Generalized Lorentz–Zygmund Spaces

• Generalized Lorentz–Zygmund spaces are function spaces that incorporate logarithmic and double-logarithmic modifications to traditional Lorentz scaling, forming a framework for critical embedding theories.
• They are defined via rearrangement-invariant norms using weighted integrals of decreasing rearrangements, which underpin sharp Sobolev, interpolation, and Hardy–Sobolev results.
• These spaces bridge classical Lebesgue, Orlicz, and Grand Lebesgue frameworks, providing precise logarithmic correction tools for endpoint phenomena in analysis.

Searching arXiv for recent and foundational papers on generalized Lorentz–Zygmund spaces and related Sobolev/interpolation embeddings. arxiv_search(query="generalized Lorentz-Zygmund spaces Sobolev embeddings interpolation Grand Lebesgue", max_results=10) arxiv_search(query="Lorentz-Zygmund generalized Gamma interpolation grand small Lebesgue", max_results=10) Generalized Lorentz–Zygmund spaces are families of function spaces in which Lorentz scaling is modified by logarithmic, double-logarithmic, or more general rearrangement weights. In the literature represented here, they appear in several equivalent or closely related forms: as weighted rearrangement spaces $[S]$ on $(0,1)$, as Orlicz–Lorentz–Zygmund spaces $L[p,a]$, as Euclidean spaces $L^{p,q;\alpha,\beta}(\Omega)$ defined through $f^*$ or $f^{**}$, as homogeneous-group spaces $L_{p,q,\lambda}(\mathbb G)$ and their double-log variants, and as generalized Gamma spaces $G\Gamma$ (Formica et al., 2022, Cavaliere et al., 20 Aug 2025, Ruzhansky et al., 2016, Gogatishvili et al., 2022). Across these models, the common theme is that endpoint or critical phenomena are captured by logarithmic corrections, and the resulting scales support sharp embedding, interpolation, and Hardy–Sobolev theories.

1. Definitions and model scales

A classical Lorentz–Zygmund–Orlicz model is obtained from the Young function

$$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$

with Luxemburg norm

$$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$

and corresponding space

$(0,1)$0

It is well known that $(0,1)$1 (Formica et al., 2022).

A more general rearrangement definition fixes a weight $(0,1)$2 with $(0,1)$3, and sets

$(0,1)$4

Special cases are explicit: $(0,1)$5 gives $(0,1)$6; $(0,1)$7 gives the weak-$(0,1)$8 space $(0,1)$9; and $L[p,a]$0 recovers exactly $L[p,a]$1 (Formica et al., 2022).

On bounded Euclidean domains, a standard generalized Lorentz–Zygmund quasi-norm is

$L[p,a]$2

with

$L[p,a]$3

A companion norm replaces $L[p,a]$4 by $L[p,a]$5: $L[p,a]$6 These are equivalent to rearrangement-invariant norms precisely when the parameters $L[p,a]$7 lie in one of the admissible ranges listed in the Sobolev embedding theory, including the cases $L[p,a]$8, $L[p,a]$9, $L^{p,q;\alpha,\beta}(\Omega)$0, as well as the endpoint $L^{p,q;\alpha,\beta}(\Omega)$1 regimes with the specified logarithmic constraints (Cavaliere et al., 20 Aug 2025).

On homogeneous groups $L^{p,q;\alpha,\beta}(\Omega)$2, Lorentz–Zygmund spaces are defined directly in terms of a homogeneous quasi-norm $L^{p,q;\alpha,\beta}(\Omega)$3. For $L^{p,q;\alpha,\beta}(\Omega)$4,

$L^{p,q;\alpha,\beta}(\Omega)$5

and the double-log version is

$L^{p,q;\alpha,\beta}(\Omega)$6

This suggests that “generalized Lorentz–Zygmund space” is not a single notation but a common structural pattern realized in several settings (Ruzhansky et al., 2016).

2. Rearrangement-invariant realization and generalized Gamma spaces

If $L^{p,q;\alpha,\beta}(\Omega)$7 is a rearrangement-invariant function norm, the representative realization on a bounded open set $L^{p,q;\alpha,\beta}(\Omega)$8 is

$L^{p,q;\alpha,\beta}(\Omega)$9

In particular,

$f^*$0

and similarly for the parenthesized spaces $f^*$1. The fundamental function of an r.i. space is

$f^*$2

and the dilation operator $f^*$3 for $f^*$4, $f^*$5 otherwise, is bounded on every r.i. space (Cavaliere et al., 20 Aug 2025).

Associate spaces also have explicit descriptions. Up to equivalence,

$f^*$6

while for the $f^*$7 scale the associate space takes endpoint forms such as

$f^*$8

depending on the parameter regime (Cavaliere et al., 20 Aug 2025).

A broader framework is provided by generalized Gamma spaces. Given weights $f^*$9, exponents $f^{**}$0, and

$f^{**}$1

one defines

$f^{**}$2

with quasi-norm

$f^{**}$3

For power-log choices of $f^{**}$4 and $f^{**}$5, this recovers the classical Lorentz–Zygmund norm up to equivalence (Gogatishvili et al., 2022).

The embedding problem

$f^{**}$6

is characterized in the convex case $f^{**}$7 by weighted Hardy-type suprema. In the prototypical case $f^{**}$8, $f^{**}$9, $L_{p,q,\lambda}(\mathbb G)$0, the best constant is equivalent to

$L_{p,q,\lambda}(\mathbb G)$1

where $L_{p,q,\lambda}(\mathbb G)$2 are explicit suprema built from the primitives $L_{p,q,\lambda}(\mathbb G)$3. The proof proceeds by restriction to nonincreasing rearrangements, discretization on a covering sequence, solution of the discrete Hardy-type problem, and antidiscretization, thereby avoiding duality (Gogatishvili et al., 2022).

3. Homogeneous groups, Euler operators, and double-log Sobolev structure

Let $L_{p,q,\lambda}(\mathbb G)$4 be a homogeneous group of homogeneous dimension $L_{p,q,\lambda}(\mathbb G)$5, equipped with a homogeneous quasi-norm $L_{p,q,\lambda}(\mathbb G)$6. The radial derivative is

$L_{p,q,\lambda}(\mathbb G)$7

and the Euler operator of degree zero is

$L_{p,q,\lambda}(\mathbb G)$8

It satisfies $L_{p,q,\lambda}(\mathbb G)$9 if and only if $G\Gamma$0 is positively homogeneous of order $G\Gamma$1 (Ruzhansky et al., 2016).

The Sobolev–Lorentz–Zygmund space is defined by

$G\Gamma$2

with norm

$G\Gamma$3

A vanishing-at-radius version subtracts the boundary value

$G\Gamma$4

and defines

$G\Gamma$5

through the norm based on $G\Gamma$6; similarly one defines $G\Gamma$7 with double logarithms plus the splitting on the annulus (Ruzhansky et al., 2016).

The main critical embedding theorem states that if $G\Gamma$8 and $G\Gamma$9, then

$$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$0

continuously for

$$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$1

For every $$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$2 and each $$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$3, inequality (1) in the paper holds, and the constant $$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$4 is sharp and cannot be improved (Ruzhansky et al., 2016).

The underlying mechanism is a critical-Hardy-type one-dimensional estimate in the radial variable. On the ball $$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$5, one uses

$$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$6

obtained by integrating by parts in $$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$7, absorbing boundary terms, and then applying Hölder’s inequality. Passing back via the polar decomposition

$$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$8

produces the full $$N_{p,a}(u)=|u|^p\Bigl[\ln\bigl(e+|u|\bigr)\Bigr]^a,$$9-dimensional weighted version; an identical argument handles the complementary region $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$0 (Ruzhansky et al., 2016).

In the Euclidean case $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$1 with standard dilations and Euclidean norm, these spaces coincide with the classical Lorentz–Zygmund and critical-Sobolev-type spaces studied by Machihara–Ozawa–Wadade and others. In a general homogeneous group, one may choose any homogeneous quasi-norm, and the same sharp constants persist while $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$2 and $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$3 adapt to that choice. This covers anisotropic dilations on $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$4 as well as stratified groups such as the Heisenberg group (Ruzhansky et al., 2016).

4. Sobolev embeddings on bounded Euclidean domains

For bounded John domains $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$5, $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$6, and $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$7, the Sobolev space

$$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$8

has a detailed GLZ embedding theory (Cavaliere et al., 20 Aug 2025).

If $$\|f\|_{L[N_{p,a}]}=\inf\Bigl\{\lambda>0:\int_\Omega N_{p,a}(f(\omega)/\lambda)\,P(d\omega)\le1\Bigr\},$$9 and $(0,1)$00 are admissible, then

$(0,1)$01

where $(0,1)$02. Moreover each target is optimal among all rearrangement-invariant spaces (Cavaliere et al., 20 Aug 2025).

If instead $(0,1)$03, then for $(0,1)$04,

$(0,1)$05

These embeddings are optimal r.i.-wise for $(0,1)$06; for $(0,1)$07 the true optimal range is not a GLZ space (Cavaliere et al., 20 Aug 2025).

For continuity, if $(0,1)$08 is a bounded Jones domain, then

$(0,1)$09

if and only if $(0,1)$10, $(0,1)$11, $(0,1)$12; or $(0,1)$13, $(0,1)$14; or $(0,1)$15. By contrast,

$(0,1)$16

for all admissible $(0,1)$17 (Cavaliere et al., 20 Aug 2025).

The range theory also includes Hölder, Morrey, and Campanato targets. For $(0,1)$18, there is an explicit optimal near-zero modulus $(0,1)$19, given piecewise by power and logarithmic factors, such that

$(0,1)$20

For $(0,1)$21, the only non-trivial Hölder embedding occurs at $(0,1)$22, and the optimal $(0,1)$23 is given by two–three-fold logarithmic expressions. The optimal Morrey function is

$(0,1)$24

and the optimal Campanato functions are

$(0,1)$25

$(0,1)$26

Classical radial examples with log oscillations and model functions such as $(0,1)$27 show the sharpness of the critical and logarithmic exponents (Cavaliere et al., 20 Aug 2025).

5. Relations with Grand Lebesgue, small Lebesgue, and interpolation scales

Grand Lebesgue spaces provide one bridge to generalized Lorentz–Zygmund spaces. For $(0,1)$28 and a measurable gauge $(0,1)$29 with $(0,1)$30,

$(0,1)$31

If $(0,1)$32, $(0,1)$33, and $(0,1)$34, then the rearrangement–Hölder inequality yields

$(0,1)$35

Optimizing over admissible $(0,1)$36 gives

$(0,1)$37

and this constant is sharp; power-type extremals $(0,1)$38 and $(0,1)$39 attain equality up to an arbitrary $(0,1)$40 (Formica et al., 2022).

The fundamental functions are explicit: $(0,1)$41 If $(0,1)$42, then

$(0,1)$43

Limiting choices of $(0,1)$44 recover $(0,1)$45, weak-$(0,1)$46, and $(0,1)$47, and as $(0,1)$48 in $(0,1)$49 one recovers $(0,1)$50 (Formica et al., 2022).

A second bridge is provided by Peetre interpolation between grand, small, and classical Lebesgue spaces. For $(0,1)$51 and $(0,1)$52,

$(0,1)$53

For the corresponding small Lebesgue spaces,

$(0,1)$54

If one interpolates a grand with a small space, then

$(0,1)$55

with

$(0,1)$56

As a corollary, any Lorentz–Zygmund space $(0,1)$57 with $(0,1)$58 is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces (Fiorenza et al., 2017).

6. Weighted Hardy–Sobolev inequalities, extremals, and scope

Generalized Lorentz–Zygmund scales also arise as admissible spaces for weighted Sobolev inequalities. For $(0,1)$59, $(0,1)$60, and $(0,1)$61, one studies

$(0,1)$62

The admissible weight classes are described by Lorentz and Lorentz–Zygmund spaces (Anoop et al., 2020).

If $(0,1)$63, then for every $(0,1)$64,

$(0,1)$65

with

$(0,1)$66

If $(0,1)$67 and $(0,1)$68 is bounded, then

$(0,1)$69

If $(0,1)$70 and $(0,1)$71 is bounded in one direction, then $(0,1)$72 for all $(0,1)$73 (Anoop et al., 2020).

The same framework contains cylindrical and product-weight results. For sector-like $(0,1)$74, the weight $(0,1)$75 belongs to $(0,1)$76 precisely under the stated relations between $(0,1)$77, $(0,1)$78, $(0,1)$79, and $(0,1)$80. Product weights can also be matched through Hölder and interpolation parameters: $(0,1)$81 yield $(0,1)$82, while the weak-type Lorentz choice

$(0,1)$83

corresponds to $(0,1)$84 (Anoop et al., 2020).

For nonnegative $(0,1)$85, the best constant is

$(0,1)$86

Under the hypothesis $(0,1)$87 with $(0,1)$88 in the closure of $(0,1)$89 in the spaces used above, the map $(0,1)$90 is compact on $(0,1)$91, so $(0,1)$92 is attained by some $(0,1)$93, yielding a weak solution of

$(0,1)$94

In symmetric or radial cases one can identify classical constants, including Hardy’s constant $(0,1)$95 for $(0,1)$96 (Anoop et al., 2020).

Taken together, these results place generalized Lorentz–Zygmund spaces at the intersection of rearrangement-invariant analysis, endpoint Sobolev theory, interpolation, and weighted Hardy inequalities. A plausible implication is that their main role is not merely to refine $(0,1)$97-based scales, but to provide the exact logarithmic and iterated-logarithmic corrections required at critical thresholds across isotropic, anisotropic, Euclidean, and homogeneous-group settings.