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RESEARCH PRODUCT
On singular integral and martingale transforms
Stefan GeissStephen Montgomery-smithEero Saksmansubject
46B09General Mathematics46B20 (Secondary)Banach space42B15 (Primary) 42B2001 natural sciencesUpper and lower bounds010104 statistics & probabilitysymbols.namesakeCorollary60G46; 42B15 (Primary) 42B20; 46B09; 46B20 (Secondary)Classical Analysis and ODEs (math.CA)FOS: Mathematics60G460101 mathematicsMathematicsNormed vector spaceDiscrete mathematicsApplied MathematicsProbability (math.PR)010102 general mathematicsSingular integralSingular valueMathematics - Classical Analysis and ODEssymbolsHilbert transformMartingale (probability theory)Mathematics - Probabilitydescription
Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD(p)-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on the X-valued L^p-space on the plane. Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2007-01-18 | Transactions of the American Mathematical Society |