Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

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Self-improving Poincaré-Sobolev type functionals in product spaces

Carolina Alejandra Mosquera

Universidad de Buenos Aires e IMAS CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this poster 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 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\,\delta^{\frac1p}(1-\delta)^{\frac1p}\,[w]_{A_{1,\mathcal{R}}}^{\frac1p}\, \Big(a_1(R)+a_2(R)\Big), \end{equation*}

where $\delta \in (0,1)$, $w \in A_{1,\mathcal{R}}$, $\frac{1}{p} -\frac{1}{ p_{\delta,w}^* }= \frac{\delta}{n} \, \frac{1}{1+\log [w]_{A_{1,\mathcal{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{\delta,p}(Q)}$. This is a biparameter weighted version of the celebrated fractional Poincaré-Sobolev estimates with the gain $\delta^{\frac1p}(1-\delta)^{\frac1p}$.

Joint work with Eugenia Cejas (Universidad de La Plata, Argentina), Carlos Pérez (Universidad del País Vasco y BCAM, España) and Ezequiel Rela (Universidad de Buenos Aires e IMAS-CONICET, Argentina).

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