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### Propagation of smallness for real analytic functions and solutions to Generalized Cauchy--Riemann systems.

**Colloquium**

Speaker: | Eugenia MALINNIKOVA, University of Tronheim (Norway) and UCDavis |

Location: | 693 Kerr |

Start time: | Mon, Nov 17 2003, 4:10PM |

Hadamard's three-circle theorem is a classical example of propagation of smallness for analytic functions. Given an analytic function in the unit disk that is a priori bounded and is small on a small disk, one can write a majorant for this function. We will discuss various generalizations of this result and its applications. In particular, we will show how to do quanitative propagation of smallness for the gradients of harmonic functions form small (zero Lebesgue measure) sets. Open problems on Carleman type inequalities and boundary unique continuation for harmonic functions are related to propagation of smallness and willbe formulated in the talk.