Degenerate Poincar\'e-Sobolev inequalities via fractional integration

We present a local weighted estimate for the Riesz potential in $\mathbb{R}^n$, which improves the main theorem of Alberico, Cianchi, and Sbordone [C. R. Math. Acad. Sci. Paris \textbf{347} (2009)] in several ways. As a consequence, we derive weighted Poincar\'e-Sobolev inequalities with sharp dependence on the constants. We answer positively to a conjecture proposed by P\'erez and Rela [Trans. Amer. Math. Soc. 372 (2019)] related to the sharp exponent in the $A_1$ constant in the $(p^*,p)$ Poincar\'e-Sobolev inequality with $A_1$ weights. Our approach is versatile enough to prove Poincar\'e-Sobolev inequalities for high-order derivatives and fractional Poincar\'e-Sobolev inequalities with the BBM extra gain factor $(1-\delta)^{1/p}$. In particular, we improve one of the main results from Hurri-Syrj\"anen, Mart\'inez-Perales, P\'erez, and V\"ah\"akangas [Int. Math. Res. Not. 20 (2023)].