The subject has as its beginning the observation by A. Zygmund that important classical operators, notably the maximal function and the Hilbert transform, which map \(L^ p\) to itself for \(p>1\), map L log\({}^+ L\) to \(L^ 1\). It was discovered later that this is not an independent fact, and that operators which map \(L^ p\) to itself for \(p>1\), with a control on the increase of the norm as \(p\to 1\), can be extrapolated to an Orlicz space. The control on the norm of the operators which map \(L^ 1\) to weak \(L^ 1\) and, say, \(L^ 2\) to \(L^ 2\), given by the Marcinkiewicz interpolation theorem, forces operators such as the Hilbert transform, to map L log\({}^+ L\) to \(L^ 1\). This is Yano’s theorem.
Another point of view is to observe that the condition on the weights in the general form of Hardy’s inequality, (Talenti-Tomaselli-Muckenhoupt) has a certain leeway which enables one to prove the L log\({}^+ L\) result, and in fact is sufficient to handle trivially the related spaces considered by Bennett and Rudnick.
In this paper the authors generalize the extrapolation techniques to general interpolation scales. They also give a number of very interesting applications.