Mini workshop of Harmonic Analysis

Programme

講演者 10:30 --12:00 Sanghyuk Lee氏 Unique continuation for the heat operator with potentials in weak spaces 13:30 -- 15:00 Po-Lam Yung氏 Hardy spaces for Fourier integral operators 15:30 -- 16:00 鈴木聡一郎氏 Optimal constants of smoothing estimates for Dirac equations

Lee 先生からアブストラクトをいただきました．

This talk concerns the strong unique continuation property for the differential inequality

\[|(\partial_t +\Delta)u(x,t)|\le V(x,t)|u(x,t)|,\]

where the potential $V$ belongs to suitable function spaces.  In particular, we establish the strong unique continuation property for

$V\in L^\infty_t L^{d/2,\infty}_x$, which has remained open  since the earlier works of Escauriaza and Escauriaza--Vega.

Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces, which in turn are built on the spectral projection estimates for the Hermite operator.

Yung先生からアブストラクトをいただきました．

Abstract: In spatial dimension 2 or above, solutions of the wave equation can lose regularity over time when measured in the scale of L^p norms, if p is not equal to 2. The number of derivatives one can lose over time is captured by a classical theorem of Peral and Miyachi, which was later extended by Seeger, Sogge and Stein to more general Fourier integral operators. Subsequently a more conceptual proof of such results emerged from the work of Smith, and Hassell, Portal and Rozendaal. Recently I joined the latter group in extending some of these results using ideas from Fourier decoupling. In this talk, I will explain some of the ideas involved, by looking at some simple examples. Joint work with Andrew Hassell, Pierre Portal and Jan Rozendaal.

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