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Sharp exponential integrability of conjugate functions
Abstract
We prove that if a real-valued function f∈ L1 on the complex unit circle has a gap of width at least π in its essential range, then exp(f̃) is not integrable, where f̃ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function Y satisfying ∫0∞ Y(x) e−x dx = ∞. We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, Y(f̃) fails to be integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Furthermore, we obtain corresponding results for complex-valued functions and a Y-integrability version of Zygmund's result.
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BibTeX
@misc{norrbo2026sharp,
title = {{Sharp exponential integrability of conjugate functions}},
author = {David Norrbo and Jani Virtanen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.05348},
eprint = {2609.05348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05348},
}
RIS
TY - UNPB
TI - Sharp exponential integrability of conjugate functions
AU - David Norrbo
AU - Jani Virtanen
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.05348
UR - https://arxiv.org/abs/2609.05348
AB - We prove that if a real-valued function f∈ L^1 on the complex unit circle has a gap of width at least π in its essential range, then exp(f̃) is not integrable, where f̃ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function Y satisfying ∫_0^∞ Y(x) e^−x dx = ∞. We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, Y(f̃) fails to be integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Furthermore, we obtain corresponding results for complex-valued functions and a Y-integrability version of Zygmund's result.
ER -