The UMD constants of the summation operators

Abstract

The UMD property of a Banach space is one of the most useful properties when one thinks about possible applications. This is in particular due to the boundedness of the vector-valued Hilbert transform for functions with values in such a space. Looking at operators instead of at spaces, it is easy to check that the summation operator does not have the UMD property. The actual asymptotic behavior however of the UMD constants computed with martingales of length n is unknown. We explain, why it would be important to know this behavior, rephrase the problem of finding these UMD constants and give some evidence of how they behave asymptotically.