Résumé
The first weighted form of the continuous form of Hardy’s inequality reads:
Where f is a measurable and non-negative function on (0,∞), p > 1, α < p − 1. First I present Hardy’s original motivation from around 1915 and some historical facts until Hardy in 1925 finally proved the first version of (0.1) (with α = 0). After that I present the easiest proof of (0.1) I know, which was not discovered by Hardy himself. I continue by presenting some steps in the remarkable development of (0.1) to what today is called Hardy type inequalities, see e.g. the books [A]-[C] and the references therein. In particular, I present a choice of examples of interesting questions and results from each of the Chapters in the book [B] including the following:
- Characterizations of inequalities of type (0.1) with general weights and various parameters.
- The study of the same problem but with general kernel involved.
- Higher order Hardy inequalities.
- Fractional order Hardy inequalities.
- The study of the same problem on the cone of monotone functions.
- A remarkable relation to interpolation theory.
A number of open questions will be pointed out. In the final part of this lecture I will present some newer results, which can not be found in the cited books above and which in some cases contribute to give new light to the mentioned open questions. In particular, nowadays it is known that :
- all" powerweighted Hardy inequalities of the type (0.1) are more or less equivalent,
- the (Muckenhoupt-Bradley type) conditions found in 1. above can be replaced by infinite many alternative conditions (even four scales of conditions),
- sharp constants can now be found in many situations, which has not been known for long time,
- there exist new possibilities to develop the multidimensional theory , which was almost stopped after a result by Sawyer in 1985.
