1 Introduction

The classical sharp $ L^{p} $ Sobolev inequality in $ \mathbb {R}^n $ is stated for functions from the homogeneous Sobolev space $\dot {W}^{1,p}(\mathbb {R}^n)$ (see, e.g.,[Reference Leoni19, Chapter 11]) defined by

$$ \begin{align*} \dot{W}^{1,p}(\mathbb{R}^n)=\{f\in L^{p^*}(\mathbb{R}^n): \nabla f\in L^p(\mathbb{R}^n)\}, \end{align*} $$

where $1\le p< n $ and $p^*=\frac {np}{n-p}$ . That is, for any $f \in \dot {W}^{1,p}(\mathbb {R}^n)$ , there exists an optimal constant $a_{n, p}$ such that

(1.1) $$ \begin{align} \|\nabla f\|_p \geq a_{n, p} \|f\|_{\frac{np}{n-p}}, \end{align} $$

where

$$ \begin{align*}a_{n,p} = n^{\frac{1}{p}}\left( \frac{p-1}{n-p} \right)^{\frac{1-p}{p}} \bigg( \frac{\omega_n\Gamma(\frac{n}{p}) \Gamma(n+1-\frac{n}{p})}{\Gamma(n)} \bigg)^{\frac{1}{n}}, \end{align*} $$

$ \left \| \cdot \right \| _{p} $ denotes the $ L^{p} $ norm of the Euclidean norm of functions, and $ \nabla f $ is the gradient of $ f $ . Here, $\omega _n=\pi ^{n/2} / \Gamma (1+n/2)$ denotes the volume of the Euclidean unit ball $B^n$ in $\mathbb {R}^n$ , and $\Gamma (\cdot )$ is the Gamma function. Inequality (1.1) goes back to Federer and Fleming [Reference Federer and Fleming9] and Maz’ya[Reference Maz’ya28] for $ p=1 $ , and to Aubin[Reference Aubin2] and Talenti[Reference Talenti35] for $ 1<p<n $ . The sharp $L^1$ Sobolev inequality is known to be equivalent to the isoperimetric inequality.

A major advance regarding (1.1) was achieved by Lutwak, Yang, and Zhang [Reference Lutwak, Yang and Zhang26], which built upon the foundational work of Zhang [Reference Zhang43] for the case $p=1$ . They established the first affine-invariant form of the $ L^p $ Sobolev inequality, now called the sharp $ L^p $ affine Sobolev inequality, which replaces the $L^p$ norm of gradient in (1.1) by the $L^p$ affine energy (a suitable average of the norm of one-dimensional gradient projections). More precisely, if $ 1 \leq p < n $ and $ f \in \dot {W}^{1, p}(\mathbb {R}^n) $ , then

(1.2) $$ \begin{align} \mathcal{E}_p(f) \geq c_{n,p}^{\frac1p}a_{n,p} \|f\|_{\frac{n p}{n-p}}. \end{align} $$

Here, we write $c_{n,p}$ as

(1.3) $$ \begin{align} c_{n,p}=\frac{2 \omega_{n+p-2}}{n\omega_{n} \omega_{p-1}} \end{align} $$

for convenience and explicit expressions for the optimal constant in (1.2) were found in [Reference Zhang43] (for $p = 1$ ) and in [Reference Lutwak, Yang and Zhang26] (for $p> 1$ ). The functional $\mathcal {E}_p(f)$ is usually called the $L^p$ affine energy of f defined by

$$ \begin{align*}\mathcal{E}_p(f) = \Big( \int_{G_{n, 1}} \Big( \int_{\mathbb{R}^n} \left| \text{P}_{\xi}(\nabla f(x)) \right|{}^p dx \Big)^{-\frac{n}{p}} d\xi \Big)^{-\frac{1}{n}} , \end{align*} $$

where we denote by $G_{n, m}$ the Grassmann manifold of m-dimensional subspaces in $\mathbb {R}^n$ ( $1\leq m\leq n$ ), $ \text {P}_{\xi }(\nabla f(x)) $ the orthogonal projection of $ \nabla f(x) $ onto $ \xi \in G_{n, m} $ , and $ d\xi $ is the rotationally invariant probability measure on $ G_{n, m} $ . Note that $G_{n,1}$ can be identified as the hemisphere of the unit sphere $S^{n-1} $ and $G_{n,n}$ can be identified as $\mathbb {R}^n$ . As the name suggests, inequality (1.2) is invariant under affine transformations of $ \mathbb {R}^n $ , whereas inequality (1.1) is invariant only under rotations and translations. Subsequently, Wang [Reference Wang37] and, independently, Nguyen [Reference Nguyen32] characterized the equality conditions in (1.2) for $1 < p < n$ if and only if

$$ \begin{align*}f(x)=\pm \Big(a + |\phi(x - x_0)|^{\frac{p}{p-1}}\Big)^{\frac{p-n}{p}}\end{align*} $$

with $\phi \in \mathrm {GL}(n)$ (the group of general linear transformations), $a>0$ , and $x_0 \in \mathbb {R}^n$ . For $p=1$ , Wang [Reference Wang36] showed that the extremals are precisely the characteristic functions of ellipsoids when functions are from BV( $\mathbb {R}^n$ ) (the set of functions of bounded variation on $ \mathbb {R}^n $ ). Moreover, the Hölder inequality immediately implies that (see [Reference Lutwak, Yang and Zhang26])

(1.4) $$ \begin{align} \|\nabla f\|_p\geq c_{n,p}^{-\frac1p}\mathcal{E}_p(f), \end{align} $$

which shows that the affine form (1.2) provides a significantly stronger inequality than (and directly implies) the classical one (1.1). The $L^p$ affine Sobolev inequality has its basis in the $L^p$ affine isoperimetric inequalities [Reference Lutwak, Yang and Zhang25] and has had significant impact in both convex geometry and analysis (see, e.g., [Reference Cianchi, Lutwak, Yang and Zhang3, Reference De Nápoli, Haddad, Jiménez and Montenegro5, Reference Fang, Xu, Zhou and Zhu7, Reference Haberl and Schuster11–Reference Kniefacz and Schuster18, Reference Nguyen33, Reference Wang and Wang38–Reference Xiao41]).

Let $ C_0^\infty (\mathbb {R}^n) $ denote the set of functions on $ \mathbb {R}^n $ that are infinitely differentiable and have compact support. It is well known that $ \dot {W}^{1,p}(\mathbb {R}^n) $ is the closure of $ C_0^\infty (\mathbb {R}^n) $ under the norm $\|\nabla f\|_p$ . Another important development for (1.1) is the sharp Lorentz–Sobolev inequality [Reference Alvino1]: if $ 1 \leq p < n $ and $ f \in C_0^\infty (\mathbb {R}^n) $ , then

(1.5) $$ \begin{align} \|\nabla f\|_p^p \geq p^{-p}(n - p)^{p-1} n \omega_n^{\frac{p}{n}} \int_0^\infty V( [f]_t)^{\frac{n-p}{n}} dt^p, \end{align} $$

where $ [f]_t = \{ x \in \mathbb {R}^n : |f(x)| \geq t \} $ is the level set of f and $V(\cdot )$ denotes the n-dimensional volume (Lebesgue measure). Actually, the Lorentz–Sobolev inequality (1.5) implies (1.1) but without sharp constant unless $ p = 1 $ (see, e.g., [Reference Ludwig, Xiao and Zhang23]).

Note that level sets in general are not convex. However, a powerful technique of the $ L^p $ convexification introduced by Lutwak, Yang, and Zhang [Reference Lutwak, Yang and Zhang26, Reference Zhang43] via the solution of the $ L^p $ Minkowski problem makes level sets always convex. Using the $ L^p $ convexification of level sets $ \langle f \rangle _t$ (an origin-symmetric convex body in $\mathbb {R}^n$ ; see (2.6)), instead of $ [f]_t $ , Ludwig, Xiao, and Zhang [Reference Ludwig, Xiao and Zhang23] established a reinforcing version of inequality (1.5), named the sharp convex Lorentz–Sobolev inequality: if $1\leq p<n$ and $f\in C_{0}^{\infty }(\mathbb {R}^{n})$ , then

(1.6) $$ \begin{align} \|\nabla f\|_p^p \geq n\omega_n^{\frac{p}{n}} \int_0^{\infty} V(\langle f \rangle_t)^{\frac{n-p}{n}} dt. \end{align} $$

The sharp convex Lorentz–Sobolev inequality (1.6) immediately implies the $ L^{p} $ Sobolev inequality with the best constant by using the following sharp inequality due to Lutwak, Yang, and Zhang [Reference Lutwak, Yang and Zhang26]:

(1.7) $$ \begin{align} \int_0^\infty V(\langle f \rangle_t)^{\frac{n-p}{n}} dt \geq \frac{a_{n,p}^p}{n\omega_n^{p/n}} \|f\|_{\frac{np}{n-p}}^p, \end{align} $$

and the equality case corresponds to inequality (1.2). It was shown in [Reference Ludwig, Xiao and Zhang23] that the convex version (1.6) is stronger than the classical one (1.5). Furthermore, as noted in [Reference Ludwig, Xiao and Zhang23], inequality (1.6) can also be derived from the proof of the $L^{p}$ affine Sobolev inequality (1.2).

In fact, the sharp convex Lorentz–Sobolev inequality (1.6) can be strengthened in the following two inequalities [Reference Ludwig, Xiao and Zhang23]: if $ f \in C_0^{\infty }(\mathbb {R}^n)$ and $ 1 \leq p < n $ , then

(1.8) $$ \begin{align} \|\nabla f\|_p^p \geq \Big( \frac{p-1}{n-p} \Big)^{p-1} \int_0^{\infty} C_p(\langle f \rangle_t) \, dt \geq n \omega_n^{\frac{p}{n}} \int_0^{\infty} V(\langle f \rangle_t)^{\frac{n-p}{n}} \, dt, \end{align} $$

and

(1.9) $$ \begin{align} \|\nabla f\|_p^p \geq \int_0^{\infty} \Phi_p(\langle f \rangle_t) \, dt \geq n \omega_n^{\frac{p}{n}} \int_0^{\infty} V(\langle f \rangle_t)^{\frac{n-p}{n}} \, dt. \end{align} $$

Here, $C_p(K)$ is the $L^{p}$ variational capacity for a compact set K in $\mathbb {R}^n$ (see, e.g., [Reference Colesanti and Salani4, Reference Evans and Gariepy6, Reference Maz’ya29, Reference Maz’ya31]) defined by

$$ \begin{align*}C_p(K) = \inf\Big\{\int_{\mathbb{R}^n} |\nabla f(x)|^p dx : f\in\dot{W}^{1,p}(\mathbb{R}^n) \text{ with } f \geq \textbf{1}_K \Big\}, \end{align*} $$

where $\textbf {1}_K$ stands for the characteristic function of K. The functional $\Phi _p(K)$ is the $L^{p}$ integral affine surface area [Reference Lutwak, Yang and Zhang25] for a Lipschitz star body K (see Section 2) in $\mathbb {R}^n$ defined by

$$ \begin{align*}\Phi_p(K) = c_{n,p}^{-1} \Big( \int_{G_{n,1}} \Big( nV_p\left(K, B^n \cap \xi\right) \Big)^{-\frac{n}{p}} d\xi \Big)^{-\frac{p}{n}}, \end{align*} $$

where $V_p(\cdot ,\cdot )$ is the $L^p$ mixed volume defined in (2.3). Moreover, (1.8) and (1.9) are sharp in the sense that all inequalities approach an equality as f tends to the characteristic function of an origin-centered ball for $ p = 1 $ , and as $ f $ tends to $(a +b|x|^{\frac {p}{p-1}})^{\frac {p-n}{p}} $ with $a, b>0$ for $ 1 < p < n $ .

Recently, Xiao [Reference Xiao40] introduced an affine version of the $L^{p}$ variational capacity – the p-affine capacity (denoted by $C_{p,d}(\cdot )$ ). In this article, we prefer to use the notation $ C_{1,p}(\cdot ) $ rather than $C_{p,d}(\cdot )$ for convenience. That is, for $1 \le p < n$ and a compact set K in $\mathbb {R}^n$ ,

$$ \begin{align*} C_{1,p}(K)= \inf\Big\{c_{n,p}^{-1}\mathcal{E}_{p}^p(f):f\in{\dot{W}^{1,p}(\mathbb{R}^n) \text{ with } f \ge \textbf{1}_{K} }\Big\}. \end{align*} $$

It is derived from [Reference Lutwak, Yang and Zhang26, inequality (7.1)] and [Reference Zhang43, inequality (4.8)] that if K is an origin-symmetric convex body in $\mathbb {R}^n$ , then

(1.10) $$ \begin{align} C_p(K)\geq C_{1,p}(K). \end{align} $$

The $ p$ -affine capacity strengthens both the $L^p$ isocapacitary inequality [Reference Maz’ya30] and the $L^{p} $ affine isoperimetric inequality [Reference Lutwak, Yang and Zhang25] in Xiao’s p-affine capacity inequality [Reference Xiao40]:

(1.11) $$ \begin{align} \Phi_p(K) \geq \Big(\frac{p-1}{n-p}\Big)^{p-1} C_{1,p}(K) \geq n\omega_n^{\frac{p}{n}}V(K)^{\frac{n-p}{n}}, \end{align} $$

with equality if K is an origin-centered ball in $\mathbb {R}^n$ . Naturally, together with inequality (1.4), Xiao [Reference Xiao40] derived the following Lutwak–Xiao–Yang–Zhang’s (abbreviated as LXYZ’s) $L^p$ affine Sobolev inequality chain: if $f \in \dot {W}^{1,p}(\mathbb {R}^n)$ and $1 \leq p < n$ , then

(1.12) $$ \begin{align} \|\nabla f\|_p^p &\geq c_{n,p}^{-1}\mathcal{E}_p^p(f) \geq \int_0^\infty \Phi_p(\langle f \rangle_t) dt \geq \Big(\frac{p-1}{n-p}\Big)^{p-1} \int_0^\infty C_{1,p}(\langle f \rangle_t) dt \nonumber\\& \geq n \omega_n^{\frac{p}{n}} \int_0^\infty V(\langle f \rangle_t)^{\frac{n-p}{n}} dt\geq a_{n, p}^p \|f\|_{\frac{np}{n-p}}^p , \end{align} $$

with equality if f tends to the characteristic function of an origin-centered ball for $ p = 1 $ , and $ f $ tends to $(a +b|x|^{\frac {p}{p-1}})^{\frac {p-n}{p}} $ with $a, b>0$ for $ 1 < p < n $ . The above inequality chain was also restated by Li and Zhu [Reference Li and Zhu22], who used $L^{p}$ polar projection bodies to provide a new proof for the second inequality in (1.12). A more general version of the chain (1.12) was established by Wang and Wang [Reference Wang and Wang38] by using the general $ L^p $ affine capacity [Reference Hong and Ye17] and integral surface area, together with the asymmetric $ L^{p} $ convexification of level sets.

The aim of this article is to establish the Grassmannian form of LXYZ’s $L^p$ affine Sobolev inequality chain. To do this, we will first introduce some definitions related to Grassmann manifolds.

The Grassmannian $L^p$ affine energy $ \mathcal {E}_{m,p}(\cdot ) $ was introduced by Kniefacz and Schuster [Reference Kniefacz and Schuster18] defined by, for $ 1\le p<n $ , $ 1\le m\le n $ , and $f \in \dot {W}^{1,p}(\mathbb {R}^n)$ ,

(1.13) $$ \begin{align} \mathcal{E}_{m,p}(f) = \Big( \int_{G_{n,m}} \Big( c_{m,p} \int_{\mathbb{R}^n} |\text{P}_\xi(\nabla f(x))|^p dx \Big)^{-\frac{n}{p}} d\xi \Big)^{-\frac{1}{n}}, \end{align} $$

where $c_{m,p}$ is given in (1.3). They subsequently proved the sharp Grassmannian $L^p$ Sobolev inequality, extending the classical $ L^{p} $ affine Sobolev inequality (1.2) to the following Grassmannian form:

(1.14) $$ \begin{align} \mathcal{E}_{m,p}(f) \geq c_{n,p}^{\frac{1}{p}} a_{n,p} \|f\|_{\frac{np}{n-p}}. \end{align} $$

When $ 1<p<n$ , equality in (1.14) holds for $ m>1 $ if and only if $ f(x) = \pm (a + |b(x - x_0)|^{\frac {p}{p-1}})^{\frac {p-n}{p}}$ with $ a,b>0 $ and $ x_{0}\in {\mathbb {R}^{n}} $ ; for $ m=1 $ if and only if $f(x) = \pm (a + |\phi (x - x_0)|^{\frac {p}{p-1}})^{\frac {p-n}{p}}$ with $ \phi \in \mathrm {GL}(n) $ , $ a>0 $ , and $ x_{0}\in {\mathbb {R}^{n}} $ . When $p = 1$ , the authors [Reference Kniefacz and Schuster18] further extended (1.14) to functions from $\mathrm {BV}(\mathbb {R}^n)$ and showed that equality in (1.14) if and only if f is a multiple of the characteristic function of a ball when $m> 1$ and an ellipsoid when $m = 1$ . Actually, the case $m = 1$ of inequality (1.14) immediately reduces to the $L^p$ affine Sobolev inequality (1.2).

The Grassmannian $L^{p}$ integral affine surface area $\Phi _{m,p}(\cdot )$ , recently introduced by Ye, Lu, and Huang [Reference Ye, Lu and Huang42], extends the $L^{p}$ integral affine surface area from the case of one-dimensional subspaces to arbitrary m-dimensional subspaces. Specifically, for $p\geq 1$ , $1 \leq m \leq n$ , and a Lipschitz star body K in $\mathbb {R}^n$ ,

(1.15) $$ \begin{align} \Phi_{m,p}(K) = \frac{c_{m,p}}{c_{n,p}}\Big( \int_{G_{n,m}} \Big( n V_p(K, B^n \cap \xi) \Big)^{-\frac{n}{p}} d\xi \Big)^{-\frac{p}{n}}. \end{align} $$

The normalization above is chosen so that $\Phi _{m,p}(B^n) = n\omega _n$ (see (4.2)). By using inequality (1.14), the authors of [Reference Ye, Lu and Huang42] extended inequalities (1.9) to the following Grassmannian form: for $ 1\le p<n $ , $ 1\le m\le n $ , and $f \in C_0^\infty (\mathbb {R}^n)$ ,

(1.16) $$ \begin{align} \| \nabla f \| _p^p \geq \int_0^\infty \Phi_{m,p}(\langle f \rangle_t) dt \geq n\omega_n^{\frac{p}{n}} \int_0^\infty V(\langle f \rangle_t)^{\frac{n-p}{n}} dt, \end{align} $$

with equality when f tends to the characteristic function of an origin-centered ball for $ p = 1 $ , and when $ f $ tends to $(a +b|x|^{\frac {p}{p-1}})^{\frac {p-n}{p}} $ with $a, b>0$ for $ 1 < p < n $ .

In this article, we will introduce the following Grassmannian form of p-affine capacities defined by, for $ 1\le p<n $ , $ 1\le m\le n $ , and a compact subset $ K \subset \mathbb {R}^{n}$ ,

(1.17) $$ \begin{align} C_{m,p}(K)=\inf\Big \{c_{n,p}^{-1} \mathcal{E}_{m,p}^p(f): f\in{\dot{W}^{1,p}(\mathbb{R}^n)\text{ with } f \ge \textbf{1}_{K} }\Big \}. \end{align} $$

The case $m=1$ of $C_{m,p}(\cdot )$ is just the $ p$ -affine capacity. The case $m=n-1$ of $C_{m,p}(\cdot )$ can be seen as the sine version (the orthogonal projection of $\nabla f$ onto the $(n-1)$ -dimensional subspace) of p-affine capacities.

Firstly, we have the following reinforcing geometric inequalities, which split and then strengthen the $ L ^{p} $ affine isoperimetric inequality.

Theorem 1.1 Let $ 1\le p<n $ and $ 1\le m\le n $ . If $ K $ is a Lipschitz star body in $\mathbb {R}^n $ , then

(1.18) $$ \begin{align} \Phi_{m,p}(K)\ge\Big(\frac{p-1}{n-p}\Big) ^ {p-1} C_{m,p}(K)\geq {n \omega_n^{\frac{p}{n}} V(K)^{\frac{n-p}{n}}}, \end{align} $$

with equality if $ K $ is a ball containing the origin in its interior for $ p=1 $ , and $ K $ is an origin-centered ball for $ 1<p<n $ .

Secondly, based on Theorem 1.1 and (1.7), we establish the Grassmannian form of LXYZ’s $L^p$ affine Sobolev inequality chain in the following theorem.

Theorem 1.2 Let $ 1\le p<n $ and $ 1\le m\le n $ . If $ f\in \dot {W}^{1,p}(\mathbb {R}^n) $ , then

(1.19) $$ \begin{align} \| \nabla f \| _p^p &\geq c_{n,p}^{-1}\mathcal{E}_{m,p}^p(f)\geq \int_0^\infty \Phi_{m,p}(\langle f \rangle_t) dt \geq \Big( \frac{p-1}{n-p} \Big) ^ {p-1} \int_0^\infty C_{m,p}(\langle f \rangle_t) dt \nonumber\\ &\geq n \omega_n^{\frac{p}{n}} \int_0^\infty V(\langle f \rangle_t)^{\frac{n-p}{n}} dt\geq a_{n, p}^p \|f\|_{\frac{np}{n-p}}^p , \end{align} $$

with equality when f tends to the characteristic function of an origin-centered ball for $ p = 1 $ , and when $ f $ tends to $(a +b|x|^{\frac {p}{p-1}})^{\frac {p-n}{p}} $ with $a, b>0$ for $ 1 < p < n $ .

Clearly, the case $ m=1 $ of (1.19) is just the inequality chain (1.12). The case $m= n-1$ is the sine counterpart of (1.12) since $\mathcal {E}_{n-1,p}(\cdot )$ , $\Phi _{n-1,p}(\cdot ),$ and $C_{n-1,p}(\cdot )$ are all the corresponding sine versions (see, e.g., [Reference Li, Xi and Zhang21] for more information).

The rest of this article is organized as follows: In Section 2, the background materials are provided. Basic properties for the newly defined $C_{m,p}(\cdot )$ are listed in Section 3. The proofs of Theorems 1.1 and 1.2 are presented in Section 4.