Self-improving Poincaré-Sobolev type functionals in product spaces

In this paper we give a geometric condition which ensures that (q, p)-Poincaré-Sobolev inequalities are implied from generalized (1, 1)-Poincaré related to L1 norms in the context of product spaces. The concept eccentricity plays central role paper. We provide several type adapted different geometries and then show our self-improving method can be applied obtain special interesting Poincaré-Sobolev estimates. Among other results, prove for each rectangle R form = I1 × I2 ≢ ℝn where $${I_1} \subset {\mathbb{R}^{{n_1}}}$$ $${I_2} {\mathbb{R}^{{n_2}}}$$ cubes with sides parallel coordinate axes, have $${\left( {\frac{1}{{w(R)}}\int_R {|f - {f_R}{|^{p_{\delta ,w}^*}}wdx} } \right)^{\frac{1}{{p_{\delta ,w}^*}}}} \leqslant c{(1 \delta )^{\frac{1}{p}}}[w]_{{A_{1,\Re }}}^{\frac{1}{p}}({a_1}(R) + {a_2}(R)),$$ δ ∈(0, 1), $$\delta \in (0,1),w {A_{1,\Re }},\frac{1}{p} \frac{1}{{p_{\delta ,w}^*}} \frac{\delta }{n}\frac{1}{{1 \log [w]{A_{1,\Re }}}}$$ ai(R) bilinear analogues fractional Sobolev seminorms $${[u]_{{W^{\delta ,p}}(Q)}}$$ (see Theorem 2.18). This is biparameter weighted version celebrated estimates gain $${(1 )^{\frac{1}{p}}}$$ due Bourgain-Brezis-Minorescu.