Original scientific paper

A new class of general refined Hardy-type inequalities with kernels

Aleksandra Čižmešija ; Department of Mathematics, University of Zagreb, Bijeniˇcka cesta 30, 10000 Zagreb, Croatia
Kristina Krulić ; Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi´ca 28a, 10000 Zagreb, Croatia
Josip Pečarić ; Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi´ca 28a, 10000 Zagreb, Croatia

Abstract

Let μ1 and μ2 be positive sigma-finite measures on Omega1 and Omega2
respectively, k : Omega1 × Omega2 --> R be a non-negative function, and
K(x) = int_Omega2 k(x, y) dμ2(y), x in Omega1.
We state and prove a new class of refined general Hardy-type inequalities related to the weighted Lebesgue spaces Lp and Lq, where 0 < p <= q < infinity or −infinity < q <= p < 0, convex functions and the integral operators Ak of the form
Ak f(x) =1/K(x) int_Omega2 k(x, y)f(y) dμ2(y).
We also provide a class of new sufficient conditions for a weighted modular
inequality involving operator Ak to hold. As special cases of our
results, we obtain refinements of the classical one-dimensional Hardy’s,
Polya–Knopp’s and Hardy–Hilbert’s inequality and of related dual inequalities,
as well as a generalization and refinement of the classical
Godunova’s inequality. Finally, we show that our results may be seen
as generalizations of some recent results related to Riemann-Liouville’s
and Weyl’s operator.

Keywords

Hardy’s inequality; Hardy-Hilbert’s inequality; weights; power weights; convex functions; Hardy’s operator; kernel

Hrčak ID:

104406

URI

https://hrcak.srce.hr/104406

Publication date:

27.6.2013.

Visits: 1.482 *