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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 17:05–17:30, 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

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
and

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

$$\int^{1}_{0}\exp(-C^{p}_{1}\int_{B_{1}\setminus B_{|x|}} u^{\frac{1}{1-p}}(y) dy) v(r)r^{-np}dr<\infty, \label{N345:x6}$$
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}.