On vector-valued Fourier multiplier theorems. (English) Zbl 0686.42008

Zimmermann, Frank

Stud. Math. 93, No. 3, 201-222 (1989).

Summary: The classical Fourier multiplier theorems of Littlewood-Paley, Marcinkiewicz, and Mikhlin are generalized to the vector-valued setting in d dimensions. A direct and a tensor product approach yield slightly different results. While the direct approach works in general UMD-spaces, the tensor product technique requires some unconditional structure and it is shown that the latter results fail for the Schatten classes \(S_ p\) with \(p\neq 2\).

MSC:

42A45

Multipliers in one variable harmonic analysis

Keywords:

Fourier multiplier theorems; Littlewood-Paley; Marcinkiewicz; Mikhlin; Schatten classes