M.Sc Student M.Sc Thesis Ivtsan Alon Topics in Interpolation Spaces Department of Mathematics PROFESSOR EMERITUS Michael Cwikel

Abstract

In this work we carry out two projects in the field of interpolation spaces.

For the first project, let (B0,B1) be a Banach pair. Stafney showed that, in the definition of the norm in the Calderón complex interpolation method on the strip, one can replace the space F(B0,B1) with its subspace G(B0,B1) if the element belongs to the intersection of the spaces Bi. We shall extend this result to a more general setting, which contains several well-known interpolation methods, including the Calderón complex interpolation method on the annulus, an appropriate version of the Lions-Peetre real method, and the Peetre “plus-minus” method.

For the second project, let (A0,A1) and (B0,B1) be Banach couples such that A0 is contained in A1 and let T be a possibly nonlinear Lipschitz operator mapping Ainto B1, which also maps Ainto B0 boundedly and compactly. It is known that T maps (A0,A1)s,q boundedly into (B0,B1)s,q for each s satisfying 0<s<1  and each q satisfying 1≤q≤∞, and that this map is also compact if T is linear. We present examples which show that in general T is not compact as a map from (A0,A1)s,q into (B0,B1)s,q. However, this map will be compact if we also assume that (B0,B1) satisfies A. Persson's approximation condition (H) and if we assume that T satisfies an appropriate quantitative compactness condition.