A weighted inequality for the Hardy operator involving suprema

Hofmanová, Pavla

Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 3, pp. 317-326.

Résumé

Let $u$ be a weight on $(0, \infty)$. Assume that $u$ is continuous on $(0, \infty)$. Let the operator $S_{u}$ be given at measurable non-negative function $\varphi$ on $(0, \infty)$ by $$ S_{u}\varphi (t)= \sup_{0 \tau\leq t}u(\tau)\varphi (\tau). $$ We characterize weights $v,w$ on $(0, \infty)$ for which there exists a positive constant $C$ such that the inequality $$ \left( \int_{0}^{\infty}[S_{u}\varphi (t)]^{q}w(t)\,dt\right)^{\frac 1q} \lesssim \left( \int_{0}^{\infty}[\varphi (t)]^{p}v(t)\,dt\right)^{\frac 1p} $$ holds for every $0$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

Détail

DOI : 10.14712/1213-7243.2015.167

Classification : 26D15, 47G10
Keywords: Hardy operators involving suprema; weighted inequalities