Chebyshevskii Sbornik
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Chebyshevskii Sb.: Year: Volume: Issue: Page: Find

 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
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}