Rearrangement-Invariant Spaces in Analysis

Updated 13 December 2025

Rearrangement-invariant spaces are Banach or quasi-Banach function spaces whose norms depend exclusively on the nonincreasing rearrangement of measurable functions.
This framework unifies classical spaces like Lebesgue, Lorentz, and Orlicz, enabling optimal operator bounds, fine interpolation, and embedding results.
Recent advances extend r.i. theory to quasi-Banach, multidimensional, and weighted settings, enhancing tools for PDEs, probability, and harmonic analysis.

A rearrangement-invariant (r.i.) space is a Banach (or quasi-Banach) function space whose norm depends solely on the distribution function—or, equivalently, the nonincreasing rearrangement—of the modulus of a measurable function. The r.i. framework unifies Lebesgue, Lorentz, Orlicz, Marcinkiewicz, and many more spaces in harmonic analysis, probability, functional analysis, and PDE theory. Rearrangement-invariance underlies optimal operator bounds, fine interpolation theory, endpoint embedding problems, and structural properties such as Boyd indices and homogeneity. Current research extends r.i. theory from Banach spaces to quasi-Banach function spaces, multidimensional and probabilistic extensions, weighted and mixed-norm settings, and precise optimality questions for classical and modern analysis.

1. Formal Structure and Defining Properties

A function norm $\varrho$ on a $\sigma$ -finite measure space $(R,\mu)$ is rearrangement-invariant if $\varrho(f) = \varrho(g)$ whenever $f^* = g^*$ , where $f^*(t) = \inf\{\lambda > 0 : \mu(\{x \in R : |f(x)| > \lambda\}) \leq t\}$ gives the nonincreasing rearrangement. The Banach (or quasi-Banach) function space $X = \{f : \varrho(|f|) < \infty\}$ is called r.i. if $\varrho$ satisfies positivity, homogeneity, quasi-triangle (or triangle), lattice, Fatou, and local finiteness axioms, together with rearrangement-invariance.

Every r.i. space $X$ admits a unique “representation” space $\overline X$ on $[0,\mu(R))$ such that for all $f$ , $\|f\|_X = \|f^*\|_{\overline X}$ , reducing norm computations to the rearrangement $f^*$ (Boza et al., 26 Jan 2025, Musilová et al., 31 Mar 2024, Turčinová, 2020).

The fundamental function $\varphi_X$ of $X$ is defined by $\varphi_X(t) = \|\chi_E\|_X$ for $\mu(E) = t$ , and fundamental relations (e.g., $\varphi_X(t)\varphi_{X'}(t) = t$ for the Köthe dual $X'$ ) facilitate duality and embedding results (Boza et al., 26 Jan 2025, Edmunds et al., 2019, Musilová et al., 31 Mar 2024).

2. Core Examples and Generated Scales

Major r.i. spaces are summarized below:

Type	Notation	Norm/Functional
Lebesgue	$L^p$	$(\int_0^\infty [f^*(t)]^p\,dt)^{1/p}$
Lorentz	$L^{p,q}$	$(\int_0^\infty [t^{1/p}f^*(t)]^q\,dt/t)^{1/q}$
Orlicz	$L^\Phi$	$\inf\{\lambda>0 : \int_0^\infty\Phi(f^*(t)/\lambda)\,dt\leq1\}$
Marcinkiewicz	$M^p$	$\sup_{t>0}t^{1/p}f^{**}(t)$
Grand Lebesgue	$G(\psi)$	$\sup_{p\in(A,B)}\\|f\\|_{L^p}/\psi(p)$
Lorentz-Zygmund	$L^{p,q;\alpha}$	$(\int_0^1 [t^{1/p-1/q}(1+\ln\frac{1}{t})^\alpha f^*(t)]^q\,dt)^{1/q}$

Extensions include mixed-norm and multidimensional r.i. spaces (Ostrovsky et al., 2012, Clavero et al., 2014, Clavero et al., 2014), and scale constructions such as $X^{\langle\alpha\rangle}$ , interpolating between $X$ and Zygmund-type classes (Turčinová, 2020).

3. Fundamental Structural Properties

R.i. spaces are characterized by several invariants and operations:

Boyd indices $(p_X, q_X)$ : Reflect the scaling behavior of the space under dilations; critical for maximal function and interpolation theory (Boza et al., 26 Jan 2025, Sukochev et al., 2010, Kerman et al., 2023).
Homogeneity: A sharp classification shows that only $p$ -homogeneous spaces (i.e., $\|D_r f\|_X = r^{-1/p}\|f\|_X$ ) exist among Banach r.i. norms; these include $L^p$ , Lorentz, and Orlicz–Lorentz, but exclude more exotic scales unless renormed appropriately (Boza et al., 26 Jan 2025).
Regularity and Tchebychev characteristic: The decay of the tails of unit-norm elements is intimately tied to the fundamental function and regularity. For classical spaces, $T_X(t)\asymp \varphi_X(1/t)^{-1}$ as $t\to\infty$ ; there exist non-regular examples, notably critical Grand Lebesgue constructions (Ostrovsky et al., 2012).

4. Operator Theory and Embeddings

The r.i. framework provides universal optimal targets and domains for classical operators:

Hardy-Littlewood maximal, fractional maximal, Hilbert and Stieltjes transforms, Riesz potentials: For each, the precise optimal pair of r.i. spaces is determined, generalizing and sharpening Lebesgue, Lorentz, Orlicz, and Lorentz–Zygmund scale estimates. Operator boundedness reduces to weighted Hardy-type inequalities over rearrangements; optimal target and domain spaces are fully characterized (Edmunds et al., 2019, Mihula, 2021).
Sobolev and Poincaré embeddings: Optimal r.i. targets for $m$ -th order Sobolev spaces are obtained as rearrangement-invariant spaces $X^m$ , determined by a duality process involving Hardy-type functionals and the fundamental function. For instance, $W^{m}L^p \hookrightarrow L^{p^*,p}$ with $p^* = np/(n-mp)$ and similar for Orlicz and Lorentz cases (Mihula, 2020).
Mixed-norm enhancements: Classical Sobolev embeddings are further sharpened by embedding into mixed-norm spaces $\mathcal{R}(X, L^\infty)$ , yielding strictly smaller targets than Lorentz spaces (Clavero et al., 2014, Clavero et al., 2014).
Optimal Gagliardo–Nirenberg and interpolation inequalities: For $X=Y^{1/2}Z^{1/2}$ (Calderón–Lozanovskii space), the Gagliardo–Nirenberg inequality $\|\nabla u\|_{X}\lesssim\|\nabla^2 u\|_Y^{1/2}\|u\|_Z^{1/2}$ is not only achieved but optimal: no strictly smaller r.i. space can serve as the domain for fixed $Y,Z$ (Lesnik et al., 2021).

5. Quasi-Banach and Multidimensional Extensions

Recent advances have established the full extension of r.i. theory to quasi-Banach function spaces (r.i. q-BFS):

Every r.i. q-BFS is an interpolation space between two Lorentz spaces via the $K$ -method, with precise control via Boyd indices and Holmstedt’s formula (Doktorski, 7 Nov 2025).
Luxemburg’s representation theorem extends to r.i. q-BFS, and fundamental functions are quasiconcave, supporting extremal endpoint spaces (Lorentz and Marcinkiewicz) in the q-Banach regime (Musilová et al., 31 Mar 2024).
Multidimensional and probabilistic rearrangement-invariant spaces are defined by scalar projections over direction sets. The resulting $d$ -dimensional spaces inherit completeness, separability, duality, and interpolation properties; applications include tail bounds for sums of independent random vectors and sharpness in random field regularity (Ostrovsky et al., 2012).

6. Special Constructions and Characterizations

Several notable developments:

Oscillation and oscillation-based characterizations: Garsia–Rodemich and Bourgain–Brezis–Mironescu spaces can be described entirely via oscillation functionals, and more generally, given any r.i. Banach function space $X$ , the associated Garsia–Rodemich space $\mathrm{GaRo}_X$ is equivalent to $X$ under mild index conditions (Milman, 2016).
Endpoint and bridge spaces: Endpoint embeddings for Hardy operators and maximal operators are characterized by Marcinkiewicz and Lorentz spaces associated to the fundamental function, with precise regularity and duality reciprocities (Musilová et al., 31 Mar 2024, Mihula, 2021).
Moment r.i. spaces: The class of moment rearrangement-invariant spaces, including Grand Lebesgue and Zygmund scales, supports sharp Wirtinger and Sobolev inequalities, transferring “Brink” constants without loss (Ostrovsky et al., 2010).

7. Applications and Ongoing Directions

Applications of r.i. spaces are prolific:

Sharp operator bounds: All classical operator boundedness on $L^p$ admits sharp generalization and refinement in the r.i. framework, including limiting and borderline cases (Edmunds et al., 2019, Mihula, 2021).
Sobolev and trace embeddings, optimal regularity classes: The optimal target in PDE and potential theory is often a very specific r.i. space derived from the underlying gradient or fractional operator scale (Mihula, 2020, Breit et al., 2020).
Probability and Banach space geometry: r.i. spaces form the natural setting for Khinchin-type inequalities, Banach–Saks properties, and limit laws for sum of i.i.d. random variables, including a complete classification in Lorentz spaces (Sukochev et al., 2010).
Fractional and nonlocal spaces: Characterizations in the r.i. setting underpin modern studies of fractional Sobolev spaces, nonlocal Gagliardo norms, and potential theory (Milman, 2016).
Open problems: Further classification in the quasi-Banach setting, the structure of non-regular spaces, true multidimensional rearrangement frameworks, and connections to harmonic analysis (e.g., endpoint estimates for the Fourier transform on r.i. spaces) are major ongoing directions (Kerman et al., 2023, Doktorski, 7 Nov 2025, Musilová et al., 31 Mar 2024).