A T1 theorem for integral transformations with operator-valued kernel

作者：Hytonen Tuomas^*; Weis Lutz

来源：Journal fur die Reine und Angewandte Mathematik, 2006, 599: 155-200.

DOI：10.1515/CRELLE.2006.081

We consider generalized Calderon-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 Journe are proved for such operators on the Bochner spaces L-P (R-n, X), where 1 < p < infinity and X is a UMD-space. For some results, X is also required to have Pisier's property (alpha).
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 is an element of 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.

• 出版日期2006