Three inequalities, one constant

The purpose of the talk will be to discuss the following three estimates:

1) Bliss' inequality for real functions on positive halfline;

2) Sobolev's inequality (the imbedding theorem);

3) Sawyer's inequality which characterizes the boundedness of fractional maximal operator on weighted Lp spaces.

It turns out that the three constants appearing in the above estimates are the same. This fact is well known for the inequalities 1) and 2) (G. Talenti, 1976). The main emphasis of the talk will be put on the analysis of 3), performed with the use of martingale methods.