Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle

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In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. The elliptic results are joint work with J.Bourgain, while the remainder of the works discussed are joint works with L.Escauriaza, G.Ponce and L.Vega.
The paper is based on lectures presented at WHAPDE 2008, Merida, Mexico. To appear in Contemp. Math. volume in honor of V.Mazya

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