Unique continuation for solutions to pde's; between hörmander's theorem and holmgren' theorem

We prove a new unique continuation result for solutions to partial differential equations, “interpolating” between Holmgren's Theorem and Hormander's Theorem. More precisely, under some partial analyticity assumptions on the coefficients we obtain an intermediate unique continuation result which is in general weaker than Holmgren's Theorem (which applies to problems with analytic coefficients) but stronger than Hormander's Theorem (which applies to problems with C1 coefficients). Some applications to the wave and the Schroedinger equation are considered next. In particular we obtain a result conjectured by Hormander, namely that for the wave equation with C1 but time independent coefficients one has unique continutaion across any noncharacteristic surface.