M.Sc Thesis | |

M.Sc Student | Avni Eliran |
---|---|

Subject | Topics in Interpolation Spaces |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Michael Cwikel |

Full Thesis text |

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 (*X*_{0}^{(p)},*X*_{1}^{(p)})
is a Calderón couple under the assumption that (*X*_{0},*X*_{1})
is a Calderón couple of Banach lattices on some measure space. We find
that the answer is affirmative whenever the spaces *X*_{0}, *X*_{1}
are complete lattices and an additional “positivity” assumption is imposed
regarding (*X*_{0},*X*_{1}). 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 *i**l* 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.