RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
Video Library
Archive
Most viewed videos

Search
RSS
New in collection





You may need the following programs to see the files






International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 28, 2015 17:05–17:30, Функциональные пространства, Moscow, Steklov Mathematical Institute of RAS
 


A weighted Hardy-type inequality for $0<p<1$ with sharp constant

A. Senouci

Ibnou Khaldoun University, Algeria
Materials:
Adobe PDF 105.4 Kb

Number of views:
This page:70
Materials:31

Abstract: Let $\Omega $ be a Lebesgue measurable set in $\mathbb{R}^{n}$, $u $ be a non-negative Lebesgue measurable function on $\Omega$ (weight function), and $ 0 < p < \infty $. We denote by $ {L_{p,u}(\Omega)} $ the space of all Lebesgue measurable functions $ f $ on $\Omega$ for which
$$ \|f\|_{L_{p,u}(B_r)} = ( \int_{\Omega} \vert f(x) \vert^p u(x)  dx )^{\frac{1}{p}}<\infty, $$
and by $H$ the $n$-dimensional Hardy operator.
Theorem. Let $C_{1}>0$, $0<p<1$ and $u$, $v$ be weight functions on $\mathbb{R}^{n}$, $(0,\infty)$ respectively. Suppose that
\begin{equation} \int_{B_r}u^{\frac{1}{1-p}}(x) dx=\infty \qquad for some \quad r>0 \label{N345:x1} \end{equation}
and
\begin{equation} V(r):=\int^{\infty}_{r}v(\rho)\rho^{-np} d\rho<\infty \qquad for all \quad r>0. \label{N345:x2} \end{equation}

Consider the set of all Lebesgue measurable functions $f$ on $\mathbb{R}^{n}$ satisfying the inequality
\begin{equation} |f(x)|\leq C_{1}u^{\frac{1}{1-p}}(x)\|f\|_{L_{_{p,u}}(B_{(|x|).})} \label{N345:x3} \end{equation}
for almost all $x\in\mathbb{R}^{n}$. Then for all functions $f$ in this set
\begin{equation}\|Hf\|_{L_{_{p,v}}(0,\infty)} \leq C_{2}\|f\|_{L_{p,w}(\mathbb{R}^{n})} \label{N345:x4} \end{equation}
where
$$ w(x)=u(x) V(|x|),\qquad x\in\mathbb{R}^{n}, $$
and
$$ C_{2}=v_{n}^{-1}pC_{1}^{1-p}. $$

If, in addition,
\begin{equation} \int_{B_{r_{_{2}}}\setminus B_{r_{_{1}}}}u^{\frac{1}{1-p}}(x) dx<\infty\qquad for all \quad 0<r_{_{1}}<r_{_{2}}<\infty, \label{N345:x5} \end{equation}
and
\begin{equation} \int^{1}_{0}\exp(-C^{p}_{1}\int_{B_{1}\setminus B_{|x|}} u^{\frac{1}{1-p}}(y) dy) v(r)r^{-np}dr<\infty, \label{N345:x6} \end{equation}
then the constant $C_{2}$ is sharp and there exists a functions $f \in L_{p,w}(\mathbb{R}^{n})$ not equivalent to $0$, satisfying inequality \eqref{N345:x3} and such that there is equality in inequality \eqref{N345:x4}.
Joint work with Professor V. I. Burenkov and N. Azzouz.

Materials: abstract.pdf (105.4 Kb)

Language: English

SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru
 
Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2018