Self-improving Poincaré-Sobolev type functionals in product spaces

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 149; no. 1; pp. 1 - 48
• Main Authors: Cejas, María Eugenia, Mosquera, Carolina, Pérez, Carlos, Rela, Ezequiel
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01-04-2023; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Analysis; Cubes; Dynamical Systems and Ergodic Theory; Estimates; Functional Analysis; Inequalities; Mathematics; Mathematics and Statistics; Norms; Partial Differential Equations; Rectangles
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-022-0244-1

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.