Self-improving Poincare-Sobolev type functionals in product spaces

In this paper we give a geometric condition which ensures that $(q,p)$-Poincare-Sobolev inequalities are implied from generalized $(1,1)$-Poincare 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 $(1,1)$-Poincare type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincare-Sobolev estimates. Among other results, we prove that for each rectangle $R$ of the form $R=I_1times I_2 subset mathbb{R}^{n}$ 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 that % begin{equation*} left( frac{1}{w(R)}int_{ R } |f -f_{R}|^{p_{delta,w}^*} ,wdxright)^{frac{1}{p_{delta,w}^*}} leq c,delta^{frac1p}(1-delta)^{frac1p},[w]_{A_{1,mathfrak{R}}}^{frac1p}, Big(a_1(R)+a_2(R)Big), end{equation*} % where $delta in (0,1)$, $w in A_{1,mathfrak{R}}$, $frac{1}{p} -frac{1}{ p_{delta,w}^* }= frac{delta}{n} , frac{1}{1+log [w]_{A_{1,mathfrak{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{delta,p}(Q)}$ (See Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincare-Sobolev estimates with the gain $delta^{frac1p}(1-delta)^{frac1p}$.