Self-improving Poincaré-Sobolev type functionals in product spaces

EUGENIA CEJAS, CAROLINA MOSQUERA, CARLOS PÉREZ Y EZEQUIEL RELA

JOURNAL D4ANALYSE MATHEMATIQUE

SPRINGER

Lugar: Berlin; Año: 2021

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 $L^1$ 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 $R=I_1imes I_2 subset mathbb{R}^{n}$ where $I_1subset mathbb{R}^{n_1}$ and $I_2subset mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have thategin{equation*}left( rac{1}{w(R)}int_{ R } |f -f_{R}|^{p_{delta,w}^*} ,wdxight)^{rac{1}{p_{delta,w}^*}} leq c,(1-delta)^{rac1p},[w]_{A_{1,ccR}}^{rac1p}, Big(a_1(R)+a_2(R)Big),end{equation*}where $delta in (0,1)$, $w in A_{1,ccR}$, $rac{1}{p} -rac{1}{ p_{delta,w}^* }= rac{delta}{n} , rac{1}{1+log [w]_{A_{1,ccR}}}$and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{delta,p}(Q)}$.This is a biparameter weighted version of the celebrated fractional Poincaré-Sobolev estimates with the gain $(1-delta)^{rac1p}$ due to Bourgain-Brezis-Minorescu.