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 L¹ 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 self-improving 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 = I₁ × I₂ ≢ ℝⁿ 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

$${\left( {\frac{1}{{w(R)}}\int_R {|f - {f_R}{|^{p_{\delta ,w}^*}}wdx} } \right)^{\frac{1}{{p_{\delta ,w}^*}}}} \leqslant c{(1 - \delta )^{\frac{1}{p}}}[w]_{{A_{1,\Re }}}^{\frac{1}{p}}({a_1}(R) + {a_2}(R)),$$

where δ ∈(0, 1), $\delta \in (0,1),w \in {A_{1,\Re }},\frac{1}{p} - \frac{1}{{p_{\delta ,w}^*}} = \frac{\delta }{n}\frac{1}{{1 + \log [w]{A_{1,\Re }}}}$ and a_i(R) are bilinear analogues 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.