Calderon couples of p-convexified Banach lattices

TL;DR

This paper extends the understanding of Calderon couples to Dedekind complete Banach lattices, demonstrating that p-convexified couples preserve Calderon properties under certain conditions, with added quantitative estimates.

Contribution

It generalizes previous results from sequence spaces to Dedekind complete Banach lattices, establishing the Calderon couple property for p-convexified pairs with norm estimates.

Findings

p-convexified couples are Calderon couples in Dedekind complete Banach lattices

The result holds under a positivity assumption on the original couple

Quantitative norm estimates are provided for the p-convexified couples

Abstract

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.