2.1. Finsler manifolds

Let M be a smooth manifold with dimension $n\ge 2$. At each point $x\in M$, the tangent space T_xM consists of tangent vectors (or simply vectors) at x, while the cotangent space $T^*_xM$ consists of cotangent vectors (or covectors) at x, sometimes also referred to as one forms. These spaces are dual to each other, the canonical pairing between them is a bilinear form defined by

\begin{equation*} \langle\xi,y\rangle=\xi(y),\qquad \forall x\in M,\ y\in T_xM,\ \xi\in T^*_xM. \end{equation*}

The tangent bundle TM is the collection of all tangent spaces, consisting of bound vectors, the cotangent bundle $T^*M$ is the collection of all cotangent spaces, consisting of bound covectors. More precisely, we have

\begin{equation*} TM=\{(x,y):x\in M, y\in T_xM\} \quad\mbox{and}\quad T^*M=\{(x,\xi): x\in M, \xi\in T^*_xM\}. \end{equation*}

A vector field on M is a map that assigns to each point $x\in M$ a bound vector $(x,y)\in TM$. Similarly, a covector field assigns to each point $x\in M$ a bound covector $(x,\xi )\in T^*M$.

The couple (M, F) is said to be a Finsler manifold if the function $F\colon TM\to[0,\infty)$ is a Finsler metric satisfying the following conditions:

1. (i) Regularity: F(y) is smooth on $TM\setminus\{0\}$, which consists of non-zero bound vectors;

2. (ii) Positive homogeneity: $F(\lambda y)=\lambda F(y)$, for every $y\in TM$ and $\lambda\ge0$;

3. (iii) Strong convexity: For every $y\in TM\setminus\{0\}$, the following quadratic form is positive definite:

   (7)\begin{equation} g_{y}(v,w)\stackrel{\textrm{def}}{=}\left.\frac{\partial^2}{\partial s\partial t}\left(\frac{1}{2}F(y+sv+tw)^2\right)\right|_{s=t=0},\qquad\forall v,w\in TM. \end{equation}

According to remark 2.1, we use both the notation F(y) and $F(x,y)$ depending on the context.

The reversibility of a Finsler manifold (M, F), introduced by Rademacher [Reference Rademacher10], is defined by

\begin{equation*} \lambda_F(M)\stackrel{\textrm{def}}{=}\sup_{x\in M}\sup_{y\in T_xM} \mathbf{rev}(x,y), \quad\mbox{where}\quad \mathbf{rev}(x,y)=\frac{F(x,-y)}{F(x,y)}. \end{equation*}

Clearly $1\le\lambda_F(M)\le +\infty$. The manifold is said to be reversible, if $\lambda_F(M)=1$.

The dual Finsler metric of F is the function $F^*\colon T^*M\to[0,\infty)$, defined by

(8)\begin{equation} F^*(\xi)\stackrel{\textrm{def}}{=} \sup_{y\in TM\setminus \{0\}}\frac{\langle\xi,y\rangle}{F(y)},\qquad \forall \xi\in T^*M, \end{equation}

which is also a Finsler metric on M. The reversibility of $(M,F^*)$ can be defined similarly by

\begin{equation*} \lambda_{F^*}(M)\stackrel{\textrm{def}}{=} \sup_{x\in M}\sup_{\xi\in T_x^*M} \mathbf{rev}^*(x,y), \quad\mbox{where}\quad \mathbf{rev}^*(x,y)=\frac{F^*(x,-\xi)}{F^*(x,\xi)}. \end{equation*}

In particular, according to e.g., Huang et al. [Reference Huang, Kristály and Zhao5, Lemma 2.1], one has

(9)\begin{equation} \lambda_F(M)=\lambda_{F^*}(M). \end{equation}

Using a Finsler metric F, one can define the length of a Lipschitz continuous curve $c\colon [a,b]\to M$ by

\begin{equation*}L(c)\stackrel{\textrm{def}}{=}\int_a^b F(\dot c(t)) \,\mathrm{d} t,\end{equation*}

where the dot denotes the derivative with respect to t. Given two points $x_0,x_1\in M$, the distance from x ₀ to x ₁ is defined by $d(x_0,x_1)= \inf_c L(c),$ where the infimum is taken over all Lipschitz continuous curves $c\colon [0,1]\to M$ with $c(0)=x_0$ and $c(1)=x_1$. In general, $d(x_0,x_1)\ne d(x_1,x_0)$, unless F is reversible.

If $\rho=d(x_0,\cdot)$ is the distance from a point $x_0\in M$, then the eikonal equations hold $\!\,\mathrm{d}\mathsf{m}$-a.e. in M:

(10)\begin{equation} F(\nabla \rho)=F^*(D\rho)=1. \end{equation}

For fixed $x\in M$, the Legendre transform $\ell^*\colon T^*_xM\to T_xM$ assigns to each covector ξ the unique vector y, which minimizes the map

\begin{equation*} E^*(y)\stackrel{\textrm{def}}{=} \langle \xi, y\rangle - \frac{1}{2}F(y)^2. \end{equation*}

The minimizer $y=\ell^*(\xi)$ can also be interpreted as the unique bound vector with the properties

(11)\begin{equation} F(\ell^*(\xi))=F^*(\xi)\quad\mbox{and}\quad \langle \xi, \ell^*(\xi)\rangle =F(\ell^*(\xi))F^*(\xi). \end{equation}

Combining the properties from relation (11) yields

(12)\begin{equation} \langle \xi,\ell^*(\xi)\rangle=F^*(\xi)^2, \qquad\forall \xi\in T^*_xM. \end{equation}

Similarly, the Legendre transform $\ell\colon T_xM\to T^*_xM$ assigns to each vector y the unique covector ξ that minimizes the map

\begin{equation*} E(\xi)\stackrel{\textrm{def}}{=} \langle \xi, y\rangle - \frac{1}{2}F^*(\xi)^2. \end{equation*}

For the minimizer $\xi=\ell(y)$ similar relations hold as in (11) and (12). One also has $\ell^{-1}=\ell^*$, hence $y=\ell^*(\xi)$. Using the fact that $\frac{\partial E}{\partial \xi}=0$, we obtain

(13)\begin{equation} \ell^*(\xi)=F^*(\xi)\frac{\partial F^*(\xi)}{\partial \xi},\qquad \forall \xi\in T_x^*M. \end{equation}

Thus, $\ell^*$ is positive homogeneous. Note that the definitions and properties of the Legendre transforms $\ell$ and $\ell^*$ naturally extend to bound vectors and vector fields. Moreover, using the conventions of remark 2.1, all of the above formulas are formally valid. We only need to make sure that the positional coordinates of y and ξ coincide, otherwise the canonical pairing does not make sense.

For every $x\in M$, there exists a local coordinate system $\{x^i\}$ defined on a coordinate neighbourhood of x. Denote by $\{\frac{\partial}{\partial x^i}\}$ and $\{\!\,\mathrm{d} x^i\}$ the induced basis on T_xM and $T^*_xM$, respectively.

An FMMM $(M,F, \mathsf{m})$ is a Finsler manifold equipped with a smooth positive measure $\mathsf{m}$. Unlike the Riemannian setting, there is no canonical measure in Finsler geometry. However, one often considers the Busemann–Hausdorff measure induced by F, which is given by

(14)\begin{equation} \,\mathrm{d} \mathsf{m}_F\stackrel{\textrm{def}}{=}\sigma_F(x)\,\mathrm{d} x^1\wedge\ldots\wedge \,\mathrm{d} x^n,\quad\mbox{where}\quad \sigma_F(x)=\frac{\operatorname{vol}(\mathbb{B}^n)}{\operatorname{vol}(B_x(1))}, \end{equation}

$\operatorname{vol}(\cdot)$ denotes the Euclidean volume, $\mathbb{B}^n\subset \mathbb{R}^n$ is the Euclidean unit ball, while

\begin{equation*} B_x(1)=\{y\in T_xM: F(y) \lt 1\}. \end{equation*}

The derivative of a smooth function $u\colon M\to \mathbb{R}$ is a covector field defined by

(15)\begin{equation} D u\stackrel{\textrm{def}}{=} \frac{\partial u}{\partial x^i}\,\mathrm{d} x^i. \end{equation}

The gradient of a smooth function $u\colon M\to \mathbb{R}$ is a vector field that can be computed as

(16)\begin{equation} \nabla u\stackrel{\textrm{def}}{=}\ell^*(D u). \end{equation}

The divergence of a smooth vector field v with components $(v^i)$ is given by

\begin{equation*} \mathrm{div}(v)\stackrel{\textrm{def}}{=} \sum_i \frac{1}{\sigma}\frac{\partial \sigma v^i}{\partial x^i}. \end{equation*}

If p > 1, then the p-Laplacian of a smooth function u is

\begin{equation*} \Delta_{F,p} (u)\stackrel{\textrm{def}}{=} \mathrm{div}(F(\nabla u)^{p-2}\nabla u). \end{equation*}

We note that for p = 2, the p-Laplacian is precisely the Finsler Laplacian

\begin{equation*}\Delta_F (u) = \mathrm{div}(\nabla u).\end{equation*}

We also notice that, if ρ is a distance function, then $\Delta_{F,p}(\rho)=\Delta_F(\rho)$ holds $\!\,\mathrm{d}\mathsf{m}$-a.e. in M.

Finally, if u ₁ and u ₂ are smooth functions, both vanishing at the boundary, then the following integration by parts formula holds:

(17)\begin{equation} \int_M u_1\Delta_F (u_2)\,\mathrm{d}\mathsf{m}=-\int_M \langle D u_1,\nabla u_2\rangle\,\mathrm{d}\mathsf{m}, \end{equation}

see e.g., Ohta and Sturm [Reference Ohta and Sturm9].

2.2. Curvatures and Laplace comparison

In the sequel, we recall the notions of the flag and S-curvature of an FMMM $(M,F,\mathsf{m})$, which can be interpreted more easily using tensors. According to the notation of the previous section, the fundamental tensor from relation (7) can be rewritten as

\begin{equation*} g_{(x,y)}=g_{ij}(x,y)\,\mathrm{d} x^i\,\mathrm{d} x^j,\quad\mbox{where}\quad g_{ij}(x,y)\stackrel{\textrm{def}}{=}\frac{1}{2}\frac{\partial^2 F(x,y)^2}{\partial y^i\partial y^j},\qquad \forall (x,y)\in TM. \end{equation*}

Note that, by convention, repeated indexes are automatically summed. The geodesic spray coefficients are defined as follows:

\begin{equation*} G^i(x,y)\stackrel{\textrm{def}}{=}\frac{1}{4}g^{il}(x,y)\left(2\frac{\partial g_{jl}(x,y)}{\partial x^k}- \frac{\partial g_{jk}(x,y)}{\partial x^l}\right)y^jy^k,\qquad \forall (x,y)\in TM. \end{equation*}

Recall that a smooth curve $t\mapsto \gamma(t)$ is geodesic if it satisfies the equation

\begin{equation*} \ddot \gamma^i(t)+2G^i(\gamma(t),\dot \gamma(t))=0. \end{equation*}

The Riemannian curvature is defined by

\begin{equation*} R_{(x,y)}\stackrel{\textrm{def}}{=} R^i_k(x,y)\frac{\partial}{\partial x^i} \otimes dx^k, \qquad \forall (x,y)\in TM, \end{equation*}

where

\begin{equation*} R^i_k(x,y)\stackrel{\textrm{def}}{=} 2 \frac{\partial G^i(x,y)}{\partial x^k}-y^i\frac{\partial^2 G^i(x,y)}{\partial x^j\partial y^k}+2G^j(x,y)\frac{\partial^2 G^i(x,y)}{\partial y^j\partial y^k}-\frac{\partial G^i(x,y)}{\partial y^j}\frac{\partial G^j(x,y)}{\partial y^k}. \end{equation*}

The definition of flag curvature is as follows:

(18)\begin{equation} \mathbf{K}(x,y,v)=\frac{g_{(x,y)}(R_{(x,y)}(v),v)}{g_{(x,y)}(y,y)g_{(x,y)}(v,v)-g_{(x,y)}(y,v)^2},\qquad \forall x\in M,\ y,v\in T_xM. \end{equation}

The name ‘flag’ refers to the plane spanned by the vectors y and v; the vector y is sometimes referred to as the flagpole.

The distortion is defined by

\begin{equation*} \tau(x,y)\stackrel{\textrm{def}}{=}\log\frac{\sqrt{\det g_{ij}(x,y)}}{\sigma(x)},\qquad \forall (x,y)\in TM. \end{equation*}

The S-curvature measures the rate of change of the distortion along geodesics, more precisely

(19)\begin{equation} \mathbf{S}(x,y)=\left.\frac{\,\mathrm{d}}{\,\mathrm{d} t}\tau(\gamma(t),\dot\gamma(t))\right|_{t=0},\qquad \forall (x,y)\in TM, \end{equation}

where γ denotes the unique geodesic with $\gamma(0)=x$ and $\dot\gamma(0)=y$. For computational convenience (due to the homogeneity of degree zero), we prefer to use the reduced S-curvature defined by

\begin{equation*} \overline{\mathbf{S}}(x,y)=\frac{\mathbf{S}(x,y)}{F(x,y)},\quad \forall (x,y)\in TM,\ y\ne 0. \end{equation*}

For every $\kappa\ge 0$, define $\mathbf{ct}_\kappa\colon (0,\infty)\to \mathbb{R}$ by

\begin{equation*}{\mathbf{ct}}_\kappa(t)= \begin{cases} \frac{1}{t}, & \mbox{if } \kappa=0, \\ \kappa\coth(\kappa t), & \mbox{if } \kappa \gt 0. \\ \end{cases} \end{equation*}

Our forthcoming arguments rely deeply on the following Laplace comparison theorem, which is a slight modification of the celebrated result by Wu and Xin [Reference Wu and Xin13, Theorem 5.1]. Here we only assume curvature conditions alongside the geodesics starting from some $x_0\in M$, which does not affect the original proof.

Theorem 2.2. Let (M, F) be a Finsler manifold and $\rho=d(x_0,\cdot)$ be the distance from a point $x_0\in M$. Suppose that $\mathbf{K}\le -\kappa^2$ and $\overline{\mathbf{S}}\le (n-1)h$ along $\nabla\rho$, for some $\kappa\ge 0$ and $h\in \mathbb{R}$. Then one has

\begin{equation*} \Delta_F (\rho)\ge (n-1)\mathbf{ct}_\kappa(\rho)-(n-1)h. \end{equation*}

Moreover, equality holds if and only if $\mathbf{K}= -\kappa^2$ and $\overline{\mathbf{S}}= (n-1)h$, along $\nabla\rho$.

2.3. Special Finsler metrics

A Finsler metric on $\Omega\subseteq\mathbb{R}^n$ is said to be projectively flat if the geodesics are straight (Euclidean) lines as point sets. Due to Rapcsák [Reference Rapcsák11], we have the following characterization.

The following proposition is useful for computing the flag and S-curvature of a projectively flat metric. For the proof, we refer to Shen [Reference Shen12, Sections 6.3 and 7.3]

Proposition 2.4. Suppose that $(M,F,\mathsf{m})$ is an FMMM equipped with a projectively flat Finsler metric F and a smooth positive measure $\mathsf{m}$ having density σ. The flag and S-curvature can be computed as follows:

\begin{align*} \mathbf{K}(x,y) & =\frac{1}{F(x,y)^2}\left(P(x,y)^2-\frac{\partial P(x,y)}{\partial x^i}y^i\right), \\ \mathbf{S}(x,y) & =\frac{\partial G^i(x,y)}{\partial y^i}-\frac{y^i}{\sigma(x)}\frac{\partial \sigma(x)}{\partial x^i}, \end{align*}

where P and Gⁱ are given by relation (20). Note that in this case K only depends on x and y.

In the sequel, we consider a special class of Finsler spaces, called Randers spaces, which are important examples of Finsler manifolds: In many cases, they help us to observe and investigate conceptual differences between Riemannian and Finsler geometries. A Randers space is a differentiable manifold M endowed with a special Finsler metric $F\colon TM\to[0,\infty)$ called Randers metric, which is defined by

(22)\begin{equation} F(x,y)=\sqrt{a_{ij}(x)y^iy^j}+b_i(x)y^i, \end{equation}

where $a_{ij}(x)\,\mathrm{d} x^i\,\mathrm{d} x^j$ is a Riemannian metric, $b_i(x)\,\mathrm{d} x^i$ is a a one-form, such that

\begin{equation*} \|b(x)\|_a\stackrel{\textrm{def}}{=}\sqrt{a^{ij}(x)b_i(x)b_j(x)} \lt 1, \quad\forall x\in M, \end{equation*}

and $\{a^{ij}(x)\}$ denotes the inverse of $\{a_{ij}(x)\}$. The density of the Busemann–Hausdorff measure induced by a Raders metric F is given by

(23)\begin{equation} \sigma_F(x)=(1-\|b(x)\|_a^2)^\frac{n+1}{2}\sqrt{\det a_{ij}(x)}. \end{equation}

For later use, define the Kronecker delta by

(24)\begin{equation} \delta_{ij}\stackrel{\textrm{def}}{=} \begin{cases} 1, & \mbox{if } i=j, \\ 0, & \mbox{if } i\ne j. \end{cases} \end{equation}

The following observation proves to be useful in the sequel.

3.1. Applications

In this section, by applying theorem 3.1, we prove a Finslerian version of a weighted Hardy inequality and a McKean-type spectral gap estimate. We note that our choice of parameters agrees with the corresponding Riemannian counterparts from [Reference Kajántó, Kristály, Peter and Zhao6]. The first application can be stated as follows.

Theorem 3.3. Let $(M,F,\mathsf{m})$ be a forward complete, non-compact FMMM, with dimension $n\ge 3$, and reversibility $\lambda \lt \infty$. Let $\Omega\subseteq M$ be a domain, $x_0\in M$ and $\rho=d(x_0,\cdot)$ the distance from x ₀. Suppose that $\mathbf{K}\le 0$, $\overline{\mathbf{S}}\le 0$ along $\nabla \rho$. Let $\alpha,p\in \mathbb{R}$ such that $1 \lt p \lt n+\alpha$. Then for every $u\in C_0^\infty(\Omega)$ one has

(32)\begin{equation} \lambda^p\displaystyle\int_\Omega \rho^\alpha\cdot F^*(Du)^p\,\mathrm{d} \mathsf{m} \ge\displaystyle\int_\Omega \rho^\alpha\cdot \max\{F^*(\pm Du)\}^p\,\mathrm{d} \mathsf{m} \ge c_{n,p,\alpha}\displaystyle\int_\Omega \rho^\alpha\frac{|u|^p}{\rho^p}\,\mathrm{d} \mathsf{m}, \end{equation}

where $c_{n,p,\alpha}=\left(\frac{n+\alpha-p}{p}\right)^p$.

Proof. In theorem 3.1 let us choose $\Omega=M\setminus\{x_0\}$, as well as,

\begin{align*} &w(t)=t^\alpha,\quad L(t)=\frac{n-1}{t},\quad W(t)= c_{n,p,\alpha}\cdot \frac{1}{t^p} ,\\ & \quad \mbox{and } G(t)=(c_{n,p,\alpha})^\frac{p-1}{p}\cdot\frac{1}{t^{p-1}},\quad\forall t \gt 0.\end{align*}

Obviously, the above functions satisfy (C1). The Laplace comparison (see theorem 2.2) implies that (C2) holds as well. By a simple computation, the Riccati ODI of condition (C3) is verified with equality. Thus, theorem 3.1 implies that inequality (32) holds for every $u\in C_0^\infty(M\setminus\{x_0\})$. Finally, we use a result from capacity theory (see e.g., [Reference Kajántó, Kristály, Peter and Zhao6, Remark 4.2]): Since the p-capacity of $\rho^{-1}(0)\subset M$ is zero, thus, inequality (32) holds for every $u\in C_0^\infty(M)$ as well.

We notice that Zhao in [Reference Zhao15, Theorem 1.1] showed that if $\mathbf{K}\le 0$ (globally) and $\mathbf{S}\ge0$ along $\nabla \overleftarrow\rho$, where $\overleftarrow\rho=d(\cdot, x_0)$, then one has

\begin{equation*} \int_\Omega \overleftarrow\rho^\alpha\cdot \max\{F^*(\pm Du)\}^2\,\mathrm{d} \mathsf{m}\ge c_{n,p,\alpha}\int_\Omega\overleftarrow\rho^\alpha\frac{|u|^p}{\overleftarrow\rho^p}\,\mathrm{d} \mathsf{m},\quad \forall u\in C_0^\infty(M); \end{equation*}

additionally, if F is reversible, then the constant $c_{n,p,\alpha}$ is sharp. Since, $\mathbf{S}\ge 0$ along $\nabla\overleftarrow\rho$ if and only if $\mathbf{S}\le 0$ along $\nabla \rho$, this result perfectly aligns with inequality (32).

The second application can be stated as follows.

Theorem 3.4. Let $(M,F,\mathsf{m})$ be a forward complete, non-compact FMMM, with dimension $n\ge 3$, and reversibility $\lambda \lt \infty$. Let $\Omega\subseteq M$ be a domain, $x_0\in M$ and $\rho=d(x_0,\cdot)$ the distance from x ₀. Suppose that $\mathbf{K}\le -\kappa^2$, $\overline{\mathbf{S}}\le (n-1)h$ along $\nabla \rho$, for some $\kappa \gt h\ge0$, and let p > 1. Then for every $u\in C_0^\infty(\Omega)$ one has

(33)\begin{equation} \lambda^p\int_\Omega F^*(Du)^p\,\mathrm{d} \mathsf{m} \ge\int_\Omega \max\{F^*(\pm Du)\}^p\,\mathrm{d} \mathsf{m} \ge c_{n,p,\kappa,h} \int_\Omega |u|^p\,\mathrm{d} \mathsf{m}, \end{equation}

where $c_{n,p,\kappa,h}=\left(\frac{(n-1)(\kappa-h)}{p}\right)^p$.

Proof. In theorem 3.1 let us choose $\Omega=M$, as well as

\begin{equation*} w(t)\equiv 1, \quad L(t)\equiv (n-1)(\kappa-h) , \quad W(t)\equiv c_{n,p,\kappa,h},\quad\mbox{and}\quad G(t)\equiv (c_{n,p,\kappa,h})^\frac{p-1}{p}. \end{equation*}

These functions satisfy (C1), the Laplace comparison theorem (see theorem 2.2) implies

\begin{equation*} \Delta_F (\rho)\ge (n-1)(\kappa\coth(\kappa \rho)-h)\ge (n-1)(\kappa-h) = L, \end{equation*}

hence (C2) holds as well. The Riccati ODI from (C3) is verified with inequality, thus, theorem 3.1 proves inequality (33).

4.1. Construction

According to the proof of theorem 3.1 (see also remark 3.2(e)) an ideal space for discussing the sharpness of Hardy-type inequalities has a radially symmetric metric F and constant K, $\overline{\mathbf{S}}$, and rev along geodesics starting from a point x ₀. Unfortunately, such spaces are not available in the Finsler setting; however, we can make the following observation.

Consider the triple $(\mathbb{R}^n,F,\mathsf{m})$, where $F\colon \mathbb{R}^n\times \mathbb{R}^n\to [0,\infty)$ is defined by

(34)\begin{equation} F(x,y)=|y|-\frac{\langle x,y\rangle\theta}{|x|},\quad\text{if } x\ne O\quad\text{and}\quad F(O,y)=|y|, \end{equation}

for some $\theta\in(0,1)$ and $\mathsf{m}$ stands for the Busemann–Hausdorff measure induced by F. Here, $O\in \mathbb{R}^n$ is the origin, $|\cdot|$ and $\langle\cdot,\cdot \rangle$ denote the norm and the inner product in $\mathbb{R}^n$. The function F is not a Finsler metric, since it is not continuous at the origin. However, it has various remarkable properties (away from O).

On the one hand, since relation (21) holds, theorem 2.3 yields projectively flatness, i.e., all the geodesics are straight (Euclidean) lines. Denote $\rho=d(O,\cdot)$ the Finslerian distance from the origin. Since the geodesics starting from the origin are precisely the integral curves of $\nabla \rho$, it follows that x is parallel with $\nabla\rho$. Using proposition 2.4 and the homogeneity of the flag curvature and the reduced S-curvature we obtain

(35)\begin{equation} \mathbf{K}(x,\nabla\rho(x))=\mathbf{K}(x,x)=0\quad\text{and}\quad \overline{\mathbf{S}}(x,\nabla\rho(x))=\overline{\mathbf{S}}(x,x)=0,\quad \forall x\in \mathbb{R}^n, x\ne O, \end{equation}

which precisely means that $\mathbf{K}=0$, $\overline{\mathbf{S}}=0$ along $\nabla \rho$.

On the other hand, by an easy computation we obtain for every $x,y\in \mathbb{R}^n$ that

(36)\begin{equation} \textbf{rev}(x,y)\le \textbf{rev}(x,x)=\frac{1+\theta}{1-\theta}\stackrel{\textrm{def}}{=} \lambda,\quad\text{if }x\ne O,\quad\text{and}\quad \mathbf{rev}(O,y)=1, \end{equation}

hence $\lambda_F(\mathbb{R}^n)=\lambda\in(1,\infty)$. If $u=v(\rho)$ is a radially symmetric function with $v' \lt 0$, then one has equality in (31). The latter provides the difference between the two common forms of Hardy inequalities in the Finsler setting, compare e.g., inequalities (2) and (3).

To remedy the discontinuity at the origin, one could simply puncture the space (considering only $\mathbb{R}^n\setminus\{O\}$), which results in a loss of computational convenience: By construction, the distance from O is easier to compute than the distance from any other base point. An other option (what is chosen here) is to ‘smooth out’ the jump at the origin, which can be done by a careful approximation of F, partially retaining the aforementioned favourable properties. We have the following result.

Theorem 4.1. Let $\lambda\in (1,\infty)$ and $\theta=\frac{\lambda-1}{\lambda+1}$. Construct the family $\mathcal{F}_{0,0,\lambda}=\{(\mathbb{R}^n,F_\varepsilon,\mathsf{m}_\varepsilon)\}_{\varepsilon \gt 0}$ of FMMMs, where $F_\varepsilon\colon T\mathbb{R}^n\simeq\mathbb{R}^n\times\mathbb{R}^n\to [0,\infty)$ is defined by

(37)\begin{equation} F_\varepsilon(x,y)=|y|-\frac{\langle x,y\rangle(|x|+2\varepsilon)\theta}{(|x|+\varepsilon)^2}, \end{equation}

and $\mathsf{m}_\varepsilon$ is the Busemann–Hausdorff measure induced by F_ɛ. The following statements hold:

1. (i) For every ɛ > 0, the triple $(\mathbb{R}^n, F_\varepsilon,\mathsf{m}_\varepsilon)$ is an FMMM. In addition, F_ɛ is a projectively flat Randers metric, with reversibility $\lambda_{F_\varepsilon}(\mathbb{R}^n)=\lambda$.

2. (ii) For every $x\in \mathbb{R}^n$ with $x\ne 0$ one has

   \begin{equation*} \mathbf{K}_\varepsilon(x,x)\to 0,\quad \overline{\mathbf{S}}_\varepsilon(x,x)\to 0,\quad\mbox{and}\quad \mathbf{rev}_\varepsilon(x,x)\to\lambda,\quad\text{as}\quad \varepsilon\to 0. \end{equation*}

3. (iii) For every ɛ > 0 and $x\in \mathbb{R}^n$, one has

   \begin{equation*} \mathbf{K}_\varepsilon(x,x)\le 0,\quad \overline{\mathbf{S}}_\varepsilon(x,x)\le 0,\quad\mbox{and}\quad \mathbf{rev}_\varepsilon(x,x)\le\lambda. \end{equation*}

Proof. Denote $r=|x|$. Using the standard notation of Randers metrics (see §2.3) we have

\begin{equation*} F_\varepsilon(x,y)=\sqrt{a_{ij}(x)y^iy^j}+b_i(x)y^i,\quad\mbox{where}\quad a_{ij}(x)=\delta_{ij}\quad\mbox{and}\quad b_i(x)=-\frac{x^i(r+2\varepsilon)\theta}{(r+\varepsilon)^2}, \end{equation*}

while δ_ij is the Kronecker delta (see (24)). Since $a^{ij}(x)=\{a_{ij}(x)\}^{-1}=\delta_{ij}$, we have

\begin{equation*}\|b(x)\|_a = \sqrt{a^{ij}(x)b_i(x)b_j(x)}=\frac{r(r+2\varepsilon)\theta}{(r+\varepsilon)^2} \lt \theta \lt 1.\end{equation*}

Obviously F_ɛ is smooth, hence it is a Randers metric on $\mathbb{R}^n$. In addition, according to remark 2.5, the metric F_ɛ is projectively flat. Indeed, the Euclidean metric is projectively flat, and one has

\begin{equation*} b_i\,\mathrm{d} x^i=D\beta(x),\quad\mbox{where}\quad \beta(x)=-\left(r+\frac{\varepsilon^2}{r+\varepsilon}\right)\theta. \end{equation*}

Let $O\in \mathbb{R}^n$ be the origin. For $\rho_\varepsilon(x)=d(O,x)$ and $\overleftarrow{\rho_\varepsilon} (x)=d(x,O)$ we obtain:

\begin{equation*} \rho_\varepsilon(x)=r\cdot \frac{r(1-\theta)+ \varepsilon }{r+\varepsilon}\quad\mbox{and}\quad \overleftarrow{\rho_\varepsilon} (x) =r\cdot \frac{r(1+\theta)+ \varepsilon }{r+\varepsilon}. \end{equation*}

In addition, one has

(38)\begin{equation} D\rho_\varepsilon(x)=\left(\frac{1}{r}-\frac{ (r+2 \varepsilon )\theta}{(r+\varepsilon )^2}\right)\cdot x\quad\mbox{and}\quad\nabla\rho_\varepsilon(x)=\frac{(r+\varepsilon)^2}{r(r+\varepsilon)^2-r^2(r+2\varepsilon)\theta}\cdot x. \end{equation}

According to relation (23), the density of the Busemann–Hausdorff measure $\mathsf{m}_\varepsilon$ induced by F_ɛ is

\begin{equation*} \sigma_{F_\varepsilon}(x)=\left(1-\frac{r^2 (r+2 \varepsilon)^2\theta^2}{(r+\varepsilon)^4}\right)^\frac{n+1}{2}. \end{equation*}

By proposition 2.4, for every ɛ > 0 and $x\in \mathbb{R}^n$ we obtain

\begin{align*} \mathbf{K}_{\rho_\varepsilon}(r) & \stackrel{\textrm{def}}{=}\mathbf{K}_\varepsilon(x,x)=-\frac{3 \varepsilon ^2 (r+\varepsilon)^4 \theta (1-\theta)}{\left((r+\varepsilon)^2- r (r+2 \varepsilon)\theta \right)^4}\le 0, \\ \overline{\mathbf{S}}_{\rho_\varepsilon}(r) & \stackrel{\textrm{def}}{=} \overline{\mathbf{S}}_\varepsilon(x,x)=-\frac{(n+1)\varepsilon ^2(r+\varepsilon ) \theta }{\left(r+\varepsilon \right)^4- r^2 \left(r+2 \varepsilon\right)^2\theta^2}\le 0, \\ \mathbf{rev}_{\rho_\varepsilon}(r) & \stackrel{\textrm{def}}{=}\mathbf{rev}_\varepsilon(x,x)=\frac{1+\|b(x)\|_a}{1-\|b(x)\|_a}=\frac{1+\frac{r(r+2\varepsilon)\theta}{(r+\varepsilon)^2}}{1-\frac{r(r+2\varepsilon)\theta}{(r+\varepsilon)^2}}\le \frac{1+\theta}{1-\theta}=\lambda. \end{align*}

Considering the limit cases of the above inequalities, we obtain

\begin{equation*} \lim_{r\to\infty}\mathbf{K}_{\rho_\varepsilon}(r)=0,\quad \lim_{r\to\infty}\overline{\mathbf{S}}_{\rho_\varepsilon}(r)=0,\quad \lim_{r\to\infty}\mathbf{rev}_{\rho_\varepsilon}(r)=\lambda,\quad \forall \varepsilon \gt 0, \end{equation*}

and

\begin{equation*} \lim_{\varepsilon\to0}\mathbf{K}_{\rho_\varepsilon}(r)=0,\quad \lim_{\varepsilon\to0}\overline{\mathbf{S}}_{\rho_\varepsilon}(r)=0,\quad \lim_{\varepsilon\to0}\mathbf{rev}_{\rho_\varepsilon}(r)=\lambda,\quad \forall r \gt 0. \end{equation*}

Moreover, for every ɛ > 0 and $x,y\in \mathbb{R}^n$ one has

(39)\begin{align} \mathbf{rev}_\varepsilon(x,y)&=\frac{F_\varepsilon(x,-y)}{F_\varepsilon(x,y)}\le \frac{F_\varepsilon(x,-x)}{F_\varepsilon(x,x)}\\ & \nonumber =\mathbf{rev}_\varepsilon(x,x)\le \lambda,\quad\text{if }x\ne O\quad\text{and}\quad \mathbf{rev}_\varepsilon(O,y)=1, \end{align}

thus, on also has $\lambda_{F_\varepsilon}(\mathbb{R}^n)=\lambda$, for every ɛ > 0.

We make the following observations.

4.2. Sufficient conditions for sharpness

In this section, we present sufficient conditions for the sharpness of Hardy inequalities provided by Riccati pairs on the spaces of $\mathcal{F}_{0,0,\lambda}$, introduced in theorem 4.1.

Let $(\mathbb{R}^n,F_\varepsilon,\mathsf{m}_\varepsilon)\in\mathcal{F}_{0,0,\lambda}$ for some ɛ > 0 and $\lambda\in(1,\infty)$. Then for every $x\in \mathbb{R}^n$, one has

\begin{equation*} \mathbf{K}_\varepsilon(x,x)\le 0,\quad \overline{\mathbf{S}}_\varepsilon(x,x)\le 0,\quad\mbox{and}\quad \mathbf{rev}_\varepsilon(x,x)\le\lambda=\lambda_{F_\varepsilon}(\mathbb{R}^n). \end{equation*}

Denote $\rho=d(O,\cdot)$ the distance from the origin. The Laplace comparison theorem (see theorem 2.2) implies

\begin{equation*} \Delta_{F_\varepsilon}(\rho_\varepsilon)\ge \frac{n-1}{\rho_\varepsilon}\stackrel{\textrm{def}}{=} L^{0,0}(\rho_\varepsilon). \end{equation*}

Let $R\in(0,+\infty]$, $L(t)\le L^{0,0}(t)$ for every $t\in(0,R)$, and define

\begin{equation*}\Omega_\varepsilon=\{x\in \mathbb{R}^n : \rho_\varepsilon(x) \lt R\}.\end{equation*}

Suppose that (L, W) is a $(p,\rho_\varepsilon,w)$-Riccati pair in $(0,R)$, that is $W\colon(0,R)\to (0,\infty)$ is continuous, $w\colon(0,R)\to (0,\infty)$ is of class C ¹, and there exists $G\colon(0,R)\to(0,\infty)$ of class C ¹ such that

(40)\begin{equation} (G(t)w(t))'+G(t)w(t)L(t)-(p-1)G(t)^{p'}w(t)\geq W(t)w(t), \quad\forall t\in (0,R). \end{equation}

Then by theorem 3.1, for every $u\in C_0^\infty(\Omega_\varepsilon)$ one has:

(41)\begin{align} \lambda^p\int_{\Omega_\varepsilon} w(\rho_\varepsilon) F^*_\varepsilon(Du)^p\,\mathrm{d} \mathsf{m}_\varepsilon & \ge\int_{\Omega_\varepsilon} w(\rho_\varepsilon) W(\rho_\varepsilon) |u|^p\,\mathrm{d} \mathsf{m}_\varepsilon. \end{align}

As in remark 3.2(e), suppose that a function G > 0 verifies the Riccati ODI (40) with equality, and define $v\colon (0,R)\to\mathbb{R}$ by

(42)\begin{equation} -(\log v(t))'=-\frac{v'(t)}{v(t)}=G(t)^\frac{1}{p-1}. \end{equation}

Since G is smooth and positive, it follows that v is smooth, positive, and decreasing. Introduce the notation

(43)\begin{equation} \mathcal{I}(f;a,b)=\int_a^b f(t)t^{n-1}\,\mathrm{d} t, \end{equation}

for arbitrary continuous functions $f\colon[a,b]\to\mathbb{R}$, with $0 \lt a \lt b \lt R$.

We have the following result.

Theorem 4.3. Fix $\delta_0 \gt 0$ and let $t_1=t_1(\delta),t_2=t_2(\delta),t_3=t_3(\delta),t_4=t_4(\delta)$ be continuous functions admitting a (possibly infinite) limit at infinity, such that for every $\delta \gt \delta_0$ one has $0 \lt t_1 \lt t_2 \lt t_3 \lt t_4 \lt R$. Suppose that the following limits hold true:

\begin{align*} \lim_{\delta\to\infty} l_0(\delta) & =0,\quad\mbox{where}\\ &\quad l_0(\delta)\stackrel {\textrm {def}} {=} \frac{v(t_2)^p\cdot \mathcal{I}(w;t_1,t_2)}{(t_2-t_1)^p\cdot \mathcal{I}(wWv^p;t_2,t_3)}+ \frac{v(t_3)^p\cdot \mathcal{I}(w;t_3,t_4)}{(t_4-t_3)^p\cdot \mathcal{I}(wWv^p;t_2,t_3)}, \\ \lim_{\delta\to\infty} l_1(\delta) & =1,\quad\mbox{where}\quad l_1(\delta)\stackrel{\textrm{def}}{=}\max_{t\in [t_2,t_3]} \frac{G(t)^{p'}}{W(t)}. \end{align*}

Then inequality (41) is sharp on $\mathcal{F}_{0,0,\lambda}$, in the sense that it no longer holds on each FMMM of $\mathcal{F}_{0,0,\lambda}$ if we multiply its right-hand side by a constant c > 1.

Proof. Let ɛ > 0 and define the limit function $u^\star_\varepsilon\colon \Omega\to (0,\infty)$ of inequality (41) by

\begin{equation*}u^\star_\varepsilon(x)=v(\rho_\varepsilon(x)).\end{equation*}

Denote $T=(t_1,t_2,t_3,t_4)$ and construct

\begin{equation*}u^\star_{\varepsilon,T}=v_T(\rho_\varepsilon),\end{equation*}

where the function v_T and its derivative satisfy

(44)\begin{align} v_T(t)=\begin{cases} 0, & \mbox{if }0 \lt t\le t_1, \\ v(t_2)\frac{t-t_1}{t_2-t_1}, & \mbox{if }t_1\le t\le t_2, \\ v(t), & \mbox{if }t_2\le t\le t_3, \\ v(t_3)\frac{t_4-t}{t_4-t_3}, & \mbox{if }t_3\le t\le t_4, \\ 0, & \mbox{if }t_4\le t \lt R, \end{cases}\quad\mbox{and}\quad v_T'(t)=\begin{cases} 0, & \mbox{if }0 \lt t \lt t_1, \\ \frac{v(t_2)}{t_2-t_1}, & \mbox{if }t_1 \lt t \lt t_2, \\ v'(t), & \mbox{if }t_2 \lt t \lt t_3, \\ -\frac{v(t_3)}{t_4-t_3}, & \mbox{if }t_3 \lt t \lt t_4, \\ 0, & \mbox{if }t_4 \lt t \lt R. \end{cases} \end{align}

Note that $v'_T$ is not defined in $t_1,t_2,t_3$ and t ₄. In addition, both v_T and $v'_T$ are supported in $[t_1,t_4]$. For a schematic illustration of v_T we refer to figure 1.

Figure 1.

Illustration of the ‘tent-like’ function $v_T(t)$: coincides with v(t) on $[t_2,t_3]$, linearized on both $[t_1,t_2]$ and $[t_3,t_4]$, and zero elsewhere.

To prove the sharpness of inequality (41), we show that under a suitable limiting procedure, one has

\begin{equation*} \frac{\lambda^p\displaystyle\int_{\Omega_\varepsilon} w(\rho_\varepsilon) F^*_\varepsilon(Du^\star_{\varepsilon,T})^p\,\mathrm{d} \mathsf{m}_\varepsilon}{ \displaystyle\int_{\Omega_\varepsilon} w(\rho_\varepsilon) W(\rho_\varepsilon) (u_{\varepsilon,T}^\star)^p\,\mathrm{d} \mathsf{m}_\varepsilon}\to 1. \end{equation*}

As a first step, we use polar transforms to reduce the above integrals to one dimension. We also establish some fine estimates on the distance and the density of the measure.

Using the notations $r=|x|$ and $t=\rho_\varepsilon(x)$, it follows that

\begin{equation*}t=r\cdot \frac{r(1-\theta)+ \varepsilon }{r+\varepsilon}\quad\iff\quad r=\frac{t-\varepsilon+\sqrt{(t+\varepsilon)^2-4t\varepsilon\theta}}{2(1-\theta)}\stackrel{\textrm{def}}{=} \varphi_\varepsilon(t),\end{equation*}

hence we obtain

(45)\begin{equation} \sigma_\varepsilon(r)r^{n-1}\,\mathrm{d} r=\sigma_\varepsilon(\varphi_\varepsilon(t))\varphi_\varepsilon(t)^{n-1}\varphi_\varepsilon'(t)\,\mathrm{d} t\stackrel{\textrm{def}}{=}\psi_\varepsilon(t)\,\mathrm{d} t. \end{equation}

Since for every $t\in(0,R)$ one has

\begin{equation*} t\le \varphi_\varepsilon(t)\le\frac{t}{1-\theta},\quad 1\le \varphi'_\varepsilon(t)\le \frac{1}{1-\theta},\quad\mbox{and}\quad\left(1-\theta^2\right)^\frac{n+1}{2}\le \sigma_\varepsilon(\varphi_\varepsilon(t))\le 1, \end{equation*}

we obtain

(46)\begin{equation} ct^{n-1}\stackrel{\textrm{def}}{=}\left(1-\theta^2\right)^\frac{n+1}{2}t^{n-1}\le \psi_\varepsilon(t)\le \frac{t^{n-1}}{(1-\theta)^n}\stackrel{\textrm{def}}{=} Ct^{n-1}. \end{equation}

Note that c and C are independent of ɛ.

Since v is positive and decreasing, the positive homogeneity of $F^*_\varepsilon$ and the relations from (44) imply

(47)\begin{align} F^*_\varepsilon(Du^\star_{\varepsilon,T})=F^*_\varepsilon(v_T'(\rho_\varepsilon)D\rho_\varepsilon)=\begin{cases} \frac{v(t_2)}{t_2-t_1}F^*_\varepsilon(D\rho_\varepsilon), & \text{if } \rho_\varepsilon\in(t_1,t_2), \\ |v'(\rho_\varepsilon)|F^*_\varepsilon(-D\rho_\varepsilon) & \text{if } \rho_\varepsilon\in(t_2,t_3), \\ \frac{v(t_3)}{t_4-t_3}F^*_\varepsilon(-D\rho_\varepsilon) & \text{if } \rho_\varepsilon\in(t_3,t_4), \\ 0, & \text{if }\rho_\varepsilon\in(0,t_1)\cup(t_4,R). \end{cases} \end{align}

According to remark 4.2(d), one has

\begin{align*}&F^*_\varepsilon(D\rho_\varepsilon)=1,\quad\mbox{and}\quad F^*_\varepsilon(-D\rho_\varepsilon)=\frac{1-\frac{r(r+2\varepsilon)}{(r+\varepsilon)^2}\theta}{1+\frac{r(r+2\varepsilon)}{(r+\varepsilon)^2}\theta}\stackrel{\textrm{def}}{=} h_\varepsilon(r), \\ & \quad\mbox{where}\quad \frac{1-\theta}{1+\theta}\le h_\varepsilon(r)\le1.\end{align*}

By a simple computation, we obtain that $h_\varepsilon(r)$ is decreasing in r, increasing in ɛ, and approaches $\frac{1-\theta}{1+\theta}=\frac{1}{\lambda}$ as ɛ → 0. Since $r=\varphi_\varepsilon(t)\ge t$, we also have

(48)\begin{equation} h_\varepsilon(\varphi_\varepsilon(t))\le h_\varepsilon(t)\le 1. \end{equation}

In the sequel, denote by ω the volume of the Euclidean unit ball of dimension n.

On the one hand, relations (45) and (47) yield

\begin{align*} &I\stackrel{\textrm{def}}{=}\frac{1}{n\omega} \int_{\Omega_\varepsilon} w(\rho_\varepsilon) F^*_\varepsilon(Du^\star_{\varepsilon,T})^p\,\mathrm{d} \mathsf{m}_\varepsilon =\frac{v(t_2)^p}{(t_2-t_1)^p}\int_{t_1}^{t_2} w(t) \psi_\varepsilon(t)\,\mathrm{d} t \\ & \quad + \int_{t_2}^{t_3} w(t)|v'(t)|^p h_\varepsilon^p(\varphi_\varepsilon(t)) \psi_\varepsilon(t)\,\mathrm{d} t +\frac{v(t_3)^p}{(t_4-t_3)^p}\int_{t_3}^{t_4} w(t) h_\varepsilon^p(\varphi_\varepsilon(t))\psi_\varepsilon(t)\,\mathrm{d} t. \end{align*}

Applying relations (46) and (48), and using the definition of $\mathcal{I}$ from (43), it follows that

\begin{align*} I & \le \frac{v(t_2)^p}{(t_2-t_1)^p}\int_{t_1}^{t_2} w(t) \psi_\varepsilon(t)\,\mathrm{d} t+ \int_{t_2}^{t_3} w(t)|v'(t)|^p h_\varepsilon^p(t) \psi_\varepsilon(t)\,\mathrm{d} t +\frac{v(t_3)^p}{(t_4-t_3)^p}\\ & \quad \times \int_{t_3}^{t_4} w(t) h_\varepsilon^p(t)\psi_\varepsilon(t)\,\mathrm{d} t \le \frac{Cv(t_2)^p}{(t_2-t_1)^p}\int_{t_1}^{t_2} w(t) t^{n-1}\,\mathrm{d} t+ h_\varepsilon^p(t_2)\int_{t_2}^{t_3} w(t)|v'(t)|^p \psi_\varepsilon(t)\,\mathrm{d} t \\ & \quad +\frac{C h_\varepsilon^p(t_3)v(t_3)^p}{(t_4-t_3)^p}\int_{t_3}^{t_4} w(t)t^{n-1}\,\mathrm{d} t =\frac{Cv(t_2)^p\cdot \mathcal{I}(w;t_1,t_2)}{(t_2-t_1)^p}+ h_\varepsilon^p(t_2)\\ &\quad \times \int_{t_2}^{t_3} w(t)|v'(t)|^p \psi_\varepsilon(t)\,\mathrm{d} t +\frac{C v(t_3)^p\cdot\mathcal{I}(w;t_3,t_4)}{(t_4-t_3)^p}. \end{align*}

By relation (42), for every $t\in[t_2,t_3]$ one has

\begin{equation*} \frac{1}{W(t)}\left|\frac{v'(t)}{v(t)}\right|^p=\frac{1}{W(t)}\left(-\frac{v'(t)}{v(t)}\right)^p=\frac{G(t)^{p'}}{W(t)}. \end{equation*}

The above function is continuous; thus, it attains its maximum on $[t_2,t_3]$. Hence, we have

\begin{align*} \int_{t_2}^{t_3} w(t)|v'(t)|^p \psi_\varepsilon(t)\,\mathrm{d} t&=\int_{t_2}^{t_3} \frac{1}{W(t)}\left|\frac{v'(t)}{v(t)}\right|^p w(t) W(t)|v(t)|^p \psi_\varepsilon(t)\,\mathrm{d} t\\ &\le \max_{t\in[t_2,t_3]} \frac{G(t)^{p'}}{W(t)}\cdot \int_{t_2}^{t_3} w(t) W(t) (v(t))^p \psi_\varepsilon(t)\,\mathrm{d} t. \end{align*}

On the other hand, we obtain

\begin{align*} \nonumber J&\stackrel{\textrm{def}}{=} \frac{1}{n\omega} \displaystyle\int_{\Omega_\varepsilon} w(\rho_\varepsilon) W(\rho_\varepsilon) (u_{\varepsilon,T}^\star)^p\,\mathrm{d} \mathsf{m}_\varepsilon=\int_{0}^{R} w(t) W(t) (v_T(t))^p \psi_\varepsilon(t)\,\mathrm{d} t \end{align*}

(49)\begin{align} &\,\ge \int_{t_2}^{t_3} w(t) W(t) (v(t))^p \psi_\varepsilon(t)\,\mathrm{d} t\ge c\cdot \mathcal{I}(wWv^p;t_2,t_3). \end{align}

Combining the above results and using both estimates of (49), it follows that

\begin{align*} \frac{\lambda^p I}{J} & \le \lambda^p h_\varepsilon^p(t_2)\cdot \max_{t\in[t_2,t_3]} \frac{G(t)^{p'}}{W(t)}+\frac{\lambda^p C}{c}\left(\frac{v(t_2)^p\cdot \mathcal{I}(w;t_1,t_2)}{(t_2-t_1)^p\cdot \mathcal{I}(wWv^p;t_2,t_3)}\right. \nonumber\\ & \left. \quad +\frac{v(t_3)^p\cdot \mathcal{I}(w;t_3,t_4)}{(t_4-t_3)^p\cdot \mathcal{I}(wWv^p;t_2,t_3)}\right) \\ & =\lambda^p h_\varepsilon^p(t_2) l_1(\delta)+\frac{\lambda^p C}{c}l_0(\delta). \end{align*}

Observe that the above inequality holds for every ɛ > 0 and $\delta \gt \delta_0$, in addition, δ, $T=(t_1,t_2,t_3,t_3)$, c, C, λ, l ₀, and l ₁ are independent of ɛ. To prove the sharpness of (41), it is enough to choose for every $\delta \gt \delta_0$ an $\varepsilon=\varepsilon(\delta)$ such that $h_{\varepsilon}(t_2(\delta))\to\frac{1}{\lambda}$ as $\delta\to\infty$. Indeed, in this case, since $l_0\to0$ and $l_1\to1$ as $\delta\to\infty$ we obtain $\lambda^pI/J\to1$ as $\delta\to \infty$.

By assumption there exists

\begin{equation*}\lim_{\delta\to\infty} t_2(\delta)\stackrel{\textrm{def}}{=} l\in[0,R].\end{equation*}

We distinguish two cases: If l > 0, then choose $\varepsilon=1/\delta$, to obtain

\begin{equation*} \lim_{\delta\to \infty} h_{\varepsilon}(t_2(\delta))= \lim_{\delta\to \infty}\frac{1-\frac{t_2(\delta)(t_2(\delta)+2/\delta)}{(t_2(\delta)+1/\delta)^2}\theta}{1+\frac{t_2(\delta)(t_2(\delta)+2/\delta)}{(t_2(\delta)+1/\delta)^2}\theta}=\frac{1-\theta}{1+\theta}=\frac{1}{\lambda}. \end{equation*}

If l = 0, then choosing $\varepsilon=t_2(\delta)^2$ yields

\begin{equation*} \lim_{\delta\to \infty} h_{\varepsilon}(t_2(\delta))= \lim_{t_2\to 0}\frac{1-\frac{t_2(t_2+2t_2^2)}{(t_2+t_2^2)^2}\theta}{1+\frac{t_2(t_2+2t_2^2)}{(t_2+t_2^2)^2}\theta}=\lim_{t_2\to 0}\frac{1-\frac{(1+2t_2)}{(1+t_2)^2}\theta}{1+\frac{(1+2t_2)}{(1+t_2)^2}\theta}=\frac{1-\theta}{1+\theta}=\frac{1}{\lambda}, \end{equation*}

which concludes the proof.

4.3. The sharpness of a weighted Hardy inequality

In this section, we present a sharpness result concerning a weighted Hardy inequality on $\mathcal{F}_{0,0,\lambda}$. As a consequence, we also obtain theorem 1.1. Our result can be stated as follows.

Theorem 4.4. Let $n\ge 3$, $\lambda\in (1,\infty)$, and $\alpha,p\in \mathbb{R}$ such that $1 \lt p \lt n+\alpha$. For every $(\mathbb{R}^n,F_\varepsilon,\mathsf{m}_\varepsilon)\in \mathcal{F}_{0,0,\lambda}$ and $u\in C_0^\infty(\mathbb{R}^n)$ one has

(50)\begin{align} \lambda^p\displaystyle\int_{\mathbb{R}^n} \rho^\alpha \cdot F_\varepsilon^*(Du)^p\,\mathrm{d} \mathsf{m}_\varepsilon\ge c_{n,p,\alpha}\int_{\mathbb{R}^n}\rho^\alpha\frac{u^p}{\rho_\varepsilon^p}\,\mathrm{d} \mathsf{m}_\varepsilon,\quad\text{where}\quad c_{n,p,\alpha}=\left(\frac{n+\alpha-p}{p}\right)^p. \end{align}

Moreover, $c_{n,p,\alpha}$ is sharp in $\mathcal{F}_{0,0,\lambda}$, in the sense that it is the greatest constant with this property.

Proof. The proof of inequality (50) follows from theorem 3.3. To prove the sharpness, we use theorem 4.3. According to the relation (42), we obtain

\begin{equation*}v(t)^p=\frac{1}{t^{n+\alpha-p}},\quad\forall t\in(0,\infty).\end{equation*}

For every δ > 1 define

\begin{equation*}t_1(\delta)=\delta/2,\quad t_2(\delta)=\delta,\quad t_3(\delta)=\delta^2,\quad\mbox{and}\quad t_4(\delta)=2\delta^2.\end{equation*}

By simple computations, we obtain

\begin{align*} \frac{v(t_2)^p\cdot \mathcal{I}(w;t_1,t_2)}{(t_2-t_1)^p} & =\frac{2^{n+\alpha}-1}{(n+\alpha)2^{n+\alpha-p}}=\text{const}, \\ \frac{v(t_3)^p\cdot \mathcal{I}(w;t_3,t_4)}{(t_4-t_3)^p} & =\frac{2^{n+\alpha}-1}{n+\alpha}=\text{const}, \\ \mathcal{I}(wWv^p;t_2,t_3) & =c_{n,p,\alpha}\log(\delta)\to\infty,\quad\text{as } \delta\to\infty. \end{align*}

It follows that $l_0\to 0$ as $\delta\to\infty$. Since $l_1=\frac{G(t)^{p'}}{W(t)}=1$, for every t > 0, thus, theorem 4.3 yields the sharpness of the inequality (50).

We observe that choosing α = 0 and p = 2 in the above result yields theorem 1.1.

5.1. Construction

Consider the triple $(\mathbb{B}^n,F, \mathsf{m})$, where $\mathbb{B}^n$ denotes the Euclidean (open) unit ball, the function $F\colon \mathbb{B}^n\times \mathbb{R}^n\to [0,\infty)$ is defined by

(51)\begin{equation} F(x,y) =\frac{\sqrt{|y|^2-|x|^2|y|^2+\langle x,y\rangle^2}}{\kappa(1-\theta)(1-|x|^2)}-\frac{\theta\langle x,y\rangle}{\kappa(1-\theta) |x|(1-|x|^2)}, \end{equation}

for some $\theta\in(0,1)$ and κ > 0; while $\mathsf{m}$ is a measure with density

\begin{equation*} \sigma= \exp(-(n-1)h \rho)\cdot \sigma_{F}, \end{equation*}

for some $h\in \mathbb{R}$. As before, ρ is the distance from the origin and σ_F is the density of the Busemann–Hausdorff measure induced by F. We observe that F is not a Finsler metric, since it is not continuous at the origin. However, for every $x\ne O$, one has

\begin{align*} \mathbf{K}(x,x)=-\kappa^2,\quad \overline{\mathbf{S}}(x,x)=(n-1)h,\quad\mbox{and}\quad \mathbf{rev}(x,x)=\frac{1+\theta}{1-\theta}\stackrel{\textrm{def}}{=}\lambda=\lambda_F(\mathbb{B}^n). \end{align*}

The following result provides an approximation of the above structure by a family of Randers spaces.

Theorem 5.1. Let $\lambda\in(1,\infty)$, $\theta=\frac{\lambda-1}{\lambda+1}$, κ > 0 and $h\in \mathbb{R}$. Construct the family $\mathcal{F}_{\kappa,h,\lambda}=\{(\mathbb{B}^n, F_\varepsilon,\mathsf{m}_\varepsilon)\}_{\varepsilon \gt 0}$ of FMMMs, where $F_\varepsilon\colon T\mathbb{B}^n\simeq\mathbb{B}^n\times\mathbb{R}^n\to [0,\infty)$ is defined by

\begin{align*} F_\varepsilon(x,y)& =\frac{\sqrt{|y|^2-|x|^2|y|^2+\langle x,y\rangle^2}}{k_{\varepsilon}(1-\theta)(1-|x|^2)} -\frac{\theta \langle x,y\rangle}{k_{\varepsilon}(1-\theta) |x|(1-|x|^2)} \nonumber\\ & \quad \cdot \frac{\operatorname{arctanh}(|x|)\left(\operatorname{arctanh}(|x|)+2\varepsilon\right)}{\left(\operatorname{arctanh}(|x|)+\varepsilon\right)^2}, \end{align*}

k_ɛ is given by relation (54), $\mathsf{m}_\varepsilon$ is a measure with density $\sigma_\varepsilon=\exp(-(n-1)h \rho_\varepsilon)\cdot \sigma_{F_\varepsilon}$, and $\sigma_{F_\varepsilon}$ denotes the density of the Busemann–Hausdorff measure induced by F_ɛ. The following statements hold:

1. (i) For every ɛ > 0, the triple $(\mathbb{B}^n, F_\varepsilon,\mathsf{m}_\varepsilon)$ is an FMMM. In addition, F_ɛ is a projectively flat Randers metric, with reversibility $\lambda_{F_\varepsilon}(\mathbb{B}^n)=\lambda$.

2. (ii) For every $x\in \mathbb{B}^n$ with $x\ne 0$, one has

   (52)\begin{align} \mathbf{K}_\varepsilon(x,x)\to -\kappa^2,\quad \overline{\mathbf{S}}_\varepsilon(x,x)\to (n-1)h,\quad\mbox{and}\quad \mathbf{rev}_\varepsilon(x,x)\to\lambda,\quad\text{as}\quad \varepsilon\to 0. \end{align}

3. (iii) For every ɛ > 0 and $x\in \mathbb{B}^n$, one has

   (53)\begin{equation} \mathbf{K}_\varepsilon(x,x)\le -\kappa^2,\quad \overline{\mathbf{S}}_\varepsilon(x,x)\le (n-1)h,\quad\mbox{and}\quad \mathbf{rev}_\varepsilon(x,x)\le\lambda. \end{equation}

Proof. For simplicity of presentation, denote $r=|x|$, $\overline{r}=\operatorname{arctanh}(|x|)$, and $\vartheta_\varepsilon=k_\varepsilon(1-\theta)$. Using the standard notation of Randers metrics (see §2.3) one has

\begin{equation*} F_\varepsilon(x,y)=\sqrt{a_{ij}(x)y^iy^j}+b_i(x)y^i, \end{equation*}

where

\begin{equation*} a_{ij}(x)=\frac{\delta_{ij}}{\vartheta_{\varepsilon}^2(1-r^2)}+\frac{x^ix^j}{\vartheta_{\varepsilon}^2(1-r^2)^2} \quad\mbox{and}\quad b_i(x)=-\frac{\theta \overline{r}(\overline{r}+2\varepsilon)x^i}{\vartheta_{\varepsilon} r(1-r^2)(\overline{r}+\varepsilon)^2}. \end{equation*}

Let $a^{ij}(x)=\{a_{ij}(x)\}^{-1}$, it follows that

\begin{equation*} \|b\|_a(x)=\sqrt{a^{ij}(x)b_i(x)b_j(x)}=\frac{\theta \overline{r}(\overline{r}+2\varepsilon)}{(\overline{r}+\varepsilon)^2} \lt \theta \lt 1. \end{equation*}

Since F_ɛ is also smooth, it is a Randers metric on $\mathbb{R}^n$. According to remark 2.5, the metric F_ɛ is projectively flat. Indeed, the Klein metric (obtained from F_ɛ for θ = 0) is projectively flat, and

\begin{equation*} b_i\,\mathrm{d} x^i=D\beta(x),\quad\mbox{where}\quad \beta(x)=-\frac{\theta}{\vartheta_{\varepsilon}}\left(\overline{r}+\frac{\varepsilon^2}{\overline{r}+\varepsilon}\right). \end{equation*}

Let $O\in\mathbb{R}^n$ denote the origin, $\rho_\varepsilon(x)=d(O,x)$ and $\overleftarrow{\rho_\varepsilon} (x)=d(x,O)$. Then

\begin{equation*} \rho_\varepsilon(x)=\overline{r}\cdot \frac{\overline{r}(1-\theta)+ \varepsilon }{\vartheta_{\varepsilon}(\overline{r}+\varepsilon)} \quad\mbox{and}\quad \overleftarrow{\rho_\varepsilon} (x) =\overline{r}\cdot \frac{\overline{r}(1+\theta)+ \varepsilon }{\vartheta_{\varepsilon}(\overline{r}+\varepsilon)}. \end{equation*}

In addition, one has

\begin{equation*} D\rho_\varepsilon(x)=\left(\frac{(\overline{r}+\varepsilon)^2-\overline{r}(\overline{r}+2\varepsilon)\theta}{\vartheta_{\varepsilon}r(1-r^2)(\overline{r}+\varepsilon )^2}\right)\cdot x \quad\mbox{and}\quad \nabla\rho_\varepsilon(x)=\frac{\vartheta_{\varepsilon}(1-r^2)(\overline{r}+\varepsilon)^2}{r((\overline{r}+\varepsilon)^2-\overline{r}(\overline{r}+2\varepsilon)\theta)}\cdot x. \end{equation*}

According to relation (23), the density of the Busemann–Hausdorff measure induced by F_ɛ is

\begin{equation*} \sigma_{F_\varepsilon}= \frac{1}{\vartheta_{\varepsilon}^n}\cdot\left(\frac{1}{1-r^2}\right)^\frac{n+1}{2}\cdot\left(1-\frac{\overline{r}^2 (\overline{r}+2 \varepsilon)^2\theta^2}{(\overline{r}+\varepsilon)^4}\right)^\frac{n+1}{2} \stackrel{\textrm{def}}{=} \frac{1}{\vartheta_{\varepsilon}^n}\cdot \left(\frac{1}{1-\overline{r}^2}\right)^\frac{n+1}{2} \cdot f_\varepsilon^\frac{n+1}{2}. \end{equation*}

Observe that $1-\theta^2\le f_\varepsilon\le 1$. The density of $\mathsf{m}_\varepsilon$ is

\begin{equation*} \sigma_\varepsilon= \exp\left(-(n-1)h\rho_\varepsilon\right)\cdot \frac{1}{\vartheta_{\varepsilon}^n}\left(\frac{1}{1-\overline{r}^2}\right)^\frac{n+1}{2}f_\varepsilon^\frac{n+1}{2}. \end{equation*}

In the sequel, we compute the flag curvature, the reduced S-curvature, and the reversibility of $(\mathbb{B}^n,F_\varepsilon,\mathsf{m}_\varepsilon)$ along the geodesics starting from O. Recall that $\vartheta_\varepsilon=k_\varepsilon(1-\theta)$. Concerning the flag curvature, we obtain

\begin{align*} \mathbf{K}_{\rho_\varepsilon}(\overline{r})\stackrel{\textrm{def}}{=}\mathbf{K}_\varepsilon(x,x)= -\frac{k_{\varepsilon}^2(1-\theta)^2(\overline{r}+\varepsilon )^4 \left(e_0-e_1\theta +e_2\theta^2 \right)}{\left((\overline{r}+\varepsilon )^2-\overline{r} (\overline{r}+2 \varepsilon )\theta \right)^4}, \end{align*}

where $e_0=(\overline{r}+\varepsilon)^4$, $e_1=2\overline{r}^4+8\overline{r}^3\varepsilon+(10\overline{r}^2-3)\varepsilon^2+4\overline{r}\varepsilon^3$, and $e_2=\overline{r}^4+4\overline{r}^3\varepsilon+(4\overline{r}^2-3)\varepsilon^2$.

Our goal is to choose k_ɛ such that relations (52) and (53) hold. For every ɛ > 0, one has

\begin{align*} &\lim_{\overline{r}\to\infty} \mathbf{K}_{\rho_\varepsilon}(\overline{r}) =-k_{\varepsilon}^2,\quad\text{and}\quad \lim_{\overline{r}\to0} \mathbf{K}_{\rho_\varepsilon}(\overline{r}) =-k_{\varepsilon}^2\cdot K_{\varepsilon,0},\nonumber\\ & \quad \text{where}\quad K_{\varepsilon,0}=(1-\theta)^2\left(1+\frac{3\theta(1-\theta)}{\varepsilon^2}\right). \end{align*}

By a simple derivative test, we obtain that if $\varepsilon\le \sqrt{\frac{3(1-\theta)}{8\theta}}\stackrel{\textrm{def}}{=} \varepsilon_0$, then $\overline{r}\mapsto\mathbf{K}_{\rho_\varepsilon}(\overline{r})$ admits a unique maximum on $(0,\infty)$, which is given by

\begin{align*} & -k_{\varepsilon}^2\cdot K_{\varepsilon}, \quad\text{where}\quad K_{\varepsilon}\nonumber\\ & \quad =\frac{6(1-\theta)^2\left((1-\theta ) \left(3 (1-\theta )-2 \varepsilon ^2 \theta \right)+\sqrt{3 (1-\theta)^3 \left(3(1-\theta)-8 \varepsilon ^2 \theta \right)}\right)^3}{\left(3 (1-\theta )^2+\sqrt{3 (1-\theta)^3 \left(3(1-\theta)-8 \varepsilon ^2 \theta \right)}\right)^4}\le 1. \end{align*}

For simplicity define $K_{\varepsilon}=\infty$, when $\varepsilon \gt \varepsilon_0$. To ensure $\mathbf{K}_{\rho_\varepsilon}(\overline{r})\le-\kappa^2$, $\forall \varepsilon \gt 0,\overline{r}\ge 0$, choose

(54)\begin{equation} k_{\varepsilon}=\max\left\{\kappa,\frac{\kappa}{\sqrt{K_{\varepsilon,0}}},\frac{\kappa}{\sqrt{K_{\varepsilon}}}\right\}. \end{equation}

If ɛ → 0, then $K_{\varepsilon,0}\to\infty$, $K_\varepsilon\to1$, and $k_{\varepsilon}\to\kappa$. Thus, for every $\overline{r} \gt 0$ we obtain $\mathbf{K}_{\rho_\varepsilon}(\overline{r})\to-\kappa^2$ as ɛ → 0.

The reduced S-curvature along ρ_ɛ can be computed as

\begin{align*} \overline{\mathbf{S}}_{\rho_\varepsilon}(\overline{r})\stackrel{\textrm{def}}{=}\overline{\mathbf{S}}(x,x) & = (n-1)h-\frac{\vartheta_{\varepsilon}(n+1)(\overline{r}+\varepsilon)\theta \varepsilon^2}{(\overline{r}+\varepsilon)^4-\overline{r}^2(\overline{r}+2\varepsilon)^2\theta^2}\le (n-1)h. \end{align*}

We easily obtain that if $\overline{r} \gt 0$, then $\overline{\mathbf{S}}_{\rho_\varepsilon}(\overline{r})\to (n-1)h$ as ɛ → 0. If $\overline{r}=0$, then $\overline{\mathbf{S}}_{\rho_\varepsilon}(\overline{r})\to -\infty$ as ɛ → 0. In addition, for every fixed ɛ > 0 one has $\overline{\mathbf{S}}_{\rho_\varepsilon}(\overline{r})\to (n-1)h$ as $\overline{r}\to \infty$.

Finally, for the reversibility we obtain

\begin{equation*} \mathbf{rev}_{\rho_\varepsilon}(\overline{r})\stackrel{\textrm{def}}{=}\mathbf{rev}(x,x)=\frac{1+\frac{\overline{r}(\overline{r}+2\varepsilon)\theta}{(\overline{r}+\varepsilon)^2}}{1-\frac{\overline{r}(\overline{r}+2\varepsilon)\theta}{(\overline{r}+\varepsilon)^2}}\le \frac{1+\theta}{1-\theta}=\lambda. \end{equation*}

It follows that, if $\overline{r} \gt 0$, then $\mathbf{rev}_{\rho_\varepsilon}(\overline{r})\to\lambda$ as ɛ → 0, and for every fixed ɛ > 0 one has $\mathbf{rev}_{\rho_\varepsilon}(\overline{r})\to \lambda$ as $\overline{r}\to \infty$, and $\mathbf{rev}_{\rho_\varepsilon}(\overline{r})\to 1$ as $\overline{r}\to 0$. The latter limit ensures the continuity of F_ɛ at the origin. In addition, $\lambda_{F_\varepsilon}(\mathbb{B}^n)=\lambda$, for every ɛ > 0. Indeed, for every $x\in \mathbb{B}^n$, and $y\in \mathbb{R}^n$ one has

\begin{equation*}\mathbf{rev}_\varepsilon(x,y)=\frac{F_\varepsilon(x,-y)}{F_\varepsilon(x,y)}\le \frac{F_\varepsilon(x,-x)}{F_\varepsilon(x,x)}=\mathbf{rev}_\varepsilon(x,x)\le \lambda,\quad \text{if }x\ne O, \quad\text{and}\quad \mathbf{rev}_\varepsilon(O,y)=1,\end{equation*}

which concludes the proof.

We make the following observations.

5.2. Sufficient conditions for sharpness

In this section, we present sufficient conditions for the sharpness of Hardy inequalities on the spaces of $\mathcal{F}_{\kappa,h,\lambda}$, introduced in theorem 5.1.

Let $(\mathbb{B}^n,F_\varepsilon,\mathsf{m}_\varepsilon)\subset \mathcal{F}_{\kappa,h,\lambda}$ for some ɛ > 0, $\lambda\in(1,\infty)$, κ > 0 and $h\in \mathbb{R}$. Then for every $x\in \mathbb{B}^n$, one has

\begin{equation*} \mathbf{K}_{\varepsilon}(x,x)\le-\kappa^2,\quad \overline{\mathbf{S}}_{\varepsilon}(x,x)\le (n-1)h,\quad\text{and}\quad \mathbf{rev}_{\varepsilon}(x,x)\le \lambda=\lambda_{F_\varepsilon}(\mathbb{B}^n). \end{equation*}

Denote by ρ_ɛ the distance from the origin. By the Laplace comparison theorem (see theorem 2.2) one has

\begin{equation*} \Delta_{F_\varepsilon} (\rho_\varepsilon)\ge (n-1)\kappa\coth(\kappa\rho_\varepsilon)-(n-1)h\stackrel{\textrm{def}}{=} L^{\kappa,h}(\rho_\varepsilon). \end{equation*}

Let $R\in(0,+\infty]$, $L(t)\le L^{\kappa,h}(t)$ for every $t\in(0,R)$, and define

\begin{equation*} \Omega_\varepsilon=\{x\in \mathbb{B}^n\mid \rho_\varepsilon(x) \lt R\}. \end{equation*}

Suppose that (L, W) is a $(p,\rho_\varepsilon,w)$-Riccati pair in $(0,R)$, that is $W\colon(0,R)\to (0,\infty)$ is continuous, $w\colon(0,R)\to (0,\infty)$ is of class C ¹, and there exists $G\colon(0,R)\to(0,\infty)$ of class C ¹ such that

(57)\begin{equation} (G(t)w(t))'+G(t)w(t)L(t)-(p-1)G(t)^{p'}w(t)\geq W(t)w(t), \quad\forall t\in (0,R). \end{equation}

By theorem 3.1, for every $u\in C_0^\infty(\Omega_\varepsilon)$ one has

(58)\begin{align} \lambda^p\int_{\Omega_\varepsilon} w(\rho_\varepsilon) F^*_\varepsilon(Du)^p\,\mathrm{d} \mathsf{m}_\varepsilon & \ge\int_{\Omega_\varepsilon} w(\rho_\varepsilon) W(\rho_\varepsilon) |u|^p\,\mathrm{d} \mathsf{m}_\varepsilon. \end{align}

Suppose that G > 0 satisfies the Riccati ODI (57) with equality, and define $v\colon (0,R)\to\mathbb{R}$ by

(59)\begin{equation} -(\log v(t))'=-\frac{v'(t)}{v(t)}=G(t)^\frac{1}{p-1}. \end{equation}

Since G is smooth and positive, v is smooth, positive, and decreasing. Introduce the notation

(60)\begin{equation} \mathcal{I}^{k,h}(f;a,b)=\int_a^b f(t)\sinh(k t)^{n-1}e^{-(n-1)h t}\,\mathrm{d} t, \end{equation}

where $k,h\in \mathbb{R}$ and $f\colon[a,b]\to\mathbb{R}$ is a continuous function, with $0 \lt a \lt b \lt R$. We have the following result.

Theorem 5.3. Fix $\delta_0 \gt 0$ and let $t_1=t_1(\delta),t_2=t_2(\delta),t_3=t_3(\delta),t_4=t_4(\delta)$ be continuous functions admitting a (possibly infinite) limit at infinity, such that for every $\delta \gt \delta_0$ one has $0 \lt t_1 \lt t_2 \lt t_3 \lt t_4 \lt R$. Let $\varepsilon=\varepsilon(\delta)$ such that ɛ → 0, as $\delta\to\infty$, and suppose that the following limits hold true:

\begin{align*} \lim_{\delta\to\infty} l_0^{k_\varepsilon,h}(\delta) & =0,\quad\mbox{where}\quad l_0^{k_\varepsilon,h}(\delta)\stackrel{\textrm{def}}{=} \frac{v(t_2)^p\cdot \mathcal{I}^{k_\varepsilon,h}(w;t_1,t_2)}{(t_2-t_1)^p\cdot \mathcal{I}^{k_\varepsilon,h}(wW v^p;t_2,t_3)}\nonumber\\ & \quad +\frac{v(t_3)^p\cdot \mathcal{I}^{k_\varepsilon,h}(w;t_3,t_4)}{(t_4-t_3)^p\cdot \mathcal{I}^{k_\varepsilon,h}(wW v^p;t_2,t_3)}, \\ \lim_{\delta\to\infty} l_1^{k_\varepsilon,h}(\delta) & =1,\quad\mbox{where}\quad l_1^{k_\varepsilon,h}(\delta)\stackrel{\textrm{def}}{=}\max_{t\in [t_2,t_3]} \frac{G(t)^{p'}}{W (t)},\\ \lim_{\delta\to\infty}l_2^{k_\varepsilon,h}(\delta) &=\frac{1}{\lambda}\quad\text{where}\quad l_2^{k_\varepsilon,h}(\delta)\stackrel{\textrm{def}}{=} h_\varepsilon((1-\theta)\kappa t_2),\quad\textrm{(see relation~{eq:k:he})}. \end{align*}

Then inequality (58) is sharp on $\mathcal{F}_{\kappa,h,\lambda}$ in the sense that it no longer holds on each FMMM of $\mathcal{F}_{\kappa,h,\lambda}$ if we multiply its right-hand side by a constant c > 1.

Proof. Let ɛ > 0 and $T=(t_1,t_2,t_3,t_4)$. As before, define the limit function $u^\star_\varepsilon \colon \Omega\to (0,\infty)$ of (58) and its truncation $u^\star_{\varepsilon,T}$ by

\begin{equation*}u^\star_\varepsilon(x)=v(\rho_\varepsilon(x)),\quad\mbox{and}\quad u^\star_{\varepsilon,T}(x)=v_{\varepsilon,T}(\rho_\varepsilon(x)),\end{equation*}

where

(61)\begin{equation} v_{\varepsilon,T}(t)=\begin{cases} 0, & \mbox{if }0 \lt t\le t_1, \\ v(t_2)\frac{t-t_1}{t_2-t_1}, & \mbox{if }t_1\le t\le t_2, \\ v(t), & \mbox{if }t_2\le t\le t_3, \\ v(t_3)\frac{t_4-t}{t_4-t_3}, & \mbox{if }t_3\le t\le t_4, \\ 0, & \mbox{if }t_4\le t \lt R, \end{cases}\quad\mbox{and}\quad v_{\varepsilon,T}'(t)=\begin{cases} 0, & \mbox{if }0 \lt t \lt t_1, \\ \frac{v(t_2)}{t_2-t_1}, & \mbox{if }t_1 \lt t \lt t_2, \\ v'(t), & \mbox{if }t_2 \lt t \lt t_3, \\ -\frac{v(t_3)}{t_4-t_3}, & \mbox{if }t_3 \lt t \lt t_4, \\ 0, & \mbox{if }t_4 \lt t \lt R. \end{cases} \end{equation}

Proceeding similarly to the proof of theorem 4.3, we reduce the following integrals to one dimension:

\begin{equation*} I\stackrel{\textrm{def}}{=} \frac{1}{n\omega}\int_{\Omega_\varepsilon} w(\rho_\varepsilon) F^*_\varepsilon(Du^\star_{\varepsilon,T})^p\,\mathrm{d} \mathsf{m}_\varepsilon\quad\mbox{and}\quad J\stackrel{\textrm{def}}{=}\frac{1}{n\omega}\int_{\Omega_\varepsilon} w(\rho_\varepsilon) W(\rho_\varepsilon) (u_{\varepsilon,T}^\star)^p\,\mathrm{d} \mathsf{m}_\varepsilon. \end{equation*}

For simplicity, denote $r=|x|$, $\overline{r}=\operatorname{arctanh}(r)$, and $\vartheta_{\varepsilon}=k_\varepsilon(1-\theta)$. Introduce the following notations:

\begin{align*} t\stackrel{\textrm{def}}{=}\rho_\varepsilon(x)=\overline{r}\cdot \frac{\overline{r}(1-\theta)+ \varepsilon }{\vartheta_{\varepsilon}(\overline{r}+\varepsilon)} \quad\text{and}\quad e(t)\stackrel{\textrm{def}}{=} \exp(-(n-1)ht). \end{align*}

By a simple computation, we obtain that

\begin{align*} r & =\tanh \varphi_\varepsilon(t), & \text{where}\quad \varphi_\varepsilon(t) & =\frac{t\vartheta_\varepsilon -\varepsilon+\sqrt{(t\vartheta_\varepsilon -\varepsilon)^2+4t\vartheta_\varepsilon\varepsilon(1-\theta)}}{2(1-\theta)}, \\ \,\mathrm{d} r & =\frac{1}{(\cosh\varphi_\varepsilon(t))^2}\varphi_\varepsilon'(t)\,\mathrm{d} t, & \text{where}\quad \varphi_\varepsilon'(t) & =\frac{k}{2}\left(1+\frac{t\vartheta_\varepsilon-\varepsilon+2\varepsilon(1-\theta)}{\sqrt{(t\vartheta_\varepsilon -\varepsilon)^2+4t\vartheta_\varepsilon\varepsilon(1-\theta)}}\right) , \\ \sigma_\varepsilon & = e(t)\cdot \frac{1}{\vartheta_\varepsilon^n} (\cosh\varphi_\varepsilon(t))^{n+1} (f_\varepsilon(t))^\frac{n+1}{2}, & \text{where}\quad f_\varepsilon(t) & =1-\frac{\varphi_\varepsilon^2(t) (\varphi_\varepsilon(t)+2 \varepsilon)^2\theta^2}{(\varphi_\varepsilon(t)+\varepsilon)^4}. \end{align*}

It follows that

(62)\begin{align} \sigma_\varepsilon r^{n-1}\,\mathrm{d} r & = e(t)\cdot\varphi'_\varepsilon(t) \frac{1}{\vartheta_\varepsilon^n}(f_\varepsilon(t))^\frac{n+1}{2} (\sinh \varphi_\varepsilon(t))^{n-1}\,\mathrm{d} t \stackrel{\textrm{def}}{=} e(t)\cdot\psi_\varepsilon(t)\,\mathrm{d} t. \end{align}

In the sequel, we provide suitable bounds for $\psi_\varepsilon(t)$. First, we observe that

\begin{equation*} (1-\theta)k_\varepsilon t \le \varphi_\varepsilon(t)\le k_\varepsilon t,\quad (1-\theta)k_\varepsilon \le \varphi_\varepsilon'(t)\le k_\varepsilon,\quad\mbox{and}\quad 1-\theta^2 \le f_\varepsilon(t)\le 1,\quad \forall t\ge 0. \end{equation*}

Next, we show that there exists $\widetilde{\varepsilon} \gt 0$ and $\widetilde c \gt 0$ such that

(63)\begin{equation} \widetilde{c}\cdot \sinh(k_\varepsilon t)\le \sinh(\varphi_\varepsilon(t))\le \sinh(k_\varepsilon t),\quad \forall t\ge 0,\varepsilon\in(0,\widetilde\varepsilon). \end{equation}

Since the hyperbolic sine is increasing, we obtain

\begin{equation*} \sinh((1-\theta)k_\varepsilon t)\le \sinh(\varphi_\varepsilon(t))\le \sinh(k_\varepsilon t),\quad \forall t\ge 0, \varepsilon \gt 0, \end{equation*}

thus, the second inequality of (63) holds. To verify the first inequality, define

\begin{equation*} \psi(k,t)=\frac{\sinh((1-\theta)k t)}{\sinh(kt)},\quad \forall k \gt 0,t\ge 0. \end{equation*}

Recall from remark 5.2(c) that there exists $\overline \varepsilon \gt 0$ and $\overline\kappa \gt \kappa$ such that

\begin{equation*} \kappa \le k_{\varepsilon}\le \overline \kappa,\quad \forall\varepsilon\in(0,\overline\varepsilon).\end{equation*}

On the one hand, observe that $t\mapsto \psi(\overline \kappa,t)$ is continuous and $\psi(\overline \kappa,t)\to 1-\theta \gt 0$ as t → 0, which implies that there exists $\widetilde{c_1} \gt 0$ and $\widetilde{t} \gt 0$ such that

\begin{equation*} \sinh((1-\theta)\overline \kappa t)\ge \widetilde{c_1}\cdot \sinh(\overline \kappa t),\quad \forall t\in[0,\widetilde{t}]. \end{equation*}

On the other hand, since $k\mapsto \psi(k,t)$ is decreasing, the above estimate holds for every $k_\varepsilon\in[\kappa,\overline\kappa]$. Hence,

\begin{equation*} \sinh((1-\theta)k_{\varepsilon} t)\ge \widetilde{c_1}\cdot \sinh(k_{\varepsilon} t),\quad \forall t\in[0,\widetilde{t}], \varepsilon\in(0,\overline\varepsilon). \end{equation*}

Now we focus on the case when $t \gt \widetilde{t}$. Studying the asymptotic behaviour of φ_ɛ we obtain

\begin{equation*} \varphi_\varepsilon(t)\ge t k_\varepsilon-\frac{\varepsilon\theta}{1-\theta}\stackrel{\textrm{def}}{=} tk_\varepsilon-s_\varepsilon,\quad \forall t \gt 0, \varepsilon\in(0,\overline\varepsilon). \end{equation*}

Note that $k_\varepsilon\to\kappa$ and $s_\varepsilon\to0$ as ɛ → 0. Thus, since $\widetilde{t} \gt 0$, there exists an $\widetilde{\varepsilon}\in(0,\overline{\varepsilon})$ such that

(64)\begin{equation} \varphi_\varepsilon(\widetilde t)\ge \widetilde{t}k_\varepsilon-s_\varepsilon \gt s_\varepsilon \gt 0,\quad \forall \varepsilon\in(0,\widetilde \varepsilon). \end{equation}

Since $t\mapsto \varphi_\varepsilon(t)$ is increasing, we also have

(65)\begin{equation} \varphi_\varepsilon(t)\ge tk_\varepsilon-s_\varepsilon \gt s_\varepsilon \gt 0,\quad \forall \varepsilon\in(0,\widetilde \varepsilon),t \gt \widetilde{t}. \end{equation}

It follows that

\begin{align*} \sinh(\varphi_\varepsilon(t)) & \ge \sinh(tk_\varepsilon-s_\varepsilon)=\sinh(tk_\varepsilon)\cosh(s_\varepsilon)-\cosh(tk_\varepsilon)\sinh(s_\varepsilon) \\ & =\sinh(tk_\varepsilon)\cdot\left(\cosh(s_\varepsilon)-\coth(tk_\varepsilon)\sinh(s_\varepsilon)\right). \end{align*}

Since the hyperbolic cotangent is decreasing, using relation (65), we obtain that

\begin{align*} \sinh(\varphi_\varepsilon(t)) & \ge \sinh(tk_\varepsilon)\cdot \left(\cosh(s_\varepsilon)-\coth(2s_\varepsilon)\sinh(s_\varepsilon)\right)= \sinh(tk_\varepsilon)\cdot \frac{1}{2\cosh(s_\varepsilon)} \end{align*}

Define $s_{\widetilde{\varepsilon}}=\frac{\widetilde{\varepsilon}\theta}{1-\theta}$, the monotonicity of the hyperbolic cosine implies

\begin{align*} \sinh(\varphi_\varepsilon(t)) & \ge \sinh(tk_\varepsilon)\cdot \frac{1}{2\cosh(s_{\widetilde{\varepsilon}})}\stackrel{\textrm{def}}{=} \widetilde{c_2}\cdot \sinh(tk_\varepsilon). \end{align*}

Thus, $\widetilde{c}=\min\{\widetilde{c_1},\widetilde{c_2}\}$ and $\widetilde{\varepsilon}$ yield inequality (63), for every $t\ge 0$.

Combining the above estimates, we obtain for every $t\ge 0$ and $0 \lt \varepsilon \lt \widetilde{\varepsilon}$ that

(66)\begin{align} & c \sinh(k_\varepsilon t)^{n-1}\stackrel{\textrm{def}}{=} \frac{(1-\theta^2)^\frac{n+1}{2}(\widetilde{c})^{n-1}}{\vartheta_{\varepsilon}^{n-1}}\sinh(k_\varepsilon t)^{n-1} \le \psi_\varepsilon(t)\nonumber\\ & \quad \le \frac{k_\varepsilon}{\vartheta^n}\sinh(k_\varepsilon t)^{n-1}\stackrel{\textrm{def}}{=} C \sinh(k_\varepsilon t)^{n-1}. \end{align}

Since v is positive and decreasing, the positive homogeneity of $F^*_\varepsilon$ and the relations from (61) imply

(67)\begin{equation} F^*_\varepsilon(Du^\star_{\varepsilon,T})=F^*_\varepsilon(v_{\varepsilon,T}'(\rho_\varepsilon)D\rho_\varepsilon)=\begin{cases} \frac{v(t_2)}{t_2-t_1}F^*_\varepsilon(D\rho_\varepsilon), & \text{if } \rho_\varepsilon\in(t_1,t_2), \\ |v'(\rho_\varepsilon)|F^*_\varepsilon(-D\rho_\varepsilon) & \text{if } \rho_\varepsilon\in(t_2,t_3), \\ \frac{v(t_3)}{t_4-t_3}F^*_\varepsilon(-D\rho_\varepsilon) & \text{if } \rho_\varepsilon\in(t_3,t_4), \\ 0, & \text{if }\rho_\varepsilon\in(0,t_1)\cup(t_4,R). \end{cases} \end{equation}

According to remark 5.2(d), one has

\begin{align*}& F^*_\varepsilon(D\rho_\varepsilon)=1,\quad\mbox{and}\quad F^*_\varepsilon(-D\rho_\varepsilon)=\frac{1-\frac{\overline{r}(\overline{r}+2\varepsilon)}{(\overline{r}+\varepsilon)^2}\theta}{1+\frac{\overline{r}(\overline{r}+2\varepsilon)}{(\overline{r}+\varepsilon)^2}\theta}\stackrel{\textrm{def}}{=} h_\varepsilon(\overline{r}),\nonumber\\ & \quad\mbox{where}\quad \frac{1-\theta}{1+\theta}\le h_\varepsilon(\overline{r})\le1.\end{align*}

Observe that $h_\varepsilon(\overline{r})$ is decreasing in $ \overline{r}$, increasing in ɛ, and approaches $\frac{1-\theta}{1+\theta}=\frac{1}{\lambda}$ as ɛ → 0. Clearly one has

\begin{equation*}\overline r=\varphi_\varepsilon(t)\ge (1-\theta)k_\varepsilon t\ge(1-\theta)\kappa t,\end{equation*}

thus, we also obtain

(68)\begin{equation} h_\varepsilon(\overline{r})=h_\varepsilon(\varphi_\varepsilon(t))\le h_\varepsilon((1-\theta)\kappa t)\le 1. \end{equation}

As before, denote by ω the volume of the Euclidean unit ball of dimension n.

On the one hand, relations (62) and (67) yield

\begin{align*} I & =\frac{v(t_2)^p}{(t_2-t_1)^p}\int_{t_1}^{t_2} w(t) \psi_\varepsilon(t)e(t)\,\mathrm{d} t + \int_{t_2}^{t_3} w(t)|v'(t)|^p h_\varepsilon^p(\varphi_\varepsilon(t)) \psi_\varepsilon(t)e(t)\,\mathrm{d} t \\ & \qquad +\frac{v(t_3)^p}{(t_4-t_3)^p}\int_{t_3}^{t_4} w(t) h_\varepsilon^p(\varphi_\varepsilon(t))\psi_\varepsilon(t)e(t)\,\mathrm{d} t. \end{align*}

Applying relations (66), (68), and using the definition of $\mathcal{I}^{k,h}$ from (60), we obtain

\begin{align*} I & \le \frac{v(t_2)^p}{(t_2-t_1)^p}\int_{t_1}^{t_2} w(t) \psi_\varepsilon(t)e(t)\,\mathrm{d} t+ \int_{t_2}^{t_3} w(t)|v'(t)|^p h_\varepsilon^p((1-\theta)\kappa t) \psi_\varepsilon(t)e(t)\,\mathrm{d} t \\ & \qquad +\frac{v(t_3)^p}{(t_4-t_3)^p}\int_{t_3}^{t_4} w(t) h_\varepsilon^p((1-\theta)\kappa t)\psi_\varepsilon(t)e(t)\,\mathrm{d} t \\ &\le \frac{Cv(t_2)^p}{(t_2-t_1)^p}\int_{t_1}^{t_2} w(t) \sinh(k_\varepsilon t)^{n-1}e(t)\,\mathrm{d} t+ h_\varepsilon^p((1-\theta)\kappa t_2)\int_{t_2}^{t_3} w(t)|v'(t)|^p \psi_\varepsilon(t)e(t)\,\mathrm{d} t \\ & \qquad +\frac{C h_\varepsilon^p((1-\theta)\kappa t_3)v(t_3)^p}{(t_4-t_3)^p}\int_{t_3}^{t_4} w(t)\sinh(k_\varepsilon t)^{n-1}e(t)\,\mathrm{d} t \\ & \le \frac{Cv(t_2)^p\cdot \mathcal{I}^{k_\varepsilon,h}(w;t_1,t_2)}{(t_2-t_1)^p} + h_\varepsilon^p((1-\theta)\kappa t_2)\int_{t_2}^{t_3} w(t)|v'(t)|^p \psi_\varepsilon(t)e(t)\,\mathrm{d} t \\ & \quad +\frac{C v(t_3)^p\cdot\mathcal{I}^{k_\varepsilon,h}(w;t_3,t_4)}{(t_4-t_3)^p}. \end{align*}

By relation (59), for every $t\in[t_2,t_3]$ one has

\begin{equation*} \frac{1}{W(t)}\left|\frac{v'(t)}{v(t)}\right|^p=\frac{1}{W(t)}\left(-\frac{v'(t)}{v(t)}\right)^p=\frac{G(t)^{p'}}{W(t)}. \end{equation*}

The above function is continuous, thus, it attains it maximum on $[t_2,t_3]$, hence we have

\begin{align*} \int_{t_2}^{t_3} w(t)|v'(t)|^p \psi_\varepsilon(t)\,\mathrm{d} t&=\int_{t_2}^{t_3} \frac{1}{W(t)}\left|\frac{v'(t)}{v(t)}\right|^p w(t) W(t)(v(t))^p \psi_\varepsilon(t)\,\mathrm{d} t\\ &\le \max_{t\in[t_2,t_3]} \frac{G(t)^{p'}}{W(t)}\cdot \int_{t_2}^{t_3} w(t) W(t) (v(t))^p \psi_\varepsilon(t)e(t)\,\mathrm{d} t. \end{align*}

On the other hand, we obtain

(69)\begin{align} J& =\int_{0}^{R} w(t) W(t) (v_{\varepsilon,T}(t))^p \psi_\varepsilon(t)e(t)\,\mathrm{d} t \ge \int_{t_2}^{t_3} w(t) W(t) (v(t))^p \psi_\varepsilon(t)e(t)\,\mathrm{d} t \nonumber\\ & \ge c\cdot \mathcal{I}^{k_\varepsilon,h}(wW v^p;t_2,t_3). \end{align}

Combining the above results and using both inequalities of (69), it follows that

\begin{align*} \frac{\lambda^p I}{J} & \le \lambda^p h_\varepsilon^p((1-\theta)\kappa t_2)\cdot \max_{t\in[t_2,t_3]} \frac{G(t)^{p'}}{W(t)} \\ & \qquad +\frac{\lambda^p C}{c}\left(\frac{v(t_2)^p\cdot \mathcal{I}^{k_\varepsilon,h}(w;t_1,t_2)}{(t_2-t_1)^p\cdot \mathcal{I}^{k_\varepsilon,h}(wW v^p;t_2,t_3)}+\frac{v(t_3)^p\cdot \mathcal{I}^{k_\varepsilon,h}(w;t_3,t_4)}{(t_4-t_3)^p\cdot \mathcal{I}^{k_\varepsilon,h}(wW v^p;t_2,t_3)}\right) \\ & =\lambda^p\cdot (l_2^{k_\varepsilon,h}(\delta))^p\cdot l_1^{k_\varepsilon,h}(\delta)+\frac{\lambda^p C}{c}\cdot l_0^{k_\varepsilon,h}(\delta). \end{align*}

By our assumptions, the last relation approaches 1 as $\delta\to\infty$, thus, the inequality (58) is sharp on $\mathcal{F}_{\kappa,h,\lambda}$.

Observe that in contrast with theorem 4.3, the limits appearing in the statement of the above result are strongly dependent on ɛ. The reason for this stems from the observation of remark 5.2(a): Since the flag curvature is not increasing along $\nabla\rho$, we need to work with k_ɛ instead of κ, which explains the dependence on ɛ. Fortunately, by suitable choices of $\varepsilon(\delta)$, one can provide sharpness results in this setting, as well. Such an example is presented in the sequel.

5.3. The sharpness of a McKean-type spectral gap estimate

In this section, we prove a sharpness result concerning a McKean-type spectral gap estimate on $\mathcal{F}_{\kappa,h,\lambda}$ (which for p = 2 reduces to theorem 1.2). Our last result can be stated as follows.

Theorem 5.4. Let $n\ge 2$, $\lambda,p\in (1,\infty)$, and $\kappa \gt h\ge0$. For every $(\mathbb{B}^n,F_\varepsilon,\mathsf{m}_\varepsilon)\in \mathcal{F}_{\kappa,h,\lambda}$ and $u\in C_0^\infty(\mathbb{B}^n)$ one has

(70)\begin{align} \lambda^p\displaystyle\int_{\mathbb{B}^n} F_\varepsilon^*(Du)^p\,\mathrm{d} \mathsf{m}_\varepsilon\ge c_{n,p,\kappa,h}\int_{\mathbb{B}^n} |u|^p \,\mathrm{d} \mathsf{m}_\varepsilon,\quad\text{where}\quad c_{n,p,\kappa,h}=\left(\frac{(n-1)(\kappa-h)}{p}\right)^p. \end{align}

Moreover, $c_{n,p,\kappa,h}$ is sharp in $\mathcal{F}_{\kappa,h,\lambda}$, in the sense that it is the greatest constant with this property.

Proof. The validity of inequality (70) follows from theorem 3.4. To prove the sharpness, we use theorem 5.3. Recall the definition of $\widetilde\varepsilon$ from (64). Let δ > 1, such that $\varepsilon=\frac{1}{\delta}\in(0,\widetilde\varepsilon)$, and define

\begin{equation*} t_1(\delta)=\delta,\quad t_2(\delta)=2\delta,\quad t_3(\delta)=3\delta,\quad\mbox{and}\quad t_4(\delta)=4\delta. \end{equation*}

By a simple computation, we obtain

\begin{align*}l_2^{k_\varepsilon,h}& =h_\varepsilon((1-\theta)\kappa t_2)=\frac{1+4\delta^2(1-\theta)^2\kappa+4\delta^4(1-\theta)^3\kappa^2}{1+4\delta^2(1-\theta^2)\kappa+4\delta^4(1-\theta)^2(1+\theta)\kappa^2}\nonumber\\ & \quad \to \frac{1-\theta}{1+\theta}=\lambda,\quad\text{as}\quad \delta\to \infty.\end{align*}

Consider the following functions (see also the proof of theorem 3.4):

\begin{align*} w(t)= 1, \quad W(t)& =c_{n,p,\kappa,h}, \quad G(t)\equiv (c_{n,p,\kappa,h})^\frac{p-1}{p},\quad\mbox{and}\quad v(t)^p=e^{-(n-1)t (\kappa-h)},\nonumber\\ & \forall t \gt 0. \end{align*}

Since $k_\varepsilon \gt \kappa \gt h$, the following expressions are strictly increasing in t:

\begin{align*} w(t)\sinh(k_\varepsilon t)^{n-1}e^{-(n-1)h t} & = \sinh(k_\varepsilon t)^{n-1}e^{-(n-1)h t}, \\ w(t)W(t)v(t)^p\sinh(k_\varepsilon t)^{n-1}e^{-(n-1)h t} & =c_{n,p,\kappa,h}\sinh(k_\varepsilon t)^{n-1}e^{-(n-1)\kappa t}. \end{align*}

On the one hand, we have

\begin{align*} \frac{v(t_2)^p\cdot\mathcal{I}^{k_\varepsilon,h}(w;t_1,t_2)}{(t_2-t_1)^p\cdot \mathcal{I}^{k_\varepsilon,h}(wWv^p;t_2,t_3)} & \le \frac{v(t_2)^p}{(t_2-t_1)^p}\cdot \frac{(t_2-t_1)\sinh(k_\varepsilon t_1)^{n-1} e^{-(n-1)ht_1}}{(t_3-t_2)c_{n,p,\kappa,h}\sinh(k_\varepsilon t_3)^{n-1}e^{-(n-1)h t_3}} \\ & =\frac{e^{(n-1)\delta(\kappa+h)}\sinh(\delta k_\varepsilon)^{n-1}}{\delta^pc_{n,p,\kappa,h} \sinh(3\delta k_\varepsilon)^{n-1}}\\ &= \frac{e^{(n-1)\delta(\kappa+h-2k_\varepsilon)}}{\delta^pc_{n,p,\kappa,h}}\cdot \frac{e^{(n-1)\delta(2k_\varepsilon)}\sinh(\delta k_\varepsilon)^{n-1}}{\sinh(3\delta k_\varepsilon)^{n-1}}\\ & \quad \to 0\cdot 1=0, \quad\text{as}\quad \delta\to\infty. \end{align*}

On the other hand, observe that

\begin{align*} \frac{v(t_3)^p\cdot\mathcal{I}^{k_\varepsilon,h}(w;t_3,t_4)}{(t_4-t_3)^p\cdot \mathcal{I}^{k_\varepsilon,h}(wWv^p;t_2,t_3)} &\le \frac{v(t_3)^p}{(t_4-t_3)^p}\cdot \frac{(t_4-t_3)\sinh(\kappa t_3)^{n-1} e^{-(n-1)ht_3}}{(t_3-t_2)c_{n,p,\kappa,h}\sinh(\kappa t_3)^{n-1}e^{-(n-1)\kappa t_3}} \\ &=\frac{1}{\delta^pc_{n,p,\kappa,h} }\to 0,\quad\mbox{as }\delta\to\infty. \end{align*}

Combining the above two limits, we obtain $l_1^{k_\varepsilon,h}(\delta)\to 0$ as $\delta\to\infty$.

Finally, since

\begin{equation*} \frac{G(t)^{p'}}{W(t)}=1, \quad\forall t \gt 0, \end{equation*}

we also obtain $l_1^{k_\varepsilon,h}(\delta)=1$. Thus, theorem 5.3 yields the sharpness of the inequality (70), which concludes the paper.