Parabolic Extrapolation of Compactness with Applications to Characterizations of the Parabolic CMO Space via Parabolic Fractional Commutators
Abstract
In this article, we prove two compactness parabolic extrapolation theorems for linear operators and their commutators on parabolic weighted Lebesgue spaces. As applications, we show that b∈PCMO (parabolic CMO space) if and only if b∈ Lloc1 and the commutator [b,Iαγ+]k is compact from Lr(ωr) to Lq(ωq) for any k∈ℕ, 1<r<q<∞ with 1/r−1/q=α∈(0,1), and ω∈ Ar,q+(γ) (parabolic Muckenhoupt class), where Iαγ+ is the parabolic fractional integral. Several other equivalent characterizations of the space PCMO are also given. Our approach combines the unweighted compactness for mixed commutators, a local lower estimate for [b,Iαγ+]k, and both a quantitative Uchiyama-type description and the equivalence with infT{|[·,Iαγ+]k−T|Lr→ Lq/1k} of distPBMO(·,PCMO), where the infimum is taken over all compact operators T:Lr→ Lq. Finally, we establish two parabolic weighted Fréchet--Kolmogorov compactness criteria, which further induce a parabolic interpolation theorem of compactness.
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BibTeX
@misc{cao2026parabolic,
title = {{Parabolic Extrapolation of Compactness with Applications to Characterizations of the Parabolic CMO Space via Parabolic Fractional Commutators}},
author = {Mingming Cao and Weiyi Kong and Dachun Yang and Wen Yuan and Chenfeng Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05933},
eprint = {2611.05933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05933},
}
RIS
TY - UNPB
TI - Parabolic Extrapolation of Compactness with Applications to Characterizations of the Parabolic CMO Space via Parabolic Fractional Commutators
AU - Mingming Cao
AU - Weiyi Kong
AU - Dachun Yang
AU - Wen Yuan
AU - Chenfeng Zhu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05933
UR - https://arxiv.org/abs/2611.05933
AB - In this article, we prove two compactness parabolic extrapolation theorems for linear operators and their commutators on parabolic weighted Lebesgue spaces. As applications, we show that b∈PCMO (parabolic CMO space) if and only if b∈ L_loc^1 and the commutator [b,I_α^γ+]_k is compact from L^r(ω^r) to L^q(ω^q) for any k∈ℕ, 1<r<q<∞ with 1/r−1/q=α∈(0,1), and ω∈ A_r,q^+(γ) (parabolic Muckenhoupt class), where I_α^γ+ is the parabolic fractional integral. Several other equivalent characterizations of the space PCMO are also given. Our approach combines the unweighted compactness for mixed commutators, a local lower estimate for [b,I_α^γ+]_k, and both a quantitative Uchiyama-type description and the equivalence with inf_T{|[·,I_α^γ+]_k−T|_L^r→ L^q^/1k} of dist_PBMO(·,PCMO), where the infimum is taken over all compact operators T:L^r→ L^q. Finally, we establish two parabolic weighted Fréchet--Kolmogorov compactness criteria, which further induce a parabolic interpolation theorem of compactness.
ER -