Opposed Exponent Profiles and Real Interpolation of Measurable Subspaces
Abstract
We study real interpolation of the G-measurable subspaces Yj of variable Lebesgue spaces Lpj(·). For a family of continuous exponent pairs that vary in opposite directions on the atoms of G, we prove that K-closedness is equivalent to a uniform logarithmic bound on the oscillation of the exponents. The same bound characterizes the existence of a common bounded projection and interpolation equality at a single interior parameter. If it fails, then for each 0<θ<1 there is one function belonging to all ambient interpolation spaces (X0,X1)θ,q, 0<q≤∞, and to none of the corresponding spaces (Y0,Y1)θ,q. Each endpoint subspace still admits a bounded positive projection. The exponents can be chosen arbitrarily close to prescribed constants 1<a<b<∞, with constant sum and smoothness away from zero. We also give a finite-profile criterion in terms of the endpoint maxima, compute the associated K-functional loss, and obtain a counterexample for adapted sequence spaces.
Related
- Interpolation of subspaces of infinite codimension
- Noncommutative commutant lifting, interpolation, and shift-invariant subspaces
- Minor-Order Exponent Profiles of Oscillatory Matrices
- Exponent Profiles of Oscillatory Matrices under Matrix and Inverse Sparsity
- Self measurable embeddings of mapping class groups
BibTeX
@misc{ullah2026opposed,
title = {{Opposed Exponent Profiles and Real Interpolation of Measurable Subspaces}},
author = {Asad Ullah},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24904},
eprint = {2610.24904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24904},
}
RIS
TY - UNPB
TI - Opposed Exponent Profiles and Real Interpolation of Measurable Subspaces
AU - Asad Ullah
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24904
UR - https://arxiv.org/abs/2610.24904
AB - We study real interpolation of the G-measurable subspaces Y_j of variable Lebesgue spaces L^p_j(·). For a family of continuous exponent pairs that vary in opposite directions on the atoms of G, we prove that K-closedness is equivalent to a uniform logarithmic bound on the oscillation of the exponents. The same bound characterizes the existence of a common bounded projection and interpolation equality at a single interior parameter. If it fails, then for each 0<θ<1 there is one function belonging to all ambient interpolation spaces (X_0,X_1)_θ,q, 0<q≤∞, and to none of the corresponding spaces (Y_0,Y_1)_θ,q. Each endpoint subspace still admits a bounded positive projection. The exponents can be chosen arbitrarily close to prescribed constants 1<a<b<∞, with constant sum and smoothness away from zero. We also give a finite-profile criterion in terms of the endpoint maxima, compute the associated K-functional loss, and obtain a counterexample for adapted sequence spaces.
ER -