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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

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Stefanie Petermichl (Université Paul Sabatier)

Characterizations of multi-parameter BMO spaces through boundedness of commutators

The characterisation of symbols that result in bounded Hankel or Toeplitz operators are classical and rather simple. When passing to real analysis and notably to multi-parameter real analysis, these questions become very quickly interesting and intensely complicated.

We discuss the simplicity and beauty of the classical base cases as well as the cornerstones into the world of real analysis 'away' from operator theory. In this situation one studies 'commutators' the simplest one of which takes the form $$Hb-bH$$ where $H$ is the Hilbert transform and $b$ stands for multiplication by a (bmo) function. Multi-parameter study of related objects was initiated by Sarah Ferguson and Cora Sadosky in the late 90s.

The real variable one-parameter theory we discuss includes parts of a classical article by Coifman Rochberg Weiss while the multi-parameter questions include a deep line by Ferguson, Lacey, Pipher, Wick, myself and others.
Emphasis is given to a recent result by Ou, Strouse and myself that solves an endpoint question begun by Ferguson/Sadosky. Although the proofs are 'hard analysis' exploiting several recent developments, one recognises the core that lies in elegant arguments stemming from operator theory.

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