}|F_{l,m}|^q d\mu \le C^q\Vert F_{l,m}\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^q. \end{aligned}$$

(52)

In what follows, we compute the left side of (52), using the nonincreasing rearrangement of $F_{l,m}$ implies

$$\begin{aligned} \int _{\mathbb R_{+}^d}|F_{l,m}|^q d\mu= & {} \int _{0}^{\infty }\bigg (\inf \{s>0:\ \mu (\{x\in \mathbb R_{+}^d:\ |F_{l,m}(x)|>s\})\le t\}\bigg )^q dt\nonumber \\= & {} \sum _{j\in \mathbb {Z}}\int _{2^{j-1}}^{2^j}\bigg (\inf \{ s>0:\ \mu (\{x\in \mathbb R_{+}^d:\ |F_{l,m}(x)|>s\})\le t\}\bigg )^q dt\nonumber \\\ge & {} \sum _{j=l}^{m}2^{j-1}\bigg (\inf \{s>0:\ \mu (\{x\in \mathbb R_{+}^d:\ |F_{l,m}(x)|>s\})\le 2^j\}\bigg )^q.\nonumber \\ \end{aligned}$$

(53)

For every $x\in K_j$ with $l\le j\le m$, we can check that

$$\begin{aligned} F_{l,m}(x)\ge \gamma _{j}f_j(x)\ge \gamma _j. \end{aligned}$$

(54)

For any small number $\eta >0$, via (54) we have

$$\begin{aligned} \mu (\{x\in \mathbb R_{+}^d:\ |F_{l,m}(x)|>\gamma _{j}-\eta \})\ge \mu (K_j)\ge 2^j, \end{aligned}$$

therefore,

$$\begin{aligned} \inf \{s>0:\ \mu (\{x\in \mathbb R_{+}^d:\ |F_{l,m}(x)|>s\})\le 2^j\}\ge \gamma _{j}-\eta . \end{aligned}$$

Thereby, letting $\eta \rightarrow 0$ gives

$$\begin{aligned} \inf \{s>0:\ \mu (\{x\in \mathbb R_{+}^d:\ |F_{l,m}(x)|>s\})\le 2^j\}\ge \gamma _{j}\quad \forall \ l\le j\le m. \end{aligned}$$

This, along with (53) deserves

$$\begin{aligned} \int _{\mathbb R_{+}^d}|F_{l,m}|^q d\mu \ge \sum _{j=l}^{m}2^{j-1}\gamma _{j}^q=2^{-1} \sum _{j=l}^m\frac{2^{j\frac{p}{p-q}}}{(h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}. \end{aligned}$$

(55)

For the right side of (52), it is easy to see that

$$\begin{aligned} \Vert F_{l,m}\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^q&=\bigg (\Vert F_{l,m}\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p +\sum _{i=1}^d\Vert \delta _i F_{l,m}\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\bigg )^{\frac{q}{p}}\\&\le \bigg (\sum _{j=l}^m\gamma _{j}^p\Vert f_j\Vert _{L^{p}(\mathbb R_{+}^d,d\mu _{\alpha })}^p +\sum _{i=1}^d\sum _{j=l}^{m}\gamma _{j}^p\Vert \delta _i f_j\Vert _{L^p(\mathbb R_{+}^d,d\mu _{\alpha })}^p\bigg )^{\frac{q}{p}}\\&=\bigg (\sum _{j=l}^m\gamma _{j}^p\Vert f_j\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p \bigg )^{\frac{q}{p}}. \end{aligned}$$

It follows from (51) and (50) that

$$\begin{aligned} \Vert f_j\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p\le \textrm{Cap}_{0,p}^{\mathcal {L}^{\alpha }}(K_j,\mathbb R_{+}^d)+2^{-j}\epsilon \le h_{\mu ,p}(2^j)+\epsilon h_{\mu ,p}(2^j)+2^{-j}\epsilon . \end{aligned}$$

Letting $\epsilon \rightarrow 0$, we obtain

$$\begin{aligned} \Vert f_j\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p\le h_{\mu ,p}(2^j). \end{aligned}$$

This, together with the definition of $\gamma _{j}$ indicates

$$\begin{aligned} \sum _{j=l}^m\gamma _{j}^p\Vert f_j\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^p\le \sum _{j=l}^m\gamma _{j} ^p h_{\mu ,p}(2^j)=(1+2\epsilon )\sum _{j=l}^m\frac{2^{j\frac{p}{p-q}}}{( h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}, \end{aligned}$$

so that

$$\begin{aligned} \Vert F_{l,m}\Vert _{W^{1,p}_{\mathcal {L}^{\alpha }}(\mathbb R_{+}^d)}^q\le \bigg ((1+2\epsilon ) \sum _{j=l}^m\frac{2^{j\frac{p}{p-q}}}{(h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}\bigg )^{\frac{q}{p}}. \end{aligned}$$

(56)

Inserting the estimates of (55) and (56) into (52), it follows that

$$\begin{aligned} 2^{-1}\sum _{j=l}^m\frac{2^{j\frac{p}{p-q}}}{(h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}\le C^q\bigg ((1+2\epsilon )\sum _{j=l}^m\frac{2^{j\frac{p}{p-q}}}{(h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}\bigg )^{\frac{q}{p}}. \end{aligned}$$

Since $q<p$, we obtain

$$\begin{aligned} \bigg (\sum _{j=l}^m\frac{2^{j\frac{p}{p-q}}}{(h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}\bigg )^{\frac{p-q}{p}}\le 2C^q(1+2\epsilon )^{\frac{q}{p}}. \end{aligned}$$

By taking supremum over all $l,m\in \mathbb {Z}$, we conclude that

$$\begin{aligned} -\infty<l\le m <J_0+1. \end{aligned}$$

By (49),

$$\begin{aligned} \Vert h_{\mu ,p}\Vert \le C_{p,q}\bigg (\sum _{j=-\infty }^{J_0}\frac{2^{j\frac{p}{p-q}}}{(h_{\mu ,p}(2^j))^{\frac{q}{p-q}}}\bigg )^{\frac{p-q}{p}}\le 2C_{p,q}C^q(1+2\epsilon )^{\frac{q}{p}}. \end{aligned}$$

We take $\epsilon \rightarrow 0$ to obtain (i), which completes the proof of the theorem. $\square$