Compact sets in Banach lattices over spaces of homogeneous type
Abstract
We prove a Kolmogorov--Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for L1. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov--Riesz theorem in L1, which avoids reflexivity and yields a unified treatment of Lp-spaces for 1≤ p<∞. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.
Related
- Compact sets in quasi-Banach function spaces and multilinear extrapolation of compactness on Lp(·)(w)
- DW-L-Weakly Compact Operators Between Banach Lattices
- Norm attainment for multilinear operators and polynomials on Banach Spaces and Banach lattices
- Polish spaces of separable Banach lattices
- Spectral theorem for compact self-adjoint operators on smooth Banach spaces
BibTeX
@misc{aydogdu2026compact,
title = {{Compact sets in Banach lattices over spaces of homogeneous type}},
author = {Ayşenur Aydoğdu and Amiran Gogatishvili and Yoshihiro Sawano and Daiki Takesako},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26153},
eprint = {2609.26153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26153},
}
RIS
TY - UNPB
TI - Compact sets in Banach lattices over spaces of homogeneous type
AU - Ayşenur Aydoğdu
AU - Amiran Gogatishvili
AU - Yoshihiro Sawano
AU - Daiki Takesako
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26153
UR - https://arxiv.org/abs/2609.26153
AB - We prove a Kolmogorov--Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for L^1. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov--Riesz theorem in L^1, which avoids reflexivity and yields a unified treatment of L^p-spaces for 1≤ p<∞. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.
ER -