PreprintCC BY 4.0arXiv
The Trudinger inequality is true for M1,s spaces
Abstract
We prove the Trudinger inequality with exponent s/(s−1) for M1,s Sobolev spaces on metric measure spaces under the sole measure growth assumption μ(B(x,r))≥ brs. This improves the previously known exponential integrability with power one. Our argument also yields sharp asymptotic bounds for the Sobolev constants as p↑ s. On doubling spaces, we further remove the connectedness assumption from the Trudinger inequality associated with a (1,p)-Poincaré inequality when p<s. An Ahlfors regular counterexample shows that this extension fails at the critical exponent p=s.
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BibTeX
@misc{alvarado2026trudinger,
title = {{The Trudinger inequality is true for $M^1,s$ spaces}},
author = {Ryan Alvarado and Ahmed Dughayshim and Piotr Hajłasz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09101},
eprint = {2609.09101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09101},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Trudinger inequality is true for M^1,s spaces
AU - Ryan Alvarado
AU - Ahmed Dughayshim
AU - Piotr Hajłasz
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09101
UR - https://arxiv.org/abs/2609.09101
AB - We prove the Trudinger inequality with exponent s/(s−1) for M^1,s Sobolev spaces on metric measure spaces under the sole measure growth assumption μ(B(x,r))≥ br^s. This improves the previously known exponential integrability with power one. Our argument also yields sharp asymptotic bounds for the Sobolev constants as p↑ s. On doubling spaces, we further remove the connectedness assumption from the Trudinger inequality associated with a (1,p)-Poincaré inequality when p<s. An Ahlfors regular counterexample shows that this extension fails at the critical exponent p=s.
ER -