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Extremals for linear functionals on Hp are outer when 0<p<1

  • Ole Fredrik Brevig
  • Øistein Søvik
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Abstract

Let L be a nontrivial bounded linear functional on the Hardy space Hp of the unit disc. We prove that if 0<p<1 and if f in the unit sphere of Hp satisfies L(f)=|L|, then f is an outer function. The proof is short and relies on a construction due to de Leeuw and Rudin. Our result complements work of Duren, Romberg, and Shields on the failure of the Hahn--Banach theorem for Hp. When applied to the special case Lk(f) = f(k)(0)/k! for k=1,2,3,…, it resolves two conjectures of the first-named author and Saksman and one conjecture of Suffridge.

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Cite

BibTeXRIS
BibTeX
@misc{brevig2026extremals,
  title = {{Extremals for linear functionals on $H^p$ are outer when $0<p<1$}},
  author = {Ole Fredrik Brevig and Øistein Søvik},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21788},
  eprint = {2610.21788},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21788},
}
RIS
TY  - UNPB
TI  - Extremals for linear functionals on H^p are outer when 0<p<1
AU  - Ole Fredrik Brevig
AU  - Øistein Søvik
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21788
UR  - https://arxiv.org/abs/2610.21788
AB  - Let L be a nontrivial bounded linear functional on the Hardy space H^p of the unit disc. We prove that if 0<p<1 and if f in the unit sphere of H^p satisfies L(f)=|L|, then f is an outer function. The proof is short and relies on a construction due to de Leeuw and Rudin. Our result complements work of Duren, Romberg, and Shields on the failure of the Hahn--Banach theorem for H^p. When applied to the special case L_k(f) = f^(k)(0)/k! for k=1,2,3,…, it resolves two conjectures of the first-named author and Saksman and one conjecture of Suffridge.
ER  -