Tuomas Hytnen, Lutz Weis
A T1 theorem for integral transformations with operator-valued kernel
We consider generalized Calderón-Zygmund operators whose kernel K(x, y) takes values in ℒ(X) (continuous linear operators on the Banach space X) and satisfies variants of the classical standard estimates involving R-boundedness, which has recently become a crucial notion in connection with operator-valued singular integrals. Boundedness criteria in the spirit of the T1 theorem of David and Journé are proved for such operators on the Bôchner spaces Lp(ℝn, X), where 1 < p < ∞ and X is a UMD-space. For some results, X is also required to have Pisier's property (α).
In the special case T1 = T′1 = 0, we obtain an essentially complete analogue of the scalar-valued theorem. We also provide sufficient conditions for the general case, but they are stronger in general than the necessary “T1, T′1 ∈ BMO”-type conditions, although they reduce to them in the classical situation. A counterexample is given to show that the natural necessary conditions are not sufficient in general.
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2006, 10/2006
Pages: 155 - 200
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