---
title: Generalized Lorentz–Zygmund Spaces
url: https://www.emergentmind.com/topics/generalized-lorentz-zygmund-spaces
type: topic
---

# Generalized Lorentz–Zygmund Spaces

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 \(G\Gamma\) [2212.08938] [2508.14777] [1610.03379] [2210.12988]. 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
\[
L[p,a]=\{\,f:\|f\|_{L[N_{p,a}]}<\infty\}.
\]
It is well known that \(L[p,0]=L^p\) [2212.08938].

A more general rearrangement definition fixes a weight \(S:(0,1)\to[0,\infty)\) with \(\int_0^1 S(t)\,dt<\infty\), and sets
\[
\|f\|_{[S]}=\int_0^1 f^*(t)\,S(t)\,dt,\qquad [S]=\{\,f:\|f\|_{[S]}<\infty\}.
\]
Special cases are explicit: \(S(t)=1\) gives \(L^1\); \(S(t)=t^{1/p-1}\) gives the weak-\(L^p\) space \(L^{p,\infty}\); and \(S(t)=t^{1/p-1}(1+|\ln t|)^a\) recovers exactly \(L[p,a]\) [2212.08938].

On bounded Euclidean domains, a standard generalized Lorentz–Zygmund quasi-norm is
\[
\|f\|_{L^{p,q;\alpha,\beta}(0,1)}
=\|s^{1/p-1/q}\,\ell(s)^\alpha\,\ell\ell(s)^\beta\,f^*(s)\|_{L^q(0,1)},
\]
with
\[
\ell(s)=1+|\log s|,\qquad \ell\ell(s)=1+\log \ell(s).
\]
A companion norm replaces \(f^*\) by \(f^{**}\):
\[
\|f\|_{L^{(p,q;\alpha,\beta)}(0,1)}
=\|s^{1/p-1/q}\,\ell(s)^\alpha\,\ell\ell(s)^\beta\,f^{**}(s)\|_{L^q(0,1)}.
\]
These are equivalent to rearrangement-invariant norms precisely when the parameters \((p,q,\alpha,\beta)\) lie in one of the admissible ranges listed in the Sobolev embedding theory, including the cases \(1<p<\infty\), \(1\le q\le\infty\), \(\alpha,\beta\in\mathbb R\), as well as the endpoint \(p=\infty\) regimes with the specified logarithmic constraints [2508.14777].

On homogeneous groups \(\mathbb G\), Lorentz–Zygmund spaces are defined directly in terms of a homogeneous quasi-norm \(|x|\). For \(0<p,q\le\infty\),
\[
\|f\|_{L_{p,q,\lambda}(\mathbb G)}
:=\sup_{R>0}\Biggl(\int_{\mathbb G}
\bigl(|x|^{Q/p}\,|\ln\tfrac{R}{|x|}|^\lambda\,|f(x)|\bigr)^q
\frac{dx}{|x|^Q}\Biggr)^{1/q},
\]
and the double-log version is
\[
\|f\|_{L_{p,q,\lambda_1,\lambda_2}(\mathbb G)}
:=\sup_{R>0}\Biggl(\int_{\mathbb G}
\bigl(|x|^{Q/p}\,|\ln\tfrac{eR}{|x|}|^{\lambda_1}\,
|\ln|\ln\tfrac{eR}{|x|}||^{\lambda_2}\,|f(x)|\bigr)^q
\frac{dx}{|x|^Q}\Biggr)^{1/q}.
\]
This suggests that “generalized Lorentz–Zygmund space” is not a single notation but a common structural pattern realized in several settings [1610.03379].

## 2. Rearrangement-invariant realization and generalized Gamma spaces

If \(X(0,1)\) is a rearrangement-invariant function norm, the representative realization on a bounded open set \(\Omega\subset\mathbb R^n\) is
\[
\|u\|_{X(\Omega)}=\|u^*(|\Omega|\cdot)\|_{X(0,1)}.
\]
In particular,
\[
L^{p,q;\alpha,\beta}(\Omega)=\{u\in L^0(\Omega):\|u^*(|\Omega|\cdot)\|_{L^{p,q;\alpha,\beta}(0,1)}<\infty\},
\]
and similarly for the parenthesized spaces \(L^{(p,q;\alpha,\beta)}(\Omega)\). The fundamental function of an r.i. space is
\[
\varphi_X(r)=\|\chi_{(0,r)}\|_{X(0,1)},\qquad 0<r<1,
\]
and the dilation operator \(D_\lambda f(s)=f(s/\lambda)\) for \(s\le\lambda\), \(0\) otherwise, is bounded on every r.i. space [2508.14777].

Associate spaces also have explicit descriptions. Up to equivalence,
\[
(L^{p,q;\alpha,\beta})'=L^{p',q';-\alpha,-\beta},
\]
while for the \(L^{(1,q;\alpha,\beta)}\) scale the associate space takes endpoint forms such as
\[
(L^{(1,q;\alpha,\beta)})'=L^{\infty,q';-\alpha-1,-\beta}
\quad\text{or}\quad
L^{\infty,q';-1/q',-\beta-1},
\]
depending on the parameter regime [2508.14777].

A broader framework is provided by generalized Gamma spaces. Given weights \(w,\delta:(0,L)\to(0,\infty)\), exponents \(r,q\in(0,\infty)\), and
\[
\Delta(t)=\int_0^t \delta(s)\,ds,
\]
one defines
\[
G\Gamma(r,q;w,\delta)
=
\Bigl\{f\in M(R,\mu):\|f\|_{G\Gamma(r,q;w,\delta)}<\infty\Bigr\},
\]
with quasi-norm
\[
\|f\|_{G\Gamma(r,q;w,\delta)}
=
\Bigl(
\int_0^L
\Bigl[
\frac1{\Delta(t)}\int_0^t (f^*(s))^r\,\delta(s)\,ds
\Bigr]^{q/r}
w(t)\,dt
\Bigr)^{1/q}.
\]
For power-log choices of \(w\) and \(\delta\), this recovers the classical Lorentz–Zygmund norm up to equivalence [2210.12988].

The embedding problem
\[
G\Gamma(r_1,q_1;w_1,\delta_1)\hookrightarrow G\Gamma(r_2,q_2;w_2,\delta_2)
\]
is characterized in the convex case \(q_1\le q_2\) by weighted Hardy-type suprema. In the prototypical case \(p>1\), \(q>1\), \(r>1\), the best constant is equivalent to
\[
C\approx B_1+B_2+B_3+B_4,
\]
where \(B_1,\dots,B_4\) are explicit suprema built from the primitives \(U,\Delta_i,V,W\). 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 [2210.12988].

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

Let \(\mathbb G\) be a homogeneous group of homogeneous dimension \(Q\), equipped with a homogeneous quasi-norm \(|x|\). The radial derivative is
\[
\mathcal R f(x)=\frac{d}{dr}f(r y)\big|_{r=|x|,\,y=x/|x|},
\]
and the Euler operator of degree zero is
\[
\mathbb E f(x)=|x|\,\mathcal R f(x).
\]
It satisfies \(\mathbb E(f)=\nu f\) if and only if \(f\) is positively homogeneous of order \(\nu\) [1610.03379].

The Sobolev–Lorentz–Zygmund space is defined by
\[
W^{1}L_{p,q,\lambda}(\mathbb G)
:=
\{\,f\in L_{p,q,\lambda}(\mathbb G):\tfrac1{|x|}\mathbb E f\in L_{p,q,\lambda}(\mathbb G)\},
\]
with norm
\[
\|f\|_{W^{1}L_{p,q,\lambda}}
=
\|f\|_{L_{p,q,\lambda}}
+
\bigl\|\,|x|^{-1}\mathbb E f\bigr\|_{L_{p,q,\lambda}}.
\]
A vanishing-at-radius version subtracts the boundary value
\[
f_R(x)=f\!\bigl(Rx/|x|\bigr),
\]
and defines
\[
\mathfrak L_{p,q,\lambda}(\mathbb G)
=
\Bigl\{\,f\in L^1_{\rm loc}:\|f\|_{\mathfrak L_{p,q,\lambda}}<\infty\Bigr\}
\]
through the norm based on \(|f(x)-f_R(x)|\); similarly one defines \(\mathfrak L_{p,q,\lambda_1,\lambda_2}(\mathbb G)\) with double logarithms plus the splitting on the annulus [1610.03379].

The main critical embedding theorem states that if \(1<\gamma<\infty\) and \(\max\{1,\gamma-1\}<q<\infty\), then
\[
W^{1}_{0}L_{Q,q,\tau,\sigma}(\mathbb G)\hookrightarrow
\mathfrak L_{\infty,q,\mu,\nu}(\mathbb G)
\]
continuously for
\[
\tau=\tfrac{q-1}{q},\qquad \sigma=\tfrac{q-\gamma}{q},\qquad
\mu=-\tfrac1q,\qquad \nu=-\tfrac{\gamma}{q}.
\]
For every \(f\in W^{1}_{0}L_{Q,q,\tau,\sigma}(\mathbb G)\) and each \(R>0\), inequality (1) in the paper holds, and the constant \(q/(\gamma-1)\) is sharp and cannot be improved [1610.03379].

The underlying mechanism is a critical-Hardy-type one-dimensional estimate in the radial variable. On the ball \(B(0,eR)\), one uses
\[
\int_0^R
\frac{|u(r)-u(R)|^q}
{r\,\ln(\tfrac{eR}{r})\,(\ln\ln\tfrac{eR}{r})^\gamma}\,dr
\le
\frac{q}{\gamma-1}
\int_0^R
r^{q-1}
\bigl(\ln\tfrac{eR}{r}\bigr)^{q-1}
\bigl(\ln\ln\tfrac{eR}{r}\bigr)^{q-\gamma}
|u'(r)|^q\,dr,
\]
obtained by integrating by parts in \(r\), absorbing boundary terms, and then applying Hölder’s inequality. Passing back via the polar decomposition
\[
dx=r^{Q-1}\,d\sigma(y)\,dr,\qquad y\in\{|y|=1\},
\]
produces the full \(Q\)-dimensional weighted version; an identical argument handles the complementary region \(|x|>R\) [1610.03379].

In the Euclidean case \(\mathbb G=\mathbb R^n\) 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 \(\mathcal R\) and \(\mathbb E\) adapt to that choice. This covers anisotropic dilations on \(\mathbb R^n\) as well as stratified groups such as the Heisenberg group [1610.03379].

## 4. Sobolev embeddings on bounded Euclidean domains

For bounded John domains \(\Omega\subset\mathbb R^n\), \(n\ge2\), and \(m<n\), the Sobolev space
\[
W^mX(\Omega)=\{u\in L^0(\Omega):D^k u\in X(\Omega)\ \text{for}\ k=0,\dots,m\},
\qquad
\|u\|_{W^mX}=\sum_{k=0}^m \||\nabla^k u|\|_X,
\]
has a detailed GLZ embedding theory [2508.14777].

If \(X=L^{p,q;\alpha,\beta}\) and \((p,q,\alpha,\beta)\) are admissible, then
\[
W^mL^{p,q;\alpha,\beta}(\Omega)\hookrightarrow
\begin{cases}
L^{p^*,q;\alpha,\beta}(\Omega) & \text{if } 1<p<n/m,\\
L^{\infty,q;\alpha-1,\beta}(\Omega) & \text{if } p=n/m,\ \alpha<1/q',\\
L^{\infty,q;-1/q,\beta-1}(\Omega) & \text{if } p=n/m,\ \alpha=1/q',\ \beta<1/q',\\
L^{\infty,q;-1/q,-1/q,-1}(\Omega) & \text{if } p=n/m,\ \alpha=\beta=1/q',\ q>1,\\
L^\infty(\Omega) & \text{else},
\end{cases}
\]
where \(p^*=np/(n-mp)\). Moreover each target is optimal among all rearrangement-invariant spaces [2508.14777].

If instead \(X=L^{(1,q;\alpha,\beta)}\), then for \(m<n\),
\[
W^mL^{(1,q;\alpha,\beta)}(\Omega)\hookrightarrow
\begin{cases}
L^{1^*,q;\alpha+1,\beta}(\Omega) & \text{if } \alpha>-1/q,\\
L^{1^*,q;0,\beta+1/q}(\Omega) & \text{if } \alpha=-1/q,\ \beta>-1/q,\\
L^{1^*,q;0,0,1/q}(\Omega) & \text{if } \alpha=\beta=-1/q,\ q>1.
\end{cases}
\]
These embeddings are optimal r.i.-wise for \(q=1\); for \(q>1\) the true optimal range is not a GLZ space [2508.14777].

For continuity, if \(\Omega\) is a bounded Jones domain, then
\[
W^mL^{p,q;\alpha,\beta}(\Omega)\hookrightarrow C^0_b(\Omega)
\]
if and only if \(p=n/m\), \(q=1\), \(\alpha=\beta=0\); or \(p=n/m\), \(\alpha>1/q'\); or \(p>n/m\). By contrast,
\[
W^mL^{(1,q;\alpha,\beta)}(\Omega)\not\hookrightarrow C^0_b(\Omega)
\]
for all admissible \((q,\alpha,\beta)\) [2508.14777].

The range theory also includes Hölder, Morrey, and Campanato targets. For \(p>1\), there is an explicit optimal near-zero modulus \(\hat\sigma_m(r)\), given piecewise by power and logarithmic factors, such that
\[
W^mL^{p,q;\alpha,\beta}(\Omega)\hookrightarrow C^{0,\hat\sigma_m}(\Omega).
\]
For \(p=1\), the only non-trivial Hölder embedding occurs at \(m=n\), and the optimal \(\hat\sigma_n(r)\) is given by two–three-fold logarithmic expressions. The optimal Morrey function is
\[
\tilde\phi(r)=\|s^{-1+m/n}\chi_{(r^n,1)}(s)\|_{X'(0,1)},
\]
and the optimal Campanato functions are
\[
\tilde\psi(r)=r^{-n+1}\|\chi_{(0,r^n)}\|_{X'(0,1)}\quad (m=1),
\]
\[
\tilde\psi(r)=r\|s^{-1+(m-1)/n}\chi_{(r^n,1)}\|_{X'(0,1)}\quad (m\ge2).
\]
Classical radial examples with log oscillations and model functions such as \(u(x)=|\log|x||^{-(1/q+\alpha)}\) show the sharpness of the critical and logarithmic exponents [2508.14777].

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

Grand Lebesgue spaces provide one bridge to generalized Lorentz–Zygmund spaces. For \(1<a<b\le\infty\) and a measurable gauge \(\psi:(a,b)\to(0,\infty)\) with \(\inf_{p\in(a,b)}\psi(p)>0\),
\[
\|f\|_{G\psi}=\sup_{p\in(a,b)}\frac{\|f\|_{L^p}}{\psi(p)}.
\]
If \(f\in G\psi\), \(S\in G\phi\), and \(1/p+1/p'=1\), then the rearrangement–Hölder inequality yields
\[
\|f\|_{[S]}
\le
\bigl[\psi(p)\phi(p')\bigr]\,
\|f\|_{G\psi}\,\|S\|_{G\phi}.
\]
Optimizing over admissible \((p,p')\) gives
\[
\|f\|_{[S]}
\le
C\bigl((a,b);(c,d)\bigr)\,
\|f\|_{G\psi}\,\|S\|_{G\phi},
\]
and this constant is sharp; power-type extremals \(f^*(t)\propto t^{-1/p}\) and \(S(t)\propto t^{-1/p'}\) attain equality up to an arbitrary \(\epsilon>0\) [2212.08938].

The fundamental functions are explicit:
\[
\varphi_{G\psi}(\delta)=\sup_{p\in(a,b)}\frac{\delta^{1/p}}{\psi(p)},
\qquad
\varphi_{[S]}(\delta)=\int_0^\delta S(t)\,dt.
\]
If \(S\in G\phi\), then
\[
\varphi_{[S]}(\delta)\le \|S\|_{G\phi}\,\varphi_{G\phi}(\delta).
\]
Limiting choices of \(S\) recover \(L^1\), weak-\(L^p\), and \(L[p,a]\), and as \(a\to0\) in \(L[p,a]\) one recovers \(L^p\) [2212.08938].

A second bridge is provided by Peetre interpolation between grand, small, and classical Lebesgue spaces. For \(1<p<q<\infty\) and \(a>0\),
\[
\bigl(L^{p),a},L^{q),a}\bigr)_{\theta,r}
=
L^{p_\theta,r}(\log L)^{-(1-\theta)a},
\qquad
\frac1{p_\theta}=(1-\theta)\frac1p+\theta\frac1q.
\]
For the corresponding small Lebesgue spaces,
\[
\bigl(L_{\,p),a},L_{\,q),a}\bigr)_{\theta,r}
=
L^{p_\theta,r}(\log L)^{+(1-\theta)a}.
\]
If one interpolates a grand with a small space, then
\[
\bigl(L^{p),a},L_{\,p),a}\bigr)_{\theta,r}
=
G\Gamma\bigl(p,r;w_1,w_2\bigr),
\]
with
\[
w_1(t)=t^{-1}(1-\log t)^{ar-1},
\qquad
w_2(t)=(1-\log t)^{-1}.
\]
As a corollary, any Lorentz–Zygmund space \(L^{a,r}(\log L)^\beta\) with \(\beta\neq0\) is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces [1709.05892].

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

Generalized Lorentz–Zygmund scales also arise as admissible spaces for weighted Sobolev inequalities. For \(1<p<\infty\), \(0<q<\infty\), and \(\Omega=\Omega_1\times\Omega_2\subset\mathbb R^k\times\mathbb R^{N-k}\), one studies
\[
\int_\Omega |g_1(y)g_2(z)|\,|u(y,z)|^q\,dy\,dz
\le
C\Bigl(\int_\Omega |\nabla u(y,z)|^p\,dy\,dz\Bigr)^{q/p}.
\]
The admissible weight classes are described by Lorentz and Lorentz–Zygmund spaces [2012.04622].

If \(n>p\), then for every \(q\in[0,p^*]\),
\[
X=L^{\alpha(p,q),\gamma}(\Omega)
\quad\Longrightarrow\quad
g\in X\Rightarrow g\in H_{p,q}(\Omega),
\]
with
\[
\alpha(p,q)=\frac{Np}{N(p-q)+qp},
\qquad
\gamma=\frac{p}{p-q}\ \text{if } q<p,\quad \gamma=\infty\ \text{if } q\ge p.
\]
If \(n=p\) and \(\Omega\) is bounded, then
\[
q<1\ \text{or}\ q\ge p:\ X=L^{1,\gamma;q/p'}(\Omega),
\qquad
1\le q<p:\ X=L^{1,\gamma;q-1}(\Omega).
\]
If \(n<p\) and \(\Omega\) is bounded in one direction, then \(L^1(\Omega)\subset H_{p,q}(\Omega)\) for all \(q\) [2012.04622].

The same framework contains cylindrical and product-weight results. For sector-like \(\Omega_1\subset\mathbb R^k\), the weight \(|y|^{-s}\) belongs to \(H_{p,q}(\Omega)\) precisely under the stated relations between \(s\), \(q\), \(p\), and \(k\). Product weights can also be matched through Hölder and interpolation parameters:
\[
g_1\in L^{k/((k-p)st+p),\,1/(st)}(\Omega_1),\qquad
g_2\in L^{1/(st)}(\Omega_2)
\]
yield \(q=(1-st)p\), while the weak-type Lorentz choice
\[
g_1\in L^{k/((1-s)tp),\infty}(\Omega_1),\qquad
g_2\in L^{(n-k)/((1-t)p),\infty}(\Omega_2)
\]
corresponds to \(q=(1-st)p+st\,p^*\) [2012.04622].

For nonnegative \(g\in H_{p,q}(\Omega)\), the best constant is
\[
B_q(g)=\sup\Bigl\{\int_\Omega g|u|^q:\ u\in\mathcal D_0^{1,p}(\Omega),\ \|\nabla u\|_p=1\Bigr\}.
\]
Under the hypothesis \(g=g_1(y)g_2(z)\) with \(g_i\) in the closure of \(C_c(\Omega_i)\) in the spaces used above, the map \(u\mapsto\int g|u|^q\) is compact on \(\mathcal D_0^{1,p}(\Omega)\), so \(1/B_q(g)\) is attained by some \(u\in\mathcal D_0^{1,p}(\Omega)\), yielding a weak solution of
\[
-\Delta_p u=\lambda\,g|u|^{q-2}u.
\]
In symmetric or radial cases one can identify classical constants, including Hardy’s constant \((p/(n-p))^p\) for \(g(x)=|x|^{-p}\) [2012.04622].

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 \(L^p\)-based scales, but to provide the exact logarithmic and iterated-logarithmic corrections required at critical thresholds across isotropic, anisotropic, Euclidean, and homogeneous-group settings.

Source: https://www.emergentmind.com/topics/generalized-lorentz-zygmund-spaces