Fouriertype

Fouriertype, often written Fourier type, is a notion in Banach space theory and harmonic analysis that describes how the Fourier transform behaves on vector-valued L^p spaces. It measures the extent to which the vector-valued Fourier transform is bounded from L^p to L^{p'} for a Banach space X.

Definition: Let X be a Banach space and let p lie in the interval [1, 2]. X is said to have Fourier type p if there exists a constant C such that for every dimension d ≥ 1 and every function f in L^p(R^d; X) the (Bochner) Fourier transform hat f lies in L^{p'}(R^d; X) and the inequality

||hat f||_{L^{p'}(R^d; X)} ≤ C ||f||_{L^p(R^d; X)}

holds. Here p' is the conjugate exponent given by 1/p + 1/p' = 1. The best such exponent p is called the Fourier type of X. In the scalar case X = C, the Fourier transform maps L^p(R^d) to L^{p'}(R^d) for all p in [1, 2], so C has Fourier type p for every p in [1, 2].

Examples and remarks: Finite-dimensional Banach spaces and Hilbert spaces have Fourier type 2. If X has Fourier type p, then its dual X* has Fourier type p' (the conjugate exponent), and the property is stable under isomorphisms of Banach spaces. Fourier type is connected to the geometry of X and to vector-valued multiplier theorems, playing a role in vector-valued harmonic analysis and certain PDE analyses.

See also: Fourier transform, vector-valued L^p spaces, type and cotype of Banach spaces, vector-valued multiplier theorems.