On the BBM-Phenomenon in Fractional Poincaré–Sobolev Inequalities with Weights-Reference-Cited by-同舟云学术

On the BBM-Phenomenon in Fractional Poincaré–Sobolev Inequalities with Weights

Published:2022-09-21 Issue: Volume: Page:
ISSN:1073-7928
Container-title:International Mathematics Research Notices
language:en
Short-container-title:

Author:

Hurri-Syrjänen Ritva¹,Martínez-Perales Javier C²,Pérez Carlos³,Vähäkangas Antti V⁴

Affiliation:

1. Department of Mathematics and Statistics , University of Helsinki, Pietari Kalmin katu 5, FI-00014 Finland

2. Calle Nueva , 18, Manilva, Málaga, 29691 Spain

3. Department of Mathematics , University of the Basque Country, IKERBASQUE (Basque Foundation for Science) and BCAM–Basque Center for Applied Mathematics, Bilbao, 48009 Spain

4. University of Jyvaskyla , Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland

Abstract

Abstract In this paper, we unify and improve some of the results of Bourgain, Brezis, and Mironescu and the weighted Poincaré–Sobolev estimate by Fabes, Kenig, and Serapioni. More precisely, we get weighted counterparts of the Poincaré–Sobolev-type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behavior of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-improving property of generalized Poincaré inequalities.

Funder

Basque Government

Spanish State Research Agency

HAPDE

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Link

https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnac246/45957420/rnac246.pdf

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