Self-improving Poincaré-Sobolev type functionals in product spaces

Abstract

In this paper we give a geometric condition which ensures that (q, p)-Poincar´e-Sobolev inequalities are implied from generalized (1, 1)-Poincar´e 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)-Poincar´e type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincar´e-Sobolev estimates. Among other results, we prove that for each rectangle R of the form R = I1 ×I2 ⊂ R n where I1 ⊂ R n1 and I2 ⊂ R n2 are cubes with sides parallel to the coordinate axes, we have that

1 w(R) Z R |f − fR| p ∗ δ,w wdx 1 p∗ δ,w ≤ c (1−δ) 1 p [w] 1 p A1,R

a1(R)+a2(R)

, where δ ∈ (0, 1), w ∈ A1,R, 1 p − 1 p ∗ δ,w = δ n 1 1+log[w]A1,R and ai(R) are bilinear analog of the fractional Sobolev seminorms [u]Wδ,p (Q) (See Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincar´e-Sobolev estimates with the gain (1 − δ) 1 p due to Bourgain-Brezis-Minorescu.