Embeddings between generalized weighted Lorentz spaces
Amiran Gogatishvili, Zdeněk Mihula, Luboš Pick, Hana Turčinová, Tuğçe Ünver
Dissertationes Mathematicae (2026)
MSC: Primary 26D10; Secondary 46E30.
DOI: 10.4064/dm875-6-2025
Published online: 30 January 2026
Abstract
We give a new characterization of a continuous embedding between two function spaces of type $G\Gamma$. Such spaces are governed by functionals of type
$$\|f\|_{G\Gamma(r,q;w,\delta)} := \left(\int_{0}^{L} \left( \frac1{\Delta(t)} \int_0^t f^*(s)^r \delta(s) \,ds \right)^{{q}/{r}}w(t) \,dt \right)^{1/q},$$
where $f^*$ is the nonincreasing rearrangement of $f$, $L\in(0,\infty]$, $r,q \in (0, \infty)$, $w, \delta$ are weights on $(0,L)$ and $\Delta(t)=\int_{0}^{t}\delta(s)\,ds$ for $t\in(0,L)$.
To characterize the embedding of such a space, say $G\Gamma(r_1,q_1;w_1,\delta_1)$, into another, $G\Gamma(r_2,q_2;w_2,\delta_2)$, means to find a balance condition on the four positive real parameters and the four weights in order that an appropriate inequality holds for every admissible function. We develop a new discretization technique which enables us to get rid of restrictions on parameters imposed in earlier work such as the nondegeneracy conditions or certain relations between the $r$’s and the $q$’s. Such restrictions were caused mainly by the use of duality techniques, which we avoid in this paper. On the other hand, we consider here only the case when $q_1 \le q_2$, leaving the reverse case to future work.
Authors
- Amiran GogatishviliInstitute of Mathematics
Czech Academy of Sciences
115 67 Praha 1, Czech Republic
ORCID: 0000-0003-3459-0355
e-mail
- Zdeněk MihulaDepartment of Mathematics
Faculty of Electrical Engineering
Czech Technical University in Prague
166 27 Praha 6, Czech Republic
and
Department of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
186 75 Praha 8, Czech Republic
ORCID: 0000-0001-6962-7635
e-mail
- Luboš PickDepartment of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
186 75 Praha 8, Czech Republic
ORCID: 0000-0002-3584-1454
e-mail
- Hana TurčinováDepartment of Mathematics
Faculty of Electrical Engineering
Czech Technical University in Prague
166 27 Praha 6, Czech Republic
and
Department of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
186 75 Praha 8, Czech Republic
ORCID: 0000-0002-5424-9413
e-mail
- Tuğçe ÜnverDepartment of Mathematics
Faculty of Engineering and Natural Sciences
Kirikkale University
71450, Yahsihan, Kirikkale, Türkiye
ORCID: 0000-0003-0414-8400
e-mail
