Zhang's Affine Sobolev Inequality

• Zhang’s affine Sobolev inequality is an affine-invariant refinement of the classical L¹ Sobolev inequality, using directional L¹ norms of gradients to achieve a sharper bound.
• It employs polar projection bodies to encode affine invariance under volume-preserving transformations, thereby identifying ellipsoidal extremals rather than smooth bubble functions.
• Extensions to BV functions, fractional orders, and spectral theory underline its foundational role in linking affine geometry with advanced Sobolev-type inequalities.

Zhang’s affine Sobolev inequality is the sharp affine-invariant refinement of the $L^{1}$ Sobolev inequality on $\mathbb R^{n}$. For $n\ge2$ and $f\in C_c^{1}(\mathbb R^{n})$, it bounds the critical norm $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$ by an affine energy built from the directional $L^{1}$-norms of $\nabla f$, and the classical Euclidean Sobolev inequality appears as a strictly weaker comparison. In later terminology it is also called the Sobolev–Zhang inequality, and Wang extended it from $C_c^{1}(\mathbb R^{n})$ to $BV(\mathbb R^{n})$ (Haddad et al., 2022).

1. Statement and analytic form

In the normalization used in "Affine Fractional Sobolev and Isoperimetric Inequalities" (Haddad et al., 2022), Zhang’s inequality reads

$$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$

Here $\mathbb R^{n}$0 is the volume of the Euclidean unit ball in $\mathbb R^{n}$1, $\mathbb R^{n}$2 is the surface measure of $\mathbb R^{n}$3, and the critical exponent is

$\mathbb R^{n}$4

The first inequality is Zhang’s affine Sobolev inequality; the second is the classical Euclidean $\mathbb R^{n}$5 Sobolev inequality, which is strictly weaker (Haddad et al., 2022).

The affine energy is the directional negative-power mean

$\mathbb R^{n}$6

Its defining feature is that it is invariant under volume-preserving affine transformations, whereas $\mathbb R^{n}$7 is invariant only under Euclidean isometries and scaling (Haddad et al., 2022).

Later papers often use an equivalent normalization. For $\mathbb R^{n}$8, "On the $\mathbb R^{n}$9th-order Affine Pólya-Szegö Principle" writes Zhang’s inequality as

$n\ge2$0

where

$n\ge2$1

This is the same $n\ge2$2 affine Sobolev inequality written in the notation that later extends to $n\ge2$3 and higher-order affine energies (Langharst et al., 2024).

2. Polar projection bodies and affine invariance

The geometric content of Zhang’s inequality is encoded by the polar projection body of a function. For $n\ge2$4, the polar projection body $n\ge2$5 is defined by the gauge

$n\ge2$6

Since $n\ge2$7 is a convex body containing $n\ge2$8 in its interior,

$n\ge2$9

Substituting the gauge formula gives a geometric restatement of Zhang’s inequality: $f\in C_c^{1}(\mathbb R^{n})$0 This is the compact convex-geometric form of the Sobolev–Zhang inequality (Haddad et al., 2022).

The affine character is transparent at the level of $f\in C_c^{1}(\mathbb R^{n})$1. If $f\in C_c^{1}(\mathbb R^{n})$2, then

$f\in C_c^{1}(\mathbb R^{n})$3

hence

$f\in C_c^{1}(\mathbb R^{n})$4

By contrast, the classical Sobolev energy $f\in C_c^{1}(\mathbb R^{n})$5 is not invariant under general volume-preserving linear maps (Haddad et al., 2022).

This projection-body formulation is the bridge to affine convex geometry. Several later works explicitly describe the affine Sobolev inequality as the functional counterpart of projection inequalities. In particular, it is presented as equivalent to Petty’s projection inequality, while the same affine invariant admits a reverse inequality of Zhang type on convex bodies, with simplices as extremizers on the opposite side (Alonso-Gutiérrez et al., 2018, Alonso-Gutiérrez et al., 18 Sep 2025).

3. Equality, extremals, and the $f\in C_c^{1}(\mathbb R^{n})$6 extension

The original statement is for $f\in C_c^{1}(\mathbb R^{n})$7, but the natural endpoint space is $f\in C_c^{1}(\mathbb R^{n})$8. Wang’s extension replaces $f\in C_c^{1}(\mathbb R^{n})$9 by the measure $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$0, writing

$\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$1

and defines

$\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$2

The Sobolev–Zhang inequality remains valid on $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$3 (Leite et al., 2021).

At $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$4, the extremals are characteristic functions of ellipsoids. The fractional paper states that the limiting extremals are indicators of ellipsoids, matching Wang’s characterization for the Sobolev–Zhang inequality on $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$5 (Haddad et al., 2022). The $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$6th-order affine Pólya–Szegö paper states the same point in the language of affine energies: for $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$7, extremals are characteristic functions of ellipsoids, up to null sets (Langharst et al., 2024). The projection-average paper likewise identifies the $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$8 affine case as the unique member of its family whose $\|f\|_{L^{n/(n-1)}(\mathbb R^n)}$9 extremals are characteristic functions of ellipsoids; for $L^{1}$0, the extremals collapse to Euclidean balls (Kniefacz et al., 2019).

A standard misconception is to expect smooth Aubin–Talenti-type bubbles at $L^{1}$1. The fractional affine $L^{1}$2 theory states this explicitly: there is no “smooth bubble” extremal here; in the $L^{1}$3 case the extremals are characteristic functions (Haddad et al., 2022). Smooth affine bubbles arise only in the $L^{1}$4 affine $L^{1}$5 theory.

4. Geometric background and companion inequalities

Zhang’s affine Sobolev inequality sits inside a larger affine isoperimetric picture built around projection bodies. Petty’s projection inequality gives the sharp upper bound on the affine invariant

$L^{1}$6

with ellipsoids as extremizers. Zhang’s reverse affine isoperimetric inequality gives the sharp lower bound for the same quantity, with simplices as extremizers (Alonso-Gutiérrez et al., 2018). In the geometric chain emphasized in the discrete projection paper,

$L^{1}$7

the convex-body projection inequality is the geometric source and the affine Sobolev inequality is its functional manifestation (Alonso-Gutiérrez et al., 18 Sep 2025).

The reverse side also has a functional extension. For an integrable log-concave function $L^{1}$8, "Zhang’s inequality for log-concave functions" defines a functional polar projection body $L^{1}$9 by averaging the polar projection bodies of the level sets $\nabla f$0, and proves

$\nabla f$1

If $\nabla f$2, equality holds if and only if

$\nabla f$3

for some $\nabla f$4-simplex $\nabla f$5 containing the origin (Alonso-Gutiérrez et al., 2018). This reverse functional inequality is not Zhang’s affine Sobolev inequality itself, but it clarifies the opposite extremal geometry for the same projection-body functional.

This geometric duality explains why the affine Sobolev theory is stronger than its Euclidean counterpart. The affine energy is smaller than the Euclidean gradient norm, but it controls the same critical $\nabla f$6 norm with the sharp constant. That is the analytic shadow of the deeper affine isoperimetric inequalities for projection bodies (Nápoli et al., 2015, Haddad et al., 2017).

5. $\nabla f$7, fractional, higher-order, and $\nabla f$8-th-order extensions

Zhang’s inequality is the $\nabla f$9 member of the affine $C_c^{1}(\mathbb R^{n})$0 Sobolev family. Lutwak–Yang–Zhang extended it to all $C_c^{1}(\mathbb R^{n})$1, and later work makes this identification explicit: in the $C_c^{1}(\mathbb R^{n})$2th-order theory, Zhang’s original inequality is exactly the case $C_c^{1}(\mathbb R^{n})$3, $C_c^{1}(\mathbb R^{n})$4, $C_c^{1}(\mathbb R^{n})$5 (Langharst et al., 2024). In the projection-average framework, it appears as the $C_c^{1}(\mathbb R^{n})$6 endpoint of a monotone chain

$C_c^{1}(\mathbb R^{n})$7

and the paper emphasizes that the strongest member $C_c^{1}(\mathbb R^{n})$8 is the only affine invariant one among them (Kniefacz et al., 2019).

The fractional theory extends the same mechanism to $C_c^{1}(\mathbb R^{n})$9. For $BV(\mathbb R^{n})$0, "Affine Fractional Sobolev and Isoperimetric Inequalities" introduces the fractional polar projection body $BV(\mathbb R^{n})$1 and proves a sharp affine fractional Sobolev inequality that is stronger than the Almgren–Lieb fractional Sobolev inequality. The key limit is

$BV(\mathbb R^{n})$2

and passing to the limit in the affine fractional inequality recovers precisely Zhang’s affine Sobolev inequality with the correct optimal constant (Haddad et al., 2022). For general $BV(\mathbb R^{n})$3, "Affine fractional $BV(\mathbb R^{n})$4 Sobolev inequalities" constructs fractional $BV(\mathbb R^{n})$5 polar projection bodies $BV(\mathbb R^{n})$6 and proves affine fractional $BV(\mathbb R^{n})$7 Sobolev inequalities that are fractional counterparts of the affine $BV(\mathbb R^{n})$8 Sobolev inequalities of Lutwak–Yang–Zhang (Haddad et al., 2022).

Higher-order extensions proceed in two distinct directions. One is geometric: "Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies" defines $BV(\mathbb R^{n})$9-th-order projection bodies $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$0 in $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$1 and proves an $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$2-th-order affine Sobolev inequality for $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$3-functions,

$$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$4

with equality if and only if $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$5 for some ellipsoid $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$6 and $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$7 (Haddad et al., 2023). The other is analytic: "Higher-order affine Sobolev inequalities" defines affine energies $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$8 for all $$\|f\|_{L^{\frac n{n-1}}(\mathbb R^n)} \le \frac{\omega_n^{1/n}}{2\,\omega_{n-1}} \left( \frac1n\int_{S^{n-1}} \left( \int_{\mathbb R^n} |\langle \nabla f(x),\xi\rangle|\,dx \right)^{-n} \,d\xi \right)^{-1/n} \le \frac{1}{n\,\omega_n^{1/n}} \int_{\mathbb R^n} |\nabla f(x)|\,dx.$$9, proves their $\mathbb R^{n}$00-invariance, and derives affine Sobolev, reverse affine, and affine Gagliardo–Nirenberg inequalities for higher-order and fractional homogeneous Sobolev spaces (Bullion-Gauthier, 12 Jun 2025).

These later theories consistently present Zhang’s inequality as the prototype of a robust affine-upgrade mechanism: replace a Euclidean Sobolev energy by an affinely invariant directional energy, keep the sharp constant, and enlarge the extremal symmetry from balls to ellipsoids or their affine analogues.

6. Bounded domains, spectral theory, stability, and stochastic extensions

On bounded domains, Zhang’s energy leads to affine Poincaré–Sobolev theory. In $\mathbb R^{n}$01, the zero extension $\mathbb R^{n}$02 gives a global affine energy $\mathbb R^{n}$03, and constrained minimization problems built from this functional admit minimizers in subcritical regimes and, under strict threshold assumptions, in critical regimes as well. As a consequence, extremal functions exist for a range of affine Poincaré–Sobolev, affine Poincaré–Wirtinger–Sobolev, and generalized affine Poincaré–Wirtinger–Sobolev inequalities on bounded Lipschitz domains (Leite et al., 2021).

For $\mathbb R^{n}$04, the affine energy $\mathbb R^{n}$05 supports a full spectral theory. "From affine Poincaré inequalities to affine spectral inequalities" defines the affine Rayleigh quotient

$\mathbb R^{n}$06

the first affine eigenvalue

$\mathbb R^{n}$07

and the affine $\mathbb R^{n}$08-Laplace operator $\mathbb R^{n}$09. The corresponding affine Faber–Krahn inequality states that $\mathbb R^{n}$10 is minimized, among sets of equal volume, only when $\mathbb R^{n}$11 is an ellipsoid (Haddad et al., 2020). This is the spectral analogue of the ellipsoidal extremal structure already present in Zhang’s Sobolev inequality.

Recent work addresses quantitative stability. "Sharp Quantitative Stability for the Affine $\mathbb R^{n}$12-Sobolev Inequality, Part I" proves, for $\mathbb R^{n}$13, a sharp quantitative stability result for the affine $\mathbb R^{n}$14-Sobolev inequality, and the stability exponent is shown to be optimal and equal to $\mathbb R^{n}$15 (Fan et al., 8 Jun 2026). "Stability for the Affine Sobolev Inequality and its Critical Points for $\mathbb R^{n}$16" proves stability with best possible norm and best possible stability exponent, and also proves a corresponding result for critical points of the affine functional in the absence of bubbling (Frank et al., 7 Jul 2026). These results place the affine Sobolev inequality on the same quantitative footing as the sharp stability theory for the classical Sobolev inequality.

A different extension is probabilistic. "On stochastic forms of functional isoperimetric inequalities" introduces random $\mathbb R^{n}$17-concave models $\mathbb R^{n}$18 built from random samples from the hypograph of $\mathbb R^{n}$19, and proves that Zhang’s affine Sobolev inequality holds in expectation: $\mathbb R^{n}$20 for every rotationally invariant convex measure $\mathbb R^{n}$21. Passing to the deterministic limit yields the convex-measure generalization

$\mathbb R^{n}$22

for integrable $\mathbb R^{n}$23-concave $\mathbb R^{n}$24 (Sola, 4 Sep 2025). This suggests that the projection-body formulation of Zhang’s inequality is flexible enough to survive both randomization and replacement of Lebesgue measure by rotationally invariant convex measures.

Zhang’s affine Sobolev inequality therefore occupies a structurally central position. It is simultaneously an endpoint sharp Sobolev inequality, an affine isoperimetric statement in disguise, the $\mathbb R^{n}$25 prototype of the affine $\mathbb R^{n}$26 theory, the local limit of sharp affine fractional inequalities, and the source of later domain, spectral, stability, higher-order, and stochastic developments (Haddad et al., 2022).