arXiv subjects

Eliran Avni

Publications and source records attributed to Eliran Avni.

2 recordsLinked to original sources

arXiv2012

The periodic complex method in interpolation spaces

We discuss a question which relates to Calderon's complex interpolation method. More precisely, we will consider the so-called "periodic" complex interpolation method, studied by Peetre. (Which also corresponds to the spaces obtained by Calderon's construction using Banach space valued analytic functions, but defined on an annulus instead of the strip used by Calderon.) Cwikel showed that, using functions with a given period iλin the complex method mechanism, one obtains the same interpolation spaces as in the original version of the complex method, up to equivalence of norms. He also showed that one of the constants of this equivalence will, in some cases, "blow up" as λtends to zero. We will show that the equivalence constants tend to 1 as λtends to infinity. Intuitively, this means that when applying the complex method, it makes a very small difference if one restricts oneself to periodic functions, provided that the period is very large (or the corresponding annulus is very thin).

math.FA↗

arXiv2012

Calderon couples of p-convexified Banach lattices

This paper updates the previous version in the following ways: 1. The main result is extended from the case of sequence spaces to the case of Dedekind complete Banach lattices. 2. A new appendix is added to mention some sufficient and necessary conditions (which are probably already known) for lattices to be Dedekind complete. We deal with the question of whether or not the p-convexified couple (X_0^{(p)},X_1^{(p)}) is a Calderon couple under the assumption that (X_0,X_1) is a Calderon couple of Banach lattices on some measure space. We find that the answer is affirmative, not only in the case of sequence spaces treated in the previous version, but also in the case where X_0 and X_1 are Dedekind complete Banach lattices (and provided the same additional "positivity" assumption is imposed regarding (X_0,X_1)). We also prove a quantitative version of the result with appropriate norm estimates.

math.FA↗