Self-improvement of $(\theta,p)$ Poincar\'{e} inequality for $p>0$

A. I. Porabkovich

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 187-200

Résumé

Classical Poincaré $(\theta,p)$-inequality on $\mathbb{R}^n$ \begin{equation*} \left(\dfrac{1}{\mu(B)}\int\limits_B \left|f(y)-\dfrac{1}{\mu(B)}\int\limits_Bf\,d\mu\right|^\theta\,d\mu(y)\right)^{1/\theta} \lesssim r_B \left(\dfrac{1}{\mu(B)}\int\limits_{B}|\nabla f|^p\,d\mu\right)^{1/p}, \end{equation*} ($r_B$ is the radius of ball $B\subset \mathbb{R}^n$) has a self-improvement property, that is $(1,p)$-inequality, $1$, implies the «stronger» $(q,p)$-inequality (Sobolev-Poincaré), where $1/q=1/p-1/n$ (inequality $A\lesssim B$ means that $A\le cB$ with some inessential constant $c$). Such effect was investigated in a series of papers for the inequalities of more general type \begin{equation*} \left(\dfrac{1}{\mu(B)}\int\limits_B |f(y)-S_Bf|^\theta\,d\mu(y)\right)^{1/\theta} \lesssim\eta(r_B) \left(\dfrac{1}{\mu(B)}\int\limits_{\sigma B}g^p\,d\mu\right)^{1/p} \end{equation*} for functions on metric measure spaces. Here $f\in L^{\theta}_{\mathrm{loc}}$, $g\in L^{p}_{\mathrm{loc}}$, and $S_Bf$ is some number depending on the ball $B$ and on the function $f$, $\eta$ is some positive increasing function, $\sigma \ge 1$. Usually mean value of the function $f$ on a ball $B$ is chosen as $S_Bf$, and the case $p\ge 1$ is considered. We investigate self-improvement property for such inequalities on quasimetric measure spaces with doubling condition with parameter $\gamma>0$. Unlike previous papers on this topic we consider the case $\theta,p>0$. In this case functions are not required to be summable, and we take $S_Bf=I^{(\theta)}_Bf$. Here $I^{(\theta)}_Bf$ is the best approximation of the function $f$ in $L^{\theta}(B)$ by constants. We prove that if $\eta(t)t^{-\alpha}$ increases with some $\alpha>0$, then for $0$ and $\theta>0$ $(\theta,p)$-inequality Poincaré implies $(q,p)$-inequality with $1/q>1/p-\gamma/\alpha$. If $p\ge \gamma(\gamma+\alpha)^{-1}$ (then the function $f$ is locally integrable) then it implies also $(q,p)$-inequality with mean value instead of the best approximations $I^{(\theta)}_Bf$. Also we consider the cases $\alpha p=\gamma$ and $\alpha p>\gamma$. If $\alpha p=\gamma$, then $(q,p)$-inequality with any $q>0$ follows from Poincaré $(\theta,p)$-inequality and moreover some exponential Trudinger type inequality is true. If $\alpha p>\gamma$ then Poincaré $(\theta,p)$-inequality implies the inequality \begin{equation*} |f(x)-f(y)|\lesssim \eta(d(x,y))[d(x,y)]^{-\gamma/p}\lesssim[d(x,y)]^{\alpha-\gamma/p} \end{equation*} for almost all $x$ and $y$ from any fixed ball $B$ ($\lesssim$ does depend on $B$). Bibliography: 15 titles.

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Keywords: metric measure space, doubling condition, Poincaré inequality.