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Optimal Constant for a Smoothing Estimate of Critical Index

  • Neal BezEmail author
  • Mitsuru Sugimoto
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We generalise a result by Hoshiro [3] which considered a critical case of Kato–Yajima’s smoothing estimate
$$\parallel\, \mid x \mid^{a-1} \mid \bigtriangledown \mid^a\mathrm{exp}(-it\bigtriangleup)f\parallel_{{L{^2_{t,x}}}(\mathbb{R}\times\mathbb{R}^d)}\leq C\parallel f \parallel_{{L^{2}}(\mathbb{R}^d)}$$
for the Schorödinger propagator \(\mathrm{exp}(-it\bigtriangleup)\). An expression for the optimal constant is also given.

Keywords

Smoothing estimates optimal constants extremisers 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of Mathematics The Watson BuildingUniversity of BirminghamEdgbaston, BirminghamEngland
  2. 2.Graduate School of MathematicsNagoya UniversityChikusa-ku, NagoyaJapan

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