Harmonic Analysis e-Seminars

Next seminar:

01/04/26 - Carlos Pérez  - 5:30 p. m.

(University of the Basque Country, BCAM e Ikerbasque)

A modern geometric perspective on Poincaré-Sobolev inequalities thru harmonic analysis and without derivatives  

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Abstract. In this lecture, we present a modern perspective on function smoothness and regularity by employing sharp tools from Harmonic Analysis. The first part of the talk moves beyond classical pointwise analysis to explore functional smoothness through the lens of local averages. This approach provides a more flexible and robust framework for understanding analytical behavior, particularly in settings where traditional derivatives are not or cannot be considered. 

In the second part, we apply these harmonic analysis techniques to derive improved local Poincaré-Sobolev estimates, which are instrumental in proving the celebrated De Giorgi regularity theorem. This framework also provides a proof of the well-known John–Nirenberg theorem for BMO functions. The central theme of the discussion will be the ’self-improving’ property—a remarkable phenomenon where modest local control over oscillations leads to significantly stronger global integrability. As another application, we present an extension of the celebrated Nash inequality (which yields another proof of De Giorgi’s theorem), as well as an improved generalization of the Gagliardo–Nirenberg–Sobolev theorem using Campanato spaces.

FOCAHA: Forum of Complex And Harmonic Analysis.