Optimal Constant for a Smoothing Estimate of Critical Index

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.

Mathematics Subject Classification (2010). Primary 35B45; Secondary 35P10, 35B65.