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, Martínez-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 L p type result.