Nonlinear Lebesgue spaces: Curves and geometry
Abstract
This paper is the second in a series by the author and collaborators devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces, that is, L^p spaces of mappings taking values in arbitrary metric spaces. The present article formalizes the pointwise description of their geometric properties---their length structure, bounds on their Alexandrov curvature as well as the definition of a velocity for absolutely continuous curves despite the lack of differential structure. To obtain this pointwise description, we first prove a nonlinear analogue of the Fubini--Lebesgue theorem, which yields an identification of L^p curves in nonlinear Lebesgue spaces with mappings taking values in the space of L^p curves. This identification of L^p curves then enables a similar identification for absolutely continuous curves, from which the pointwise description of the geometric properties of nonlinear Lebesgue spaces follows.
Related
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- Characterization of the boundedness of the segment multiplier on weighted Lebesgue spaces
- Some weighted modular inequalities in variable Lebesgue spaces
- Weighted boundedness of maximal operator associated with invertible linear transformations on variable Lebesgue spaces
- Critical orthonormal Strichartz estimates for Lebesgue spaces
BibTeX
@misc{serieys2026nonlinear,
title = {{Nonlinear Lebesgue spaces: Curves and geometry}},
author = {Guillaume Sérieys},
howpublished = {HAL},
year = {2026},
month = oct,
eprint = {2603.09459},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05544460v4},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Nonlinear Lebesgue spaces: Curves and geometry
AU - Guillaume Sérieys
PY - 2026
DA - 2026/10/12
PB - HAL
UR - https://hal.science/hal-05544460v4
AB - This paper is the second in a series by the author and collaborators devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces, that is, L^p spaces of mappings taking values in arbitrary metric spaces. The present article formalizes the pointwise description of their geometric properties---their length structure, bounds on their Alexandrov curvature as well as the definition of a velocity for absolutely continuous curves despite the lack of differential structure. To obtain this pointwise description, we first prove a nonlinear analogue of the Fubini--Lebesgue theorem, which yields an identification of L^p curves in nonlinear Lebesgue spaces with mappings taking values in the space of L^p curves. This identification of L^p curves then enables a similar identification for absolutely continuous curves, from which the pointwise description of the geometric properties of nonlinear Lebesgue spaces follows.
ER -