Self-improving Poincar\'e-Sobolev type functionals in product spaces

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 $R=I_1\times I_2 \subset \mathbb{R}^{n}$ where $I_1\subset \mathbb{R}^{n_1}$ and $I_2\subset \mathbb{R}^{n_2}$ 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}^*} \,wdx\right)^{\frac{1}{p_{\delta,w}^*}} \leq c\,(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 Poincar\'e-Sobolev estimates with the gain $(1-\delta)^{\frac1p}$ due to Bourgain-Brezis-Minorescu.