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Keywords:
Hardy operators involving suprema; weighted inequalities
Hardy operators involving suprema; weighted inequalities
Summary:
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 $0<p, q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.
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