Affine Fractional Lp Pólya–Szegő Inequalities

• Affine fractional Lp Pólya–Szegő inequalities are nonlocal, affine-invariant rearrangement principles defined in fractional Sobolev spaces that decrease energy under symmetric rearrangement.
• They leverage geometric constructs such as star bodies, dual mixed volumes, and polar projection bodies to connect analysis with affine volume monotonicity and sharper Sobolev inequalities.
• The theory spans from a p=1 prototype to a comprehensive one- and two-function Lp framework, unifying classical affine Sobolev, isoperimetric results, and nonlocal interaction energies.

Affine fractional $L_p$ Pólya–Szegő inequalities are nonlocal affine-invariant rearrangement principles asserting that a suitably defined affine fractional energy decreases under symmetric decreasing rearrangement. In the literature covered here, the subject is built around fractional Sobolev spaces on $\mathbb R^n$, star-body geometry, dual mixed volumes, and fractional analogues of polar projection bodies. The theory appears in three main layers: an $L^1$-based prototype in the fractional Sobolev–Zhang setting, a one-function $L_p$ theory for $1Haddad et al., 2022, Haddad et al., 2022, Lin et al., 2 Aug 2025).

1. Historical placement and scope

Within the papers considered here, the fractional affine theory begins with Haddad and Ludwig’s $p=1$ result. That work establishes an explicit affine fractional Pólya–Szegő inequality in the $W^{s,1}(\mathbb R^n)$ setting, together with sharp affine fractional Sobolev and isoperimetric inequalities. It does not develop a full general affine fractional $L_p$ theory for arbitrary $p$, but it provides the $p=1$ model case and the geometric machinery later reused in the $\mathbb R^n$0 theory (Haddad et al., 2022).

The one-function $\mathbb R^n$1 theory is then developed for $\mathbb R^n$2 and $\mathbb R^n$3. In that setting the central object is a nonlocal $\mathbb R^n$4 polar projection body, and the main rearrangement principle is an affine fractional $\mathbb R^n$5 Pólya–Szegő inequality of the form

$\mathbb R^n$6

together with an asymmetric analogue. These inequalities are used to derive sharp affine fractional $\mathbb R^n$7 Sobolev inequalities that are stronger than the classical sharp fractional $\mathbb R^n$8 Sobolev inequalities (Haddad et al., 2022).

A further extension considers two functions on $\mathbb R^n$9. There the basic nonlocal object is a generalized fractional $L^1$0 polar projection body $L^1$1, and the main result is an affine fractional $L^1$2 Pólya–Szegő inequality for interaction energies. Setting $L^1$3 recovers the one-function theory (Lin et al., 2 Aug 2025).

These fractional results belong to a broader affine Sobolev program. In the local first-order setting, the general affine $L^1$4 Pólya–Szegő principle and the affine Orlicz Pólya–Szegő principle were already known, and the fractional theory is explicitly presented as a nonlocal counterpart of that program (Nguyen, 2015, Lin, 2017).

2. Analytic and geometric framework

The analytic background is the fractional Sobolev space

$L^1$5

with $L^1$6 and $L^1$7. For the two-function theory, the corresponding space is

$L^1$8

Symmetric decreasing rearrangement is defined by replacing superlevel sets with centered balls of equal measure; in the $L^1$9 paper this is written $L_p$0, while the later $L_p$1 papers write $L_p$2 (Haddad et al., 2022, Lin et al., 2 Aug 2025).

The geometric side is expressed through star bodies. If $L_p$3 is star-shaped, its gauge is $L_p$4 and its radial function is $L_p$5. Dual mixed volumes are defined by

$L_p$6

Because the fractional affine theory uses $L_p$7, the dual mixed volume inequality takes the form

$L_p$8

with equality iff $L_p$9 are dilates (Haddad et al., 2022).

The one-function affine body is the $1

$1

In the $1

$1

For two functions, the generalized body is

$1

and there is an asymmetric version $p=1$0 obtained by replacing $p=1$1 with $p=1$2 (Haddad et al., 2022, Haddad et al., 2022, Lin et al., 2 Aug 2025).

The basic bridge from analysis to geometry is the anisotropic identity

$p=1$3

and, in the two-function case,

$p=1$4

This identity is the structural reason that rearrangement inequalities for anisotropic kernels can be converted into affine volume monotonicity statements (Haddad et al., 2022, Lin et al., 2 Aug 2025).

3. One-function affine fractional $p=1$5 Pólya–Szegő principles

For $p=1$6, $p=1$7, and nonnegative $p=1$8, the one-function affine fractional $p=1$9 Pólya–Szegő principle is

$W^{s,1}(\mathbb R^n)$0

The asymmetric form is

$W^{s,1}(\mathbb R^n)$1

In both cases the equality characterization is rigid: equality holds for $W^{s,1}(\mathbb R^n)$2 iff $W^{s,1}(\mathbb R^n)$3 is a translate of $W^{s,1}(\mathbb R^n)$4 for some $W^{s,1}(\mathbb R^n)$5 (Haddad et al., 2022).

These rearrangement inequalities imply the sharp affine fractional $W^{s,1}(\mathbb R^n)$6 Sobolev inequality

$W^{s,1}(\mathbb R^n)$7

together with the comparison

$W^{s,1}(\mathbb R^n)$8

Thus the affine energy is strictly smaller than the Euclidean fractional seminorm and directly implies the classical sharp fractional Sobolev inequality. Equality in the first inequality holds iff $W^{s,1}(\mathbb R^n)$9 for some $L_p$0, where $L_p$1 is an extremal of the classical sharp fractional Sobolev inequality; equality in the second inequality holds if $L_p$2 is radially symmetric (Haddad et al., 2022).

The $L_p$3 prototype has the same structural form. For nonnegative $L_p$4,

$L_p$5

Here equality holds iff for almost every $L_p$6, the level set $L_p$7 has measure zero or is homothetic to an ellipsoid independent of $L_p$8, up to null sets. In the same paper, this affine $L_p$9 principle yields a sharp affine fractional Sobolev inequality stronger than the sharp Almgren–Lieb inequality (Haddad et al., 2022).

4. Two-function and asymmetric formulations

The two-function theory replaces the one-function seminorm by an interaction energy. For $p$0, $p$1, nonnegative $p$2, and with $p$3, the main symmetric affine fractional $p$4 Pólya–Szegő inequality has the form

$p$5

The asymmetric version is

$p$6

The affine core of the theory is the monotonicity

$p$7

and similarly for $p$8 (Lin et al., 2 Aug 2025).

The equality structure is also explicitly affine. Equality in the first inequality of the chain holds if $p$9 are radially symmetric. Equality in the second inequality holds iff

$p=1$0

for some $p=1$1 and $p=1$2. Setting $p=1$3 recovers the one-function theory, so the two-function result is an extension rather than a separate formalism (Lin et al., 2 Aug 2025).

A distinctive feature of this extension is that “stronger” is interpreted explicitly: the affine interaction energy sits between the original Euclidean fractional energy and the rearranged Euclidean energy. This gives a sharper lower bound than the rearranged Euclidean term alone (Lin et al., 2 Aug 2025).

5. Proof architecture and geometric consequences

The proof strategy is an overview of rearrangement theory and dual Brunn–Minkowski geometry. The first step is an anisotropic rearrangement inequality. In the one-function $p=1$4 theory, for a star body $p=1$5,

$p=1$6

with an asymmetric version using $p=1$7. In the $p=1$8 model, the corresponding statement is formulated both for functions and for fractional perimeters $p=1$9 (Haddad et al., 2022, Haddad et al., 2022).

The analytic ingredient is a layer-cake reduction together with the decomposition

$\mathbb R^n$00

which allows the application of the Riesz rearrangement inequality. Equality is then analyzed via Burchard’s theorem, forcing the kernel body to be an ellipsoid and the relevant level sets to be translates of homothetic ellipsoids (Haddad et al., 2022, Lin et al., 2 Aug 2025).

The geometric conversion is achieved through the dual mixed volume identity. Once

$\mathbb R^n$01

is known, the dual mixed volume inequality yields the affine volume monotonicity

$\mathbb R^n$02

The same pattern underlies the two-function theory (Haddad et al., 2022, Lin et al., 2 Aug 2025).

In the $\mathbb R^n$03 paper, this mechanism has a pronounced geometric output. Using the coarea formula

$\mathbb R^n$04

the affine fractional Pólya–Szegő principle yields the fractional Petty projection inequality

$\mathbb R^n$05

This is stronger than the classical fractional isoperimetric inequality because it inserts a strictly finer affine middle term between volume and Euclidean fractional perimeter (Haddad et al., 2022).

6. Limits, local antecedents, and current frontier

A central consistency requirement is the limit $\mathbb R^n$06. In the $\mathbb R^n$07 theory,

$\mathbb R^n$08

so the fractional affine theory converges to the classical polar projection body and recovers Gaoyong Zhang’s affine Sobolev inequality (Haddad et al., 2022). In the $\mathbb R^n$09 theory, the fractional affine Pólya–Szegő inequalities are explicitly stated to recover the known affine and asymmetric affine $\mathbb R^n$10 Pólya–Szegő inequalities as $\mathbb R^n$11 (Haddad et al., 2022).

These fractional results should be read against the local first-order affine background. The general affine $\mathbb R^n$12 Pólya–Szegő principle in $\mathbb R^n$13 and the affine Orlicz Pólya–Szegő principle show that the fractional bodies $\mathbb R^n$14 are nonlocal analogues of earlier affine energies built from directional derivatives (Nguyen, 2015, Lin, 2017). The local geometric philosophy is sharpened by the fact that, in a broader Euclidean-to-affine family, the Petty projection inequality is the only affine invariant member and the strongest one; the fractional affine theory mirrors that strengthening phenomenon in the nonlocal setting (Haberl et al., 2018).

The surrounding literature also clarifies the present boundary of the theory. A recent survey on the fractional Pólya–Szegő principle develops the Euclidean and anisotropic fractional theories but does not explicitly formulate an affine fractional $\mathbb R^n$15 Pólya–Szegő inequality (Carbotti, 15 Sep 2025). A later paper on higher-order affine Sobolev inequalities extends the affine Sobolev framework to arbitrary orders $\mathbb R^n$16, including $\mathbb R^n$17, but does not establish a new rearrangement theorem for higher order; instead it explicitly recalls the Haddad–Ludwig affine Pólya–Szegő inequality for $\mathbb R^n$18 as the sharp fractional benchmark (Bullion-Gauthier, 12 Jun 2025). Adjacent $\mathbb R^n$19-based affine Hardy–Littlewood–Sobolev and logarithmic endpoint theories use rearrangement monotonicity of star-body volumes such as $\mathbb R^n$20 and $\mathbb R^n$21, but are not stated as general affine fractional $\mathbb R^n$22 Pólya–Szegő principles (Haddad et al., 2022, Cai, 12 Apr 2025).

The literature therefore currently supports a clear picture. Sharp affine fractional Pólya–Szegő inequalities are fully established for $\mathbb R^n$23 in the $\mathbb R^n$24 prototype, in the one-function $\mathbb R^n$25 theory, and in the two-function interaction theory. Beyond that range, the affine energies are known, the Sobolev consequences are known in broader form, and the higher-order affine framework exists, but a comparably sharp higher-order rearrangement theory is not yet part of the results summarized here (Haddad et al., 2022, Haddad et al., 2022, Lin et al., 2 Aug 2025, Bullion-Gauthier, 12 Jun 2025).