Affine Hardy–Littlewood–Sobolev Inequality

• The affine Hardy–Littlewood–Sobolev inequality is a geometric refinement of the classical HLS inequality that replaces the Euclidean Riesz-kernel with star-shaped bodies to attain invariance under translations and volume-preserving linear maps.
• It employs dual mixed volumes and rearrangement inequalities to transform analytic kernel integrals into geometric terms, providing stronger and more detailed estimates.
• The theory’s framework extends to two-function cases and log-concave refinements, illustrating its versatility in sharpening classical Sobolev inequalities with affine-invariant structures.

Searching arXiv for recent and foundational papers on affine Hardy–Littlewood–Sobolev inequalities. Affine Hardy–Littlewood–Sobolev inequalities are affine-invariant refinements of the classical Hardy–Littlewood–Sobolev (HLS) inequality on $\mathbb R^n$. Their defining feature is that the Euclidean Riesz-kernel bilinear form is replaced, or interpolated, by a geometric quantity derived from a star-shaped body attached to one function or to a pair of functions. The resulting inequalities are invariant under translations and volume-preserving linear maps, and the recent literature treats both the diagonal case and a genuinely two-function version. In this sense, affine HLS theory strengthens the classical sharp HLS inequality by inserting an affine-geometric layer between the norm product and the kernel integral (Haddad et al., 2022, Lin et al., 2 Aug 2025).

1. Classical template and the motivation for an affine refinement

The classical HLS inequality is the analytic model for the affine theory. In one standard sharp formulation, if $0<\lambda<N$ and

$p=\frac{2N}{2N-\lambda},$

then

$$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$

with sharp constant

$$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$

and equality iff $f$ and $g$ are proportional to a common translate and dilation of

$H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$

This classical optimizer family is the bubble family (Frank et al., 2010).

The two-function HLS inequality may also be written in the form: for $p,r>1$, $0<\alpha<n$, and

$0<\lambda<N$0

one has

$0<\lambda<N$1

for nonnegative $0<\lambda<N$2 and $0<\lambda<N$3 (Lin et al., 2 Aug 2025).

What the affine theory changes is not the scaling relation but the symmetry class of the inequality. The classical sharp HLS inequality is invariant under translations, rotations, dilations, and conformal transformations, but the affine literature emphasizes that it is not invariant under volume-preserving linear maps. The affine refinement is designed precisely to restore invariance under translations and $0<\lambda<N$4-transformations, thereby capturing anisotropic information that the Euclidean kernel alone does not record (Haddad et al., 2022).

2. Geometric formulation through star bodies and dual mixed volumes

The central affine object is a star-shaped set built from the correlation structure of one or two functions. In the two-function theory, for $0<\lambda<N$5, the star-shaped set $0<\lambda<N$6 is defined by its radial function

$0<\lambda<N$7

This is the geometric object that replaces the usual convolution-type integral in the affine theory (Lin et al., 2 Aug 2025).

In the diagonal case, the corresponding body $0<\lambda<N$8 is defined by

$0<\lambda<N$9

and its volume is

$p=\frac{2N}{2N-\lambda},$0

This body transforms naturally under volume-preserving linear maps: $p=\frac{2N}{2N-\lambda},$1 and, in the two-function formulation,

$p=\frac{2N}{2N-\lambda},$2

The same reformulation is also invariant under the same translation of $p=\frac{2N}{2N-\lambda},$3 and $p=\frac{2N}{2N-\lambda},$4 (Haddad et al., 2022, Lin et al., 2 Aug 2025).

The bridge to convex geometry is the dual mixed volume

$p=\frac{2N}{2N-\lambda},$5

A key identity is

$p=\frac{2N}{2N-\lambda},$6

which converts an analytic kernel integral into a dual mixed volume. This identity is the structural reason the affine HLS theory is genuinely affine rather than merely anisotropic (Lin et al., 2 Aug 2025).

3. Sharp diagonal affine Hardy–Littlewood–Sobolev inequalities

The 2022 affine theory establishes sharp affine HLS inequalities for nonnegative functions on $p=\frac{2N}{2N-\lambda},$7 in both the direct regime $p=\frac{2N}{2N-\lambda},$8 and the reverse regime $p=\frac{2N}{2N-\lambda},$9, and states that these inequalities are significantly stronger than, and directly imply, the sharp HLS inequalities of Lieb and of Beckner, Dou, and Zhu (Haddad et al., 2022).

For $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$0, the diagonal theory is expressed through the body $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$1, the volume $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$2, and the classical Riesz-kernel integral. Equality in the affine inequality is attained precisely by an affine bubble family: $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$3 with $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$4, $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$5, and $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$6 with $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$7. In the comparison inequality with the classical kernel integral, equality holds iff $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$8 is radially symmetric (Haddad et al., 2022).

For $$\iint_{\mathbb R^N\times\mathbb R^N}\frac{f(x)\,g(y)}{|x-y|^\lambda}\,dx\,dy \le C_{N,\lambda}\,\|f\|_{p}\,\|g\|_{p},$$9, the same framework yields a sharp reverse affine HLS inequality. The extremals are again of bubble type, with the same affine freedom, and the affine result strengthens the reverse HLS inequalities previously obtained by Beckner and by Dou–Zhu (Haddad et al., 2022).

These diagonal results already show the distinctive logic of affine HLS theory. The Euclidean kernel is not discarded; rather, it is embedded into a stronger inequality whose middle term depends on the affine geometry of the associated star body. The sharp constant is still the classical sharp HLS constant, but the affine middle term contains additional geometric information (Haddad et al., 2022).

4. Generalized two-function affine inequalities and the reverse regime

A further development is the generalized affine HLS inequality for two different functions. For $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$0, $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$1, $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$2, and

$$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$3

there exists $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$4 such that for all nonnegative $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$5 and $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$6 the norm product dominates an affine-invariant term expressed by the volume of $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$7, and this affine term in turn dominates the classical HLS bilinear form. The affine strengthening is exactly that the middle term is affine-invariant and sits between the $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$8 norm product and the classical HLS bilinear form (Lin et al., 2 Aug 2025).

When $$C_{N,\lambda} = \pi^{\lambda/2}\, \frac{\Gamma\!\left(\frac{N-\lambda}{2}\right)}{\Gamma\!\left(N-\frac{\lambda}{2}\right)} \left(\frac{\Gamma(N)}{\Gamma(N/2)}\right)^{1-\lambda/N},$$9 and

$f$0

the generalized inequality recovers the earlier affine HLS inequality of Haddad–Ludwig (Lin et al., 2 Aug 2025). Thus the two-function theory is not a separate branch; it is an extension of the original diagonal affine inequality.

The equality structure is explicit in the critical case. Equality in the first inequality of Theorem 1.1 forces

$f$1

with $f$2, $f$3, $f$4, $f$5, and $f$6. Equality in the second inequality is attained precisely when $f$7 and $f$8 are radially symmetric (Lin et al., 2 Aug 2025).

The reverse regime is parallel but with reversed order. For $f$9, if $g$0 and

$g$1

then the generalized affine theory gives the reversed affine statement: the norm product is bounded above by the affine term, and the affine term is bounded above by the classical reverse HLS integral. This strengthens the reversed classical HLS inequality of Dou–Zhu and Beckner. In the critical case, the same extremal family appears, while equality in the comparison with the Euclidean integral again occurs when $g$2 and $g$3 are radially symmetric (Lin et al., 2 Aug 2025).

5. Log-concave and $g$4-concave refinements

Affine HLS theory extends beyond sharp $g$5 inequalities into reverse inequalities for log-concave and $g$6-concave functions. In the two-function setting, the reverse inequalities are formulated using $g$7 and a normalized radial mean body

$g$8

For $g$9, the same paper defines

$H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$0

(Lin et al., 2 Aug 2025).

For even log-concave $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$1, Theorem 1.3 gives reverse inequalities in terms of $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$2, $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$3, and $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$4, with one form for $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$5 and the reversed ordering for $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$6. In both regimes, equality in the first inequality occurs iff

$H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$7

where $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$8, $H(x)=(1+|x|^2)^{-(2N-\lambda)/2}.$9, and $p,r>1$0 is an $p,r>1$1-dimensional simplex with a vertex at the origin (Lin et al., 2 Aug 2025).

A major structural result is the monotonicity inclusion: for nonzero even log-concave $p,r>1$2 and $p,r>1$3,

$p,r>1$4

For nonzero even $p,r>1$5-concave $p,r>1$6 with $p,r>1$7,

$p,r>1$8

The paper states that the log-concave theorem is recovered as $p,r>1$9, and that the indicator-function limit recovers the classical radial mean body inclusions of Gardner–Zhang (Lin et al., 2 Aug 2025).

The diagonal affine theory also yields affine fractional $0<\alpha<n$0 Sobolev inequalities. For $0<\alpha<n$1, the affine fractional Sobolev body $0<\alpha<n$2 is defined by

$0<\alpha<n$3

and the reverse inequality for log-concave $0<\alpha<n$4 is

$0<\alpha<n$5

with equality for simplex-exponential functions of the form

$0<\alpha<n$6

where $0<\alpha<n$7 is an $0<\alpha<n$8-simplex with a vertex at the origin (Haddad et al., 2022).

6. Proof architecture, rearrangement, and extremal structure

The affine proofs are built on an overview of affine convex geometry and rearrangement theory. The analytic input is converted into a geometric statement through the dual mixed volume identity

$0<\alpha<n$9

or its two-function counterpart, and the geometric input is the dual mixed volume inequality. For $0<\lambda<N$00,

$0<\lambda<N$01

while for $0<\lambda<N$02 the inequality reverses, with equality iff $0<\lambda<N$03 and $0<\lambda<N$04 are dilates (Haddad et al., 2022).

The rearrangement step is equally central. For $0<\lambda<N$05, the two-function theory proves

$0<\lambda<N$06

where $0<\lambda<N$07 are symmetric decreasing rearrangements. Equality, under the strict monotonicity hypotheses on the rearrangements, is characterized by Burchard’s equality case in the Riesz rearrangement inequality: there must exist $0<\lambda<N$08 and translations such that $0<\lambda<N$09 and $0<\lambda<N$10 are translates of $0<\lambda<N$11 and $0<\lambda<N$12. For $0<\lambda<N$13, the reverse theory proves

$0<\lambda<N$14

provided $0<\lambda<N$15, again with equality characterized through ellipsoids and translates of volume-preserving linear images of the rearrangements (Lin et al., 2 Aug 2025).

This proof architecture distinguishes affine HLS from some classical sharp HLS methods. The classical sharp HLS inequality on $0<\lambda<N$16 also has a rearrangement-free proof based on stereographic projection, a center-of-mass normalization, second variation, and the Funk–Hecke theorem, with rigidity obtained from spherical harmonic eigenvalue comparisons (Frank et al., 2010). Affine HLS sharpness, by contrast, is presently tied in the cited literature to the star-body formalism, dual mixed volumes, and rearrangement inequalities (Haddad et al., 2022, Lin et al., 2 Aug 2025).

7. Relation to other HLS variants and common points of confusion

A common confusion is to treat any anisotropic or geometric HLS inequality as an affine HLS inequality. The literature separates several distinct directions.

The sharp upper-half-space HLS inequality is a conformal extension, not an affine one. It studies the extension kernel

$0<\lambda<N$17

and exploits the conformal equivalence between the upper half-space and the ball. Extremals are classified in the critical case by the method of moving spheres and have the standard bubble form. The geometric invariance here is Möbius-type rather than $0<\lambda<N$18-invariance (Dou et al., 2013).

The Heisenberg-group and Zygmund-dilation theories are likewise not affine HLS theories in the convex-geometric sense. On the Heisenberg group, the recent revisit proves HLS-type estimates for fractional integral operators adapted to the noncommutative group law and to Zygmund dilations. For the mixed kernel $0<\lambda<N$19 and the Zygmund kernel $0<\lambda<N$20, the critical scaling law is

$0<\lambda<N$21

and the sharp kernel threshold is

$0<\lambda<N$22

These are anisotropic and multi-parameter HLS results, but their geometry is Heisenberg and Zygmund rather than affine convex-geometric (Sun et al., 15 Sep 2025).

The product-space HLS theorem on rectangle-doubling measure replaces Euclidean balls by rectangles and the Euclidean kernel by

$0<\lambda<N$23

where $0<\lambda<N$24 is the $0<\lambda<N$25-volume of the smallest centered rectangle containing $0<\lambda<N$26. The scale condition remains

$0<\lambda<N$27

but the underlying structure is product geometry, not affine invariance (Wang, 2018).

The semigroup-based HLS theory for degenerate hypoelliptic operators replaces the Riesz potential by

$0<\lambda<N$28

and the Euclidean dimension by an intrinsic volume exponent $0<\lambda<N$29, yielding

$0<\lambda<N$30

Here the decisive structure is the geometry of the non-symmetric semigroup and its volume growth function $0<\lambda<N$31 (Garofalo et al., 2019).

These comparisons clarify the specific content of the term “affine” in affine HLS theory. It refers neither to conformal invariance, nor to generic anisotropy, nor to noncommutative scaling. It refers to the fact that the strengthened HLS inequality is encoded by star bodies, dual mixed volumes, and transformation laws under translations and volume-preserving linear maps. Within that precise meaning, the affine Hardy–Littlewood–Sobolev inequality is the convex-geometric refinement of the classical HLS principle (Haddad et al., 2022, Lin et al., 2 Aug 2025).