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The Two-Point Inequality for p>q

  • Di Wu
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Abstract

For 1<q<p<∞ we study the Lp→ Lq contractivity of the two-point operator Tω(a+bx)=a+ωbx with a complex parameter ω, extending the Weissler two-point condition, usually discussed in the range 1<p≤ q<∞, to the reversed order of exponents. For 1<q<p<2 we prove that the second-order necessary condition |(p−2)−(q−2)ω2|≤ p−q|ω|2, obtained near (a,b)=(1,0), is also sufficient; by duality the same description applies to 2<q<p<∞. The proof uses the method of characteristic flows. In addition, for q≤2≤ p the contractivity follows directly from L2 contraction.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026twopoint,
  title = {{The Two-Point Inequality for $p>q$}},
  author = {Di Wu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00835},
  eprint = {2611.00835},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00835},
}
RIS
TY  - UNPB
TI  - The Two-Point Inequality for p>q
AU  - Di Wu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00835
UR  - https://arxiv.org/abs/2611.00835
AB  - For 1<q<p<∞ we study the L^p→ L^q contractivity of the two-point operator T_ω(a+bx)=a+ωbx with a complex parameter ω, extending the Weissler two-point condition, usually discussed in the range 1<p≤ q<∞, to the reversed order of exponents. For 1<q<p<2 we prove that the second-order necessary condition |(p−2)−(q−2)ω^2|≤ p−q|ω|^2, obtained near (a,b)=(1,0), is also sufficient; by duality the same description applies to 2<q<p<∞. The proof uses the method of characteristic flows. In addition, for q≤2≤ p the contractivity follows directly from L^2 contraction.
ER  -