CARLEMAN ESTIMATES, OPTIMAL THREE CYLINDER INEQUALITIES AND UNIQUE CONTINUATION PROPERTIES FOR PARABOLIC OPERATORS

We prove the following unique continuation property. Let u be a solution of a second order linear parabolic equation and S be a segment parallel to the t-axis. If u has a zero of order faster than any nonconstant polynomial at each point of S then u vanishes in each point (x, t') such that the plane t = t' has nonempty intersection with S.