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

Research paper by **Martin Křepela**

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

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#### Abstract

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.