Smoothing Rearrangements

• Smoothing rearrangements are transformations that replace functions or sets with more structured, symmetric representatives while preserving their distribution and variational complexity.
• They underpin key inequalities such as the Pólya–Szegő and fractional isoperimetric inequalities, often lowering energies or seminorms in Sobolev and nonlocal settings.
• Their applications span discrete graphs, information-theoretic convolutions, image filtering, and fixed-boundary surface smoothing, yielding practical improvements in regularity and compactness.

Smoothing rearrangements are transformations that preserve distribution data while replacing a function, set, or geometric object by a more structured representative whose variational complexity is no greater than that of the original. In the classical Euclidean setting, the prototype is the symmetric decreasing rearrangement, which sends a function to a radial, radially nonincreasing, equimeasurable one and typically lowers energies or seminorms without changing $L^p$-norms. In a more abstract formulation, a rearrangement is called smoothing when it does not increase the modulus of continuity; this viewpoint unifies symmetric decreasing rearrangement, Steiner-type symmetrizations, polarizations, and anisotropic variants. The terminology also appears in adjacent literatures for one-dimensional reformulations of nonlocal filters and for curvature-based variational improvements of fixed-boundary surfaces (Frank, 2022, Bianchi et al., 2022, Galiano et al., 2014, Ahmad et al., 2012).

1. Symmetric decreasing rearrangement as a variational smoothing operation

For a measurable function $f$ vanishing at infinity, the symmetric decreasing rearrangement $f^*$ is defined by

$$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$

where $\{|f|>t\}^*$ is the centered ball with the same measure as $\{|f|>t\}$. Equivalently,

$\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$

The rearranged function is nonnegative, radial, and radially nonincreasing, and it preserves all $L^p$-norms: $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$ The smoothing effect is variational rather than classical pointwise regularization: rearrangement imposes symmetry and monotonicity and often decreases energies or seminorms while leaving distribution functions unchanged (Frank, 2022).

A basic monotonicity principle is

$$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$

for nonnegative $f$0. More generally, the Crowe–Zweibel–Rosenbloom / Brascamp–Lieb–Luttinger framework shows that certain multilinear integrals increase under rearrangement. This mechanism underlies isoperimetric inequalities, existence of minimizers, and Sobolev-type regularity improvements. In variational problems, if a functional $f$1 satisfies $f$2 and the admissible class is rearrangement-stable, one may restrict to symmetric decreasing minimizing sequences. Such sequences satisfy the decay estimate

$f$3

which yields compactness through Helly-type arguments. This strategy is emphasized for problems such as the Choquard functional and the Coulomb interaction energy; it does not imply convergence of every minimizing sequence, but it does produce symmetric minimizing sequences with useful compactness properties (Frank, 2022).

2. Sobolev regularity, continuity, and sharp obstructions

In fractional Sobolev spaces $f$4, with $f$5 and $f$6, rearrangement has a particularly strong smoothing property. The paper reviewed in (Frank, 2022) states

$f$7

so the fractional Sobolev seminorm does not increase under rearrangement. Almgren and Lieb’s continuity theorem then gives a global regularity statement:

If $f$8, $f$9, and $f^*$0, then the rearrangement map $f^*$1 is continuous on $f^*$2.

Corollary 11 implies that the map is Lipschitz at the zero function, but the theorem is stronger because it gives continuity everywhere in $f^*$3. The same framework yields the fractional isoperimetric inequality

$f^*$4

and, as $f^*$5,

$f^*$6

so the fractional statement limits to the classical isoperimetric inequality. This identifies rearrangement as a smoothing principle across nonlocal and local perimeter scales (Frank, 2022).

In the local Sobolev space $f^*$7, the fundamental inequality is the Pólya–Szegő inequality: $f^*$8 valid for $f^*$9, $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$0, and $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$1 vanishing at infinity. For $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$2, Lieb gave a short proof using the heat semigroup and the Riesz rearrangement inequality: $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$3 After taking $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$4, the gradient inequality follows. This semigroup argument exhibits smoothing by converting a rearrangement inequality into a Sobolev regularity statement (Frank, 2022).

Continuity in $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$5, however, is sharply more delicate. Almgren–Lieb theory shows that for $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$6 and $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$7, the map $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$8 is continuous in $$f^*(x)=\int_0^\infty \mathbf{1}_{\{|f|>t\}^*}(x)\,dt,$$9 precisely at those $\{|f|>t\}^*$0 for which $\{|f|>t\}^*$1 is co-area regular. The residual distribution function is

$\{|f|>t\}^*$2

and $\{|f|>t\}^*$3 is co-area regular if the absolutely continuous part of the Radon measure associated with $\{|f|>t\}^*$4 vanishes. Sufficient smoothness implies continuity at the point: if $\{|f|>t\}^*$5 and $\{|f|>t\}^*$6, then $\{|f|>t\}^*$7 is co-area regular; in dimension one this recovers Coron’s result that rearrangement is continuous in $\{|f|>t\}^*$8 for $\{|f|>t\}^*$9. At the same time, the $\{|f|>t\}$0 threshold is essentially optimal on the Hölder scale, co-area irregular functions are dense in dimensions $\{|f|>t\}$1, and for $\{|f|>t\}$2, $\{|f|>t\}$3, there exists $\{|f|>t\}$4, supported in $\{|f|>t\}$5, with

$\{|f|>t\}$6

so $\{|f|>t\}$7 is co-area irregular. The continuity theory therefore combines a monotonicity inequality with a sharp geometric characterization of where continuity actually holds (Frank, 2022).

3. Abstract smoothing rearrangements and the generalized Pólya–Szegő principle

A broad abstract framework is developed in "The Pólya-Szegő inequality for smoothing rearrangements" (Bianchi et al., 2022). The natural domain is

$\{|f|>t\}$8

that is, nonnegative measurable functions that vanish at infinity in the level-set sense. A rearrangement $\{|f|>t\}$9 is assumed to be equimeasurable and monotone: $\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$0 Its action is encoded at the set level by $\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$1 a.e.

The defining geometric condition for smoothing is the Minkowski-type inclusion

$\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$2

essentially, for bounded measurable $\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$3 and $\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$4. More generally, for a convex body $\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$5 containing the origin in its interior, $\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$6 is $\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$7-smoothing if

$\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$8

essentially. The same paper shows that smoothing rearrangements are exactly those rearrangements that reduce the appropriate modulus of continuity. For the Euclidean case,

$\{f^*>t\}=\{|f|>t\}^* \qquad (t\ge 0).$9

and for a convex body $L^p$0,

$L^p$1

When $L^p$2 is o-symmetric, Corollary 4.12 identifies the following as equivalent: reduction of the $L^p$3-modulus of continuity, reduction of the $L^p$4-modulus for each $L^p$5-contraction, and $L^p$6-smoothing (Bianchi et al., 2022).

The main consequence is a unified Pólya–Szegő principle. If $L^p$7 is smoothing, $L^p$8 is Lipschitz in $L^p$9, and $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$0 is a Young function, then

$$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$1

The theory extends to $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$2 under the finiteness assumption $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$3, and in particular yields for $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$4

$$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$5

as well as the $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$6 bound

$$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$7

Anisotropic versions replace the Euclidean norm by the support function $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$8 of a convex body $$\|f^*\|_{L^p(\mathbb{R}^d)}=\|f\|_{L^p(\mathbb{R}^d)},\qquad p>0.$$9, giving inequalities of the form

$$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$0

The proof is geometric: it passes to subgraphs $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$1, compares anisotropic outer Minkowski contents, and uses a formula of Lussardi–Villa to recover the relevant energy integrals (Bianchi et al., 2022).

This framework subsumes symmetric decreasing rearrangement, Schwarz / $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$2-Steiner rearrangements, polarization, Solynin’s continuous symmetrizations, and anisotropic $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$3-Schwarz rearrangements. A later development studies which smoothing rearrangements can be approximated by polarizations, an important question because the Pólya–Szegő inequality holds with equality for polarizations. Symmetric decreasing rearrangement and Steiner / Schwarz rearrangements are sequentially approximable in $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$4, and Solynin rearrangements are approximable in $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$5 by finite compositions of polarizations. By contrast, Brock rearrangements with parameter $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$6 are not weakly approximable by such compositions; the obstruction is expressed through associated contraction maps on $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$7 induced by the rearrangement at the level of balls (Bianchi et al., 2 Sep 2025).

4. Discrete, information-theoretic, and weighted-analytic extensions

On lattice graphs, smoothing rearrangements become graph-dependent reorderings adapted to discrete isoperimetry. For a labeling $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$8 of the vertices, the rearrangement is defined by placing the $$\int_{\mathbb{R}^d} f(x)g(x)\,dx \le \int_{\mathbb{R}^d} f^*(x)g^*(x)\,dx$$9-th largest value of $f$00 at $f$01. On $f$02, two distinguished examples are the spiral rearrangement and the Wang–Wang rearrangement. The discrete gradient is

$f$03

The paper "Rearrangement Inequalities on the Lattice Graph" proves approximate Pólya–Szegő inequalities with explicit constants: $f$04 for the spiral rearrangement, and

$f$05

for the Wang–Wang rearrangement, for all $f$06. At the endpoints, the spiral rearrangement is exact for $f$07, while the Wang–Wang rearrangement is exact for $f$08. No rearrangement can satisfy the exact inequality with constant $f$09 at $f$10: for every rearrangement on $f$11, there exists a compactly supported $f$12 such that

$f$13

The proof strategy uses a universal comparison graph $f$14 built from the vertex-isoperimetric profile, a modified coarea formula, and a path mapping from edges of the original graph into short paths in $f$15. For $f$16, the paper establishes the existence of rearrangements with

$f$17

(Gupta et al., 2022).

In information theory, rearrangement acts as a concentration-enhancing transformation for convolutions of densities. If $f$18 are probability densities on $f$19, then for Rényi order $f$20,

$f$21

and under the appropriate assumptions the same holds for Shannon entropy: $f$22 For two independent random variables with finite entropy and variance, this gives

$f$23

where $f$24 are Gaussians chosen so that $f$25. Since rearrangement preserves entropy and reduces variance, the rearranged sum supplies an intermediate lower bound stronger than the classical entropy power inequality in the sense of interpolation. The same paper establishes a majorization relation for convolutions, a divergence inequality $f$26 for $f$27, and in one dimension the Fisher-information inequality $f$28 (Wang et al., 2013).

A different analytical extension arises in weighted Fourier inequalities. The paper "Weighted Fourier inequalities via rearrangements" reduces weighted Fourier estimates to one-dimensional Hardy-type inequalities via non-increasing rearrangements. If

$f$29

then the original weighted Fourier inequality follows with the same constant. A key smoothing-like step is the Jodeit–Torchinsky estimate, which yields

$f$30

Here rearrangement is followed by an averaging operator $f$31, which functions as a regularization step. The paper also uses approximation of $f$32 by continuous non-increasing functions by convolution in $f$33, and introduces the level function $f$34 through the least concave majorant of $f$35, a regularization on the weight side rather than on the function side (Rastegari et al., 2016).

5. Relative rearrangements and nonlocal image filtering

In image analysis, rearrangements enter as an exact reformulation of a broad class of nonlocal smoothing filters. The starting operator is

$f$36

with image $f$37, similarity kernel $f$38, and weight $f$39. The decreasing rearrangement $f$40 is defined from the distribution function

$f$41

and satisfies equimeasurability: $f$42 for every Borel $f$43, together with the contractivity property

$f$44

The relative rearrangement $f$45 records the profile of a second function $f$46 along the level sets of $f$47; under suitable regularity and absence of flat regions,

$f$48

and in general it is characterized by

$f$49

This construction transports spatial weights into the rearranged one-dimensional setting (Galiano et al., 2014).

The exact rearranged form of the filter is

$f$50

Theorem 1 proves the equivalence

$f$51

under the hypotheses $f$52, $f$53, $f$54, and $f$55 without loss of generality. The multidimensional operator is thereby reduced to a one-dimensional integral over the level-set measure variable $f$56, independently of the spatial dimension of the image (Galiano et al., 2014).

The same paper develops a discretization by constant-wise interpolators. If

$f$57

and

$f$58

then for $f$59,

$f$60

Theorem 2 proves convergence $f$61 almost everywhere and strongly in $f$62. The computational complexity is reported as $f$63 in general, $f$64 for the Neighborhood filter, $f$65 for the Weighted Neighborhood filter, and $f$66 for the Yaroslavsky filter (Galiano et al., 2014).

The rearranged formulation also yields a detailed description of filter behavior. For each iteration there exists a strictly increasing function $f$67 such that

$f$68

so the filtered image is a contrast change of the original image. For Gaussian kernels $f$69, the small-$f$70 asymptotics contain a border diffusion term, a shock-filter-like sharpening term of the form $f$71, and in weighted cases an additional source term tied to the weight variation. The paper identifies these effects as responsible for staircaising and loss of contrast in iterative filtering (Galiano et al., 2014).

6. Fixed-boundary surface smoothing and string-rearrangement surfaces

A distinct geometric usage of smoothing rearrangements appears in the study of surfaces with fixed boundary. In "Variational Minimization on String-rearrangement Surfaces, Illustrated by an Analysis of the Bilinear Interpolation" (Ahmad et al., 2012), a surface is parametrized by

$f$72

with fixed boundary $f$73, and its area is

$f$74

The standard minimal-surface criterion is recalled: a surface is minimal iff its mean curvature vanishes everywhere. Instead of minimizing the full area functional directly, the paper minimizes the root-mean-square mean curvature

$f$75

using a variational ansatz based on the numerator $f$76 of the mean curvature of the initial surface: $f$77 The factor $f$78 forces the perturbation to vanish on the boundary, so the same boundary curves are preserved (Ahmad et al., 2012).

Because $f$79 depends polynomially on $f$80, the coefficients $f$81 are quadratic in $f$82, $f$83 are cubic in $f$84, and hence

$f$85

becomes a polynomial in $f$86. After integration,

$f$87

is again a polynomial; in the reported case the degree reaches $f$88 at the integrand level. The procedure is: build $f$89, compute $f$90, minimize it with respect to $f$91, and use the minimizing value $f$92 to define the improved surface. The paper emphasizes that a zero or near-zero of this polynomial corresponds to a minimal surface within the allowed ansatz class (Ahmad et al., 2012).

The principal application concerns bilinear interpolation surfaces viewed as string-rearrangement surfaces and as special Coons patches. In the bilinear case,

$f$93

with linear blending functions $f$94, $f$95, $f$96, $f$97. For the ruled$f$98 configuration with corners

$f$99

the starting surface is

$f^*$00

and its mean-curvature numerator is

$f^*$01

Thus the surface is not minimal, since $f^*$02 except on the lines $f^*$03 and $f^*$04. With the choice $f^*$05, the perturbation becomes an explicit polynomial correction (Ahmad et al., 2012).

The benchmark hemiellipsoid case,

$f^*$06

has a known minimal surface, the elliptic disk, and serves to validate the method. The reported dimensionless area decrease lies in the range

$f^*$07

for varying $f^*$08. For the ruled$f^*$09 bilinear surface, by contrast, the dimensionless decrease in area is in the range

$f^*$10

that is, less than $f^*$11 of the original area. The authors interpret this as evidence that the ruled$f^*$12 bilinear interpolation surface is already near minimal within the perturbative class defined by the ansatz. This usage differs from the Euclidean symmetrization literature, but it retains the same underlying idea: a controlled rearrangement-like transformation is employed to reduce a variational measure of roughness while keeping the prescribed boundary data fixed (Ahmad et al., 2012).