A tribute to Pola Harboure: Isoperimetric inequalities and the HMS extrapolation theorem

Abstract

We give a simpler proof of the Gagliardo estimate with a measure obtained by Franchi, Pérez, and Wheeden [Proc. London Math. Soc. (3) 80 no. 3 (2000), 665–689], and improved by Pérez and Rela [Trans. Amer. Math. Soc. 372 no. 9 (2019), 6087–6133]. This result will be further improved using fractional Poincaré type inequalities with the extra bonus of Bourgain– Brezis–Mironescu as done by Hurri-Syrjänen, Mart́ınez-Perales, Pérez, and Vähäkangas [Internat. Math. Res. Notices (2022), rnac246] with a new argument. This will be used with the HMS extrapolation theorem to get Lp type result. 1. The isoperimetric inequality and extrapolation theory It is a great pleasure for us to dedicate this article to Eleonor Harboure, Pola, who played a central role in the development of modern Harmonic Analysis in Argentina. The first author is deeply grateful for her kind support during early stages of his career. Both authors want to stress how influential the work of Pola was to the mathematical community. This paper is also a tribute to the extrapolation theorem of Pola, R. Maćıas, and C. Segovia which was published in the American Journal of Mathematics [21] (see also [20]). See Theorem 2.1 in Section 2 for an updated version. We will refer to it as the HMS extrapolation theorem. Thanks to this result we can complete some of the main results obtained in [32] in the classical setting. A fractional counterpart with the Bourgain–Brezis–Mironescu gain will be obtained in the line of results as derived in [22] or [3]. The HMS extrapolation theorem was inspired by the classical extrapolation theorem of Rubio de Francia [8, 10,18]. 2020 Mathematics Subject Classification. Primary 42B25; Secondary 42B20. C. P. was supported by grant PID2020-113156GB-I00, Spanish Government; by the Basque Government through grant IT1615-22 and the BERC 2014-2017 program; and by BCAM Severo Ochoa accreditation SEV-2013-0323, Spanish Government. He is also very grateful to the MittagLeffler Institute under the program “Geometric aspects of nonlinear partial differential equations” where part of this research was carried out. E. R. was supported the projects PICT 2019-03968, PICT 2018-3399 and UBACyT 20020190200230BA. He is also very grateful to the suuport from Guandgdong Technion Israel Institute of Technology where part of this research was carried out.