Resumen de The vector valued quartile operator

Tuomas P. Hytönen, Michael T. Lacey, Ioannis Parissis

• Certain vector-valued inequalities are shown to hold for a Walsh analog of the bilinear Hilbert transform. These extensions are phrased in terms of a recent notion of quartile type of a unconditional martingale differences (UMD) Banach space ? . Every known UMD Banach space has finite quartile type, and it was recently shown that the Walsh analog of Carleson’s Theorem holds under a closely related assumption of finite tile type. For a Walsh model of the bilinear Hilbert transform however, the quartile type should be sufficiently close to that of a Hilbert space for our results to hold. A full set of inequalities is quantified in terms of quartile type.