This is an announcement for the paper "$R$-boundedness versus $\gamma$-boundedness" by Stanislaw Kwapien, Mark Veraar, and Lutz Weis.

Abstract: It is well-known that in Banach spaces with finite cotype, the $R$-bounded and $\gamma$-bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$-boundedness implies $\gamma$-boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$-boundedness is stable under taking adjoints if and only if the underlying space is $K$-convex.

Archive classification: math.FA math.PR

Mathematics Subject Classification: 47B99 (Primary) 46B09, 46B07, 47B10 (Secondary)

Submitted from: m.c.veraar@tudelft.nl

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1404.7328

or

http://arXiv.org/abs/1404.7328