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®}\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®+a_2®\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$a_i®$ 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 $(1-\delta {)}^{\frac{1}{p}}$ due to Bourgain-Brezis-Minorescu.