Degenerate Poincaré–Sobolev inequalities

Abstract:

We study weighted Poincaré and Poincaré–Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form \begin{equation*} \left (\frac {1}{w(Q)}\int _Q|f-f_Q|^{q}w\right )^\frac {1}{q}\le C_w\ell (Q)\left (\frac {1}{w(Q)}\int _Q |\nabla f|^p w\right )^\frac {1}{p}, \end{equation*} with different quantitative estimates for both the exponent $q$ and the constant $C_w$. We derive those estimates together with a large variety of related results as a consequence of a general self-improving property shared by functions satisfying the inequality \[ -\kern -10.5pt\int _Q |f-f_Q| d\mu \le a(Q) \] for all cubes $Q\subset \mathbb {R}^n$ and where $a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev type exponent $p^*_w>p$ associated with the weight $w$ and obtain further improvements involving $L^{p^*_w}$ norms on the left-hand side of the inequality above. For the endpoint case of $A_1$ weights, we reach the classical critical Sobolev exponent $p^*=\frac {pn}{n-p}$, which is the largest possible, and provide different types of quantitative estimates for $C_w$. We also show that this best possible estimate cannot hold with an exponent on the $A_1$ constant smaller than $1/p$.

As a consequence of our results (and the method of proof), we obtain further extensions to two-weight Poincaré inequalities and to the case of higher order derivatives. Some other related results in the spirit of the work of Keith and Zhong on the open-ended condition of Poincaré inequality are obtained using extrapolation methods. We also apply our method to obtain similar estimates in the scale of Lorentz spaces.

We also provide an argument based on extrapolation ideas showing that there is no $(p,p)$, $p\geq 1$, Poincaré inequality valid for the whole class of $RH_\infty$ weights by showing their intimate connection with the failure of Poincaré inequalities $(p,p)$ in the range $0<p<1$.