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Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 187–200 (Mi cheb463)  

Self-improvement of $(\theta,p)$ Poincaré inequality for $p>0$

A. I. Porabkovich

Belarusian State University, Minsk

Abstract: Classical Poincaré $(\theta,p)$-inequality on $\mathbb{R}^n$
\begin{equation*} (\dfrac{1}{\mu(B)}\int\limits_B |f(y)-\dfrac{1}{\mu(B)}\int\limits_Bf d\mu|^\theta d\mu(y))^{1/\theta} \lesssim r_B (\dfrac{1}{\mu(B)}\int\limits_{B}|\nabla f|^p d\mu)^{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, $1<p<n$, 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*} (\dfrac{1}{\mu(B)}\int\limits_B |f(y)-S_Bf|^\theta d\mu(y))^{1/\theta} \lesssim\eta(r_B) (\dfrac{1}{\mu(B)}\int\limits_{\sigma B}g^p d\mu)^{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 $0<p<\gamma/\alpha$ 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.

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

Full text: PDF file (793 kB)
References: PDF file   HTML file
UDC: 517.5
Received: 29.12.2015
Accepted:11.03.2016

Citation: A. I. Porabkovich, “Self-improvement of $(\theta,p)$ Poincaré inequality for $p>0$”, Chebyshevskii Sb., 17:1 (2016), 187–200

Citation in format AMSBIB
\Bibitem{Por16}
\by A.~I.~Porabkovich
\paper Self-improvement of $(\theta,p)$ Poincar\'{e} inequality for $p>0$
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 1
\pages 187--200
\mathnet{http://mi.mathnet.ru/cheb463}
\elib{https://elibrary.ru/item.asp?id=25795082}


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