Extrapolation theory at the scale of $L_{p}$ spaces and applications

Extrapolation theory allows by the family of Banach spaces with common substance (Banach scale) build a new (extrapolation) space using extrapolation functor. If there is an operator acting between two Banach scales, it will also act between extrapolation spaces. Foundations of General (abstract) extrapolation theory were laid by Mario Milman and Bjorn Jawerth in the 90-ies of the last century in a series of papers. In these papers have been proved basic theorems about properties of extrapolation functors and extrapolation spaces and also presented a number of ideas for further development of the theory and possible applications. In the case of special families of Banach spaces the theory allows to prove more accurate and interesting results. The report will be discussed on the extrapolation properties of the scale of $L_p$-spaces, and applications of these properties to some problems of classical analysis.