The theory of extrapolation of operators has its roots in a theorem of S. Yano [J. Math. Soc. Jap. 3, 296-305 (1951; Zbl 0045.17901)] which states that if a linear operator maps \(L^p\) to \(L^p\) for all \(1< p< p+ \varepsilon\) i.e. if \(|Tf|_p\leq A_p|f|_p\) and if \(A_p\leq A(p- 1)^{- 1}\) then \(T\) maps \(L\log L\) to \(L\).
The theory complements work in interpolation theory, providing norm inequalities for interpolated operators at the endpoints of intervals of interpolation. Under suitable assumptions on the rate of increase of the norms of the operators several methods giving the limiting, extrapolated, inequalities have been developed. The manuscript continues the development of the theory [see also B. Jawerth and M. Milman, “Extrapolation theory with applications”, Mem. Am. Math. Soc. 440 (1991; Zbl 0733.46040)]. Several applications to concrete problems in analysis are also given.