GUSTAVO GARRIGÓS
Universidad de Murcia
Padova June 2021. Mini-course on
Bergman projections
and decoupling inequalities
Padova June 2021, Mini-courses in Mathematical Analysis: webpage
ABSTRACT
In this minicourse we discuss the problem of Lp boundedness of Bergman
projections in a special class of domains in C^n,
given by D=R^n +iO with O a
symmetric cone of R^n.
Finding
the optimal range of p is a difficult question, which has recently been settled
for light-cones, but is still open for cones of higher rank.
These
Bergman operators, which arise in classical complex analysis, turn out to have
an interesting link with modern harmonic analysis. Namely, when using the
Fourier-Laplace transform, they are given by a family of multipliers of Bochner-Riesz type associated with the cone O. The natural tool to establish their Lp-boundedness,
in the optimal range of p, are the so-called decoupling inequalities,
which were proved in their sharpest form by Bourgain
and Demeter in 2015.
During
the lectures we shall discuss the main properties of Bergman projections and
symmetric cones, establish a precise link with the decoupling inequalities, and
obtain from these a proof of the main result. If time permits, we shall also
comment on the proof of the decoupling inequalities. Finally, we shall conclude
with various open questions, which are related to seemingly unexplored problems
in harmonic analysis.
LECTURE NOTES
ˇ Lectures
1 and 2: Bergman spaces, Bergman projections, symmetric cones
ˇ Lecture
3: Boundedness of Bergman projections in tubes
over cones
ˇ Lecture
4: Decoupling inequalities (appendix: sketch
of proof decoupling ineq for the parabola)
REFERENCES
ˇ Faraut, Korányi,
Analysis on symmetric cones. Clarendon Press, Oxford 1994.
ˇ BBGNPR. Lecture Notes
on Bergman projections in tube domains over cones:an analytic and geometric viewpoint. Imhotep 5, 2004.
ˇ BBGR. Littlewood-Paley
decompositions related to symmetric cones and Bergman projections in tuve domains. PLMS (3) 89 (2004).
ˇ Slides from Bergman Workshop, Indam June 2021: slides, talk
ˇ Demeter, Fourier restriction,
decoupling and applications. Cambridge Univ Press
2020.
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Gustavo
Garrigós
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Universidad
de Murcia
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Departamento
de Matemáticas
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Campus
Espinardo
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30100 Murcia, SPAIN
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