Self-improving Poincaré-Sobolev type functionals in product spaces

Abstract

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 $(1,1)$-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 $R=I_1\times I_2\subset {\mathbb{R}}^n$ 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_{δ,w}^*} \,wdx\right)^{\frac{1}{p_{δ,w}^*}} \leq c\,(1-δ)^{\frac1p}\,[w]_{A_{1,\mathfrak{R}}}^{\frac1p}\, \Big(a_1(R)+a_2(R)\Big), \end{equation*} % where $\delta \in (0,1)$, $w\in A_{1,\mathfrak{R}}$, $\frac{1}{p}-\frac{1}{p_{\delta ,w}^{\ast }}=\frac{\delta }{n}\frac{1}{1+\log [w{]}_{A_{1,\mathfrak{R}}}}$ 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 Poincaré-Sobolev estimates with the gain $(1-\delta {)}^{\frac{1}{p}}$ due to Bourgain-Brezis-Minorescu.

In the first version of the paper there was an issue with the last two inequalities on page 5, since the factor ${\delta }^{1/p}$ on the right-hand side has to be omitted in the general case. This is not relevant to our contribution in the paper, but in order to correct this issue and not propagate the imprecision, we decided to remove the factor ${\delta }^{1/p}$ from every incorrect appearance