We prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

We characterize the weight functions $u,v,w$ on $\left( 0,\infty \right)$ such that

$${{\left( \int\limits_{0}^{\infty }{{{f}^{*}}{{\left( t \right)}^{q}}w\left( t \right)}\,dt \right)}^{1/q}}\le C\,\,\underset{t\in \left( 0,\infty \right)}{\mathop{\sup }}\,{{f}_{u}}^{**}\left( t \right)v\left( t \right),$$

where

$${{f}_{u}}^{**}\left( t \right):={{\left( \int\limits_{0}^{t}{u\left( s \right)}\,ds \right)}^{-1}}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)u\left( s \right)\,ds.$$

As an application we present a new simple characterization of the associate space to the space ${{\Gamma }^{\infty }}\left( v \right)$, determined by the norm

$${{\left\| f \right\|}_{\Gamma \infty \left( v \right)}}=\,\underset{t\in \left( 0,\infty \right)}{\mathop{\sup }}\,{{f}^{**}}\left( t \right)v\left( t \right),$$

where

$${{f}^{**}}\left( t \right):=\frac{1}{t}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)\,ds.$$

The purpose of this paper is to characterize the weight functions for which the Hardy operator , with non-decreasing function ƒ, is bounded between various weighted Lp-spaces for a wide range of indices. Our characterizations complement for the most part those of E. T. Sawyer [11] and V. D. Stepanov [15] for the Hardy operator of non-increasing function.

The nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.