Self-improving Poincaré-Sobolev type functionals in product spaces

In this paper we give a geometric condition which ensures that ( q, p )-Poincaré-Sobolev inequalities are implied from generalized (1, 1)-Poincaré inequalities related to L 1 norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several (1,...

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Bibliographic Details
Published in:Journal d'analyse mathématique (Jerusalem) 2023-04, Vol.149 (1), p.1-48
Main Authors: Cejas, María Eugenia, Mosquera, Carolina, Pérez, Carlos, Rela, Ezequiel
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Analysis
Cubes
Dynamical Systems and Ergodic Theory
Estimates
Functional Analysis
Inequalities
Mathematics
Mathematics and Statistics
Norms
Partial Differential Equations
Rectangles
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Summary:In this paper we give a geometric condition which ensures that ( q, p )-Poincaré-Sobolev inequalities are implied from generalized (1, 1)-Poincaré inequalities related to L 1 norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several (1, 1)-Poincaré type inequalities adapted to different geometries and then show that our self-improving method can be applied to obtain special interesting Poincaré-Sobolev estimates. Among other results, we prove that for each rectangle R of the form R = I 1 × I 2 ≢ ℝ n where and are cubes with sides parallel to the coordinate axes, we have that where δ ∈(0, 1), and a i ( R ) are bilinear analogues of the fractional Sobolev seminorms (see Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincaré-Sobolev estimates with the gain due to Bourgain-Brezis-Minorescu.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-022-0244-1