University of Birmingham > Talks@bham > Analysis seminar > Fourier, harmonic analysis, and spaces of homogeneous type

Fourier, harmonic analysis, and spaces of homogeneous type

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If you have a question about this talk, please contact Alessio Martini.

What do the integers, the Sierpiński gasket, compact Riemann surfaces and the Heisenberg group have in common? Each of them is a space of homogeneous type (X,d,μ): a set X equipped with a way of measuring the distance between any two points (a quasi-metric d) and a way of measuring the volume of subsets of X (a doubling measure μ). A familiar example is Euclidean space Rn equipped with the Euclidean metric and Lebesgue measure. Spaces of homogeneous type arise in many areas, including several complex variables and Riemannian geometry.

The Calderón-Zygmund theory in harmonic analysis deals with singular integral operators and the functions on which they act. Early impetus came from problems in partial differential equations and Fourier analysis. Here we focus on the generalisation from functions defined on Euclidean spaces Rn to functions defined on spaces X of homogeneous type. In particular, for general X the Fourier transform and the group structure of Rn are missing.

The goal is to build a Calderón-Zygmund theory on spaces of homogeneous type. I will survey some recent progress towards this goal.

This talk is part of the Analysis seminar series.

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