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

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

A. Senouci

Ibnou Khaldoun University, Algeria

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
$$ \|f\|_{L_{p,u}(B_r)} = \biggl( \int_{\Omega} \vert f(x) \vert^p u(x)\, dx \biggr)^{\frac{1}{p}}<\infty, $$
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 \text{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 \text{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 \text{for all} \quad 0<r_{_{1}}<r_{_{2}}<\infty, \label{N345:x5} \end{equation}
and
\begin{equation} \int^{1}_{0}\exp\biggl(-C^{p}_{1}\int_{B_{1}\setminus B_{|x|}} u^{\frac{1}{1-p}}(y)\,dy\biggr) 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.

Language: English

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