M.Sc Thesis
M.Sc Student Eliran Avni Topics in Interpolation Spaces Department of Mathematics Professor Emeritus Cwikel Michael

Abstract

In this thesis we address two problems in the field of interpolation spaces.

In the first part we deal with the question of whether the p-convexified couple (X0(p),X1(p)) is a Calderón couple under the assumption that (X0,X1) is a Calderón couple of Banach lattices on some measure space. We find that the answer is affirmative whenever the spaces X0, X1 are complete lattices and an additional “positivity” assumption is imposed regarding (X0,X1). We also prove a quantitative version of the result with appropriate norm estimates. In the last section of this part we identify some cases where appropriate assumptions on a Banach lattice X guarantee that it is indeed a complete lattice.

In the second part we consider the “periodic” variant of the complex interpolation method, apparently first studied by Peetre. Cwikel showed that using functions with a given period il in the complex method construction introduced and studied by Calderón, one may construct the same interpolation spaces as in the “regular” complex method, up to equivalence of norms. Cwikel also showed that one of the constants of this equivalence will, in some cases, “blow up” as l tends to 0. (The other constant is obviously bounded by 1.) We show that this same equivalence constant approaches 1 as l tends to infinity. Intuitively, this means that when applying the complex method of Calderón, it makes a very small difference if one restricts oneself to periodic functions, provided that the period is very large.