Wiener–Luxemburg Amalgam Spaces Overview

• Wiener–Luxemburg amalgam spaces are rearrangement-invariant modifications that split the non-increasing rearrangement f* into local ([0,1]) and global ((1,∞)) components.
• They address classical Wiener amalgams’ shortcomings by using measure-theoretic localization to ensure Banach/quasi-Banach function space properties.
• The framework extends to associate duality and Orlicz-type settings, offering robust embeddings and improved functional-analytic structures.

Searching arXiv for the core and follow-up papers on Wiener–Luxemburg amalgam spaces and closely related Orlicz-type/generalized Wiener amalgam frameworks. Wiener–Luxemburg amalgam spaces are a rearrangement-based modification of classical Wiener amalgam spaces, introduced to separate the local and global behavior of a function while remaining within the category of rearrangement-invariant Banach or quasi-Banach function spaces. Their defining feature is that locality and globality are read from the non-increasing rearrangement $f^*$: the “local” component is the part of $f^*$ on $[0,1]$, and the “global” component is the part on $(1,\infty)$. This makes the construction measure-theoretic rather than geometric and is designed to avoid the failures of classical Wiener amalgams with respect to Banach-function-space structure and rearrangement invariance (Peša, 2021).

1. Definition and conceptual framework

The basic theory is developed on

$(R,\mu)=([0,\infty),\lambda),$

with $\lambda$ the Lebesgue measure, which is sufficient because of Luxemburg representation for rearrangement-invariant Banach function spaces over resonant measure spaces (Peša, 2021).

For a measurable function $f$, the distribution function is

$\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$

and the non-increasing rearrangement is

$$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$

Two functions are equimeasurable if they have the same distribution function, equivalently the same rearrangement. The Hardy–Littlewood inequality,

$$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$

is used throughout the theory (Peša, 2021).

If $f^*$0 and $f^*$1 are rearrangement-invariant Banach function norms, the Wiener–Luxemburg quasinorm is

$f^*$2

and the corresponding space is

$f^*$3

The same formula is used in the quasi-Banach setting when $f^*$4 and $f^*$5 are rearrangement-invariant quasi-Banach function norms (Peša, 2021).

The paper explicitly calls $f^*$6 the local component and $f^*$7 the global component. The resulting notion of locality is not topological. It is, in the paper’s formulation, “a measure-theoretic notion of locality/globality, not a topological one” (Peša, 2021).

2. Motivation: why classical Wiener amalgams are inadequate in the rearrangement-invariant setting

Classical Wiener amalgams measure local behavior geometrically, for example by

$f^*$8

with the usual modification for $f^*$9. In the rearrangement-invariant setting, the paper shows that this construction does not preserve expected structural properties (Peša, 2021).

More precisely, even for Wiener amalgams of Lebesgue spaces, the resulting spaces need not be Banach function spaces and need not be rearrangement-invariant. The appendix establishes that $[0,1]$0 satisfies weakened versions of the Banach-function-space axioms only for bounded sets $[0,1]$1, not arbitrary finite-measure sets, and that in general $[0,1]$2 fails axiom $[0,1]$3 or $[0,1]$4 unless $[0,1]$5 (Peša, 2021).

For $[0,1]$6, the failure of $[0,1]$7 is witnessed by

$[0,1]$8

with $[0,1]$9. Then $(1,\infty)$0, but

$(1,\infty)$1

For $(1,\infty)$2, the failure of $(1,\infty)$3 is shown using

$(1,\infty)$4

with suitable $(1,\infty)$5, for which $(1,\infty)$6 but

$(1,\infty)$7

Consequently, $(1,\infty)$8 is a Banach function norm if and only if $(1,\infty)$9, in which case it is just the usual $(R,\mu)=([0,\infty),\lambda),$0 norm (Peša, 2021).

The rearrangement issue is equally sharp. The paper proves that $(R,\mu)=([0,\infty),\lambda),$1 is equivalent to a rearrangement-invariant norm if and only if $(R,\mu)=([0,\infty),\lambda),$2, again reducing to $(R,\mu)=([0,\infty),\lambda),$3. This is the central reason for replacing geometric localization by rearrangement localization in the Wiener–Luxemburg construction (Peša, 2021).

3. Structural properties

A basic consistency property is that every rearrangement-invariant space is a trivial Wiener–Luxemburg amalgam. If $(R,\mu)=([0,\infty),\lambda),$4 is a rearrangement-invariant Banach function norm, then

$(R,\mu)=([0,\infty),\lambda),$5

In the quasi-Banach setting, if $(R,\mu)=([0,\infty),\lambda),$6 is the modulus of concavity of $(R,\mu)=([0,\infty),\lambda),$7, then

$(R,\mu)=([0,\infty),\lambda),$8

Thus $(R,\mu)=([0,\infty),\lambda),$9 is equivalent to $\lambda$0 (Peša, 2021).

The local component is always well behaved. If $\lambda$1 is a rearrangement-invariant Banach function norm, then

$\lambda$2

is itself a rearrangement-invariant Banach function norm, and there is $\lambda$3 such that

$\lambda$4

The paper emphasizes that any pathology comes from the global component rather than the local one (Peša, 2021).

The quasi-triangle structure is derived from the rearrangement inequality

$\lambda$5

combined with boundedness of the dilation operator $\lambda$6 on rearrangement-invariant spaces. In the quasi-Banach case, the Wiener–Luxemburg quasinorms are rearrangement-invariant quasi-Banach function norms and also satisfy axiom $\lambda$7. In the Banach-input case, the resulting quasinorm is equivalent to a rearrangement-invariant Banach function norm, so $\lambda$8 is a rearrangement-invariant Banach function space up to equivalent norm (Peša, 2021).

Because the resulting spaces are quasi-Banach or Banach function spaces, they inherit the standard lattice property: if $\lambda$9 a.e., then

$f$0

Completeness follows from the general theorem that every quasi-Banach function space is complete. In this sense, Wiener–Luxemburg amalgams preserve the functional-analytic framework that classical Wiener amalgams may fail to preserve in the rearrangement-invariant category (Peša, 2021).

4. Associate spaces and quasi-Banach duality

For a functional $f$1, the associate functional is

$f$2

with the conventions $f$3 and $f$4 for $f$5. Hölder inequality then takes the form

$f$6

This is the starting point for the duality theory (Peša, 2021).

In the Banach setting, if $f$7 are rearrangement-invariant Banach function norms with associates $f$8, then there is $f$9 such that

$\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$0

Hence

$\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$1

up to equivalence of defining functionals. This is the expected amalgam-of-associates formula, but its proof is nontrivial because $\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$2 need not be non-increasing (Peša, 2021).

In the quasi-Banach setting, the ordinary associate may degenerate, so the paper introduces the integrable norm

$\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$3

with integrable subspace

$\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$4

The paper proves that $\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$5 is a rearrangement-invariant quasi-Banach function norm with property $\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$6, and

$\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$7

The integrable associate $\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$8 is then the associate of $\mu_f(s)=\mu(\{t\in R:\ |f(t)|>s\}),$9 (Peša, 2021).

For Wiener–Luxemburg amalgams built from quasi-Banach inputs $$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$0, the correct duality statement becomes

$$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$1

so that

$$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$2

up to equivalence. If $$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$3 has $$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$4, then $$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$5, and

$$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$6

up to equivalence (Peša, 2021).

The second integrable associate $$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$7 behaves more subtly than the classical Banach bidual. The paper proves four cases: $$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$8 up to equivalent norms in the Banach-function-space case, strict one-sided embeddings in two intermediate cases, and possible incomparability when neither $$f^*(t)=\inf\{s\in[0,\infty): \mu_f(s)\le t\}, \qquad t\in[0,\infty).$$9 nor $$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$0 holds (Peša, 2021).

5. Embeddings, sum/intersection structure, and the Hardy–Littlewood–Pólya principle

Embeddings between Wiener–Luxemburg amalgams separate into independent local and global conditions. For rearrangement-invariant Banach function norms $$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$1, the paper proves that

$$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$2

if and only if the local component of $$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$3 is stronger than that of $$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$4, and

$$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$5

if and only if the global component of $$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$6 is stronger than that of $$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$7. Therefore

$$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$8

if and only if both local and global component comparisons hold. The same statement remains valid in the quasi-Banach setting (Peša, 2021).

The general sum/intersection bounds are

$$\int_R |fg|\,d\mu \le \int_0^\infty f^*g^*\,d\lambda,$$9

Under the additional assumptions that the local component of $f^*$00 is stronger than that of $f^*$01 and the global component of $f^*$02 is stronger than that of $f^*$03, the 2021 paper proves

$f^*$04

up to equivalence of quasinorms, while

$f^*$05

as sets (Peša, 2021).

A later note strengthens this substantially. Peša proves that if $f^*$06 and $f^*$07 are rearrangement-invariant quasi-Banach function norms, then

$f^*$08

and under the same local/global ordering assumptions,

$f^*$09

both up to equivalence of quasinorms. This extends the sum/intersection theorem from the Banach setting to the quasi-Banach setting and upgrades the sum identity from set-theoretic equality to equivalence of quasinorms (Peša, 14 Aug 2025).

The extremal roles of $f^*$10 and $f^*$11 are reversed between local and global components. In the Banach case,

$f^*$12

and

$f^*$13

The paper interprets this as: $f^*$14 is the strongest local and weakest global space, while $f^*$15 is the weakest local and strongest global space (Peša, 2021).

The Hardy–Littlewood–Pólya principle is one of the main applications. For a rearrangement-invariant quasi-Banach function norm $f^*$16, the principle is defined by

$f^*$17

whenever

$f^*$18

The 2021 paper proves that if $f^*$19 is an r.i. Banach function norm, then HLP holds, and if HLP holds, then $f^*$20 has $f^*$21; neither implication reverses in general. It then gives a negative answer to the question whether HLP holds for every rearrangement-invariant quasi-Banach function space: the space

$f^*$22

has property $f^*$23 but does not satisfy HLP (Peša, 2021).

Peša’s 2025 improvement weakens the sufficient hypothesis for HLP on resonant measure spaces. In the notation of the Banach-function-space axioms, the note proves

$f^*$24

and, more precisely, shows that on resonant measure spaces $f^*$25 already implies $f^*$26. A corollary is that any rearrangement-invariant quasi-Banach function space $f^*$27 satisfying HLP must obey

$f^*$28

This sharpens the earlier criterion and clarifies the role of resonance in the theory (Peša, 14 Aug 2025).

6. Relation to Orlicz-type and broader Wiener amalgam theory

The conceptual framework of Wiener amalgams extends beyond Lebesgue components. A recent survey states explicitly that one may replace the local norm by “some another Banach function space norm (in the sense of Luxemburg-Zaanen), such as Lorentz space or Orlicz space norms for the local (but also the global) component.” It also presents the general architecture $f^*$29, where $f^*$30 is a localizable Banach space and $f^*$31 is a solid translation-invariant Banach function space. This places Wiener–Luxemburg constructions within the broader generalized Wiener amalgam program (Feichtinger, 28 Jun 2026).

A direct Orlicz/Luxemburg realization is given by the paper on Wiener amalgam spaces of Orlicz type. For Young functions $f^*$32, it defines

$f^*$33

through the control function

$f^*$34

with norm

$f^*$35

The paper also defines the Luxemburg norm

$f^*$36

and the corresponding Luxemburg amalgam norm

$f^*$37

These satisfy

$f^*$38

When $f^*$39 and $f^*$40, the construction reduces to the classical $f^*$41 spaces (Aris et al., 2024).

That Orlicz-type paper also proves dilation estimates and continuity of the Zak transform when the local component is Orlicz and the global component is $f^*$42. This suggests a concrete interface between rearrangement-based Wiener–Luxemburg ideas and translation-based Orlicz amalgams. A plausible implication is that “Wiener–Luxemburg” can denote two related but distinct directions: the rearrangement-based spaces $f^*$43 of Peša, and translation-based Wiener amalgams whose local or global components are Orlicz spaces equipped with Luxemburg norms (Aris et al., 2024).

Within the supplied sources, the rearrangement-based construction remains the distinctive contribution of “Wiener-Luxemburg amalgam spaces” itself: it replaces geometric localization by the split

$f^*$44

applied to $f^*$45, thereby preserving rearrangement invariance and restoring Banach/quasi-Banach function-space behavior in a setting where classical Wiener amalgams can fail (Peša, 2021).