Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $$0

Research paper by Martin Křepela

Indexed on: 22 Dec '17Published on: 11 Apr '17Published in: Revista Matemática Complutense


Let \(1< p <\infty \) and \(0<q<p\) . We prove necessary and sufficient conditions under which the weighted inequality $$\begin{aligned} \left( \int _0^\infty \left( \int _0^t f(x)U(x,t)\mathrm {\,d}x\right) ^q w(t) \mathrm {\,d}t\right) ^\frac{1}{q}\le C \left( \int _0^\infty f^p(t)v(t)\mathrm {\,d}t\right) ^\frac{1}{p}, \end{aligned}$$ where U is a so-called \(\vartheta \) -regular kernel, holds for all nonnegative measurable functions f on \((0,\infty )\) . The conditions have an explicit integral form. Analogous results for the case \(p=1\) and for the dual version of the inequality are also presented. The results are applied to close various gaps in the theory of weighted operator inequalities.