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We present a connected account of various spectral and regularity properties of the semigroup associated with a fairly general symmetric second order elliptic operator on a Riemannian manifold. Our main goal is to relate the L2 theory to the less well understood L1 theory, and hence to the...
We answer two problems posed by H. S. Shapiro in , and we show that, in general, there is no equivalence between the Cauchy problem for entire functions and the one for exponential type functions. We characterize some situations for which the equivalence holds.
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