SELF-IMPROVING POINCARE-SOBOLEV TYPE ´ FUNCTIONALS IN PRODUCT SPACES

CEJAS, MARÍA EUGENIA; MOSQUERA, CAROLINA; RELA, EZEQUIEL; PEREZ MORENO, CARLOS

JOURNAL D4ANALYSE MATHEMATIQUE

SPRINGER

Lugar: Berlin; Año: 2022

0021-7670

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 L1 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 selfimproving 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=I1×I2⊂Rn where I1⊂Rn1 and I2⊂Rn2 are cubes with sides parallel to the coordinate axes, we have that %(1w(R)∫R|f−fR|p∗δ,wwdx)1p∗δ,w≤c(1−δ)1p[w]1pA1,R(a1(R)+a2(R)),% where δ∈(0,1), w∈A1,R, 1p−1p∗δ,w=δn11+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 Poincaré-Sobolev estimates with the gain (1−δ)1p due to Bourgain-Brezis-Minorescu.