Nonexistence of Henkin type projections via a Wiener theorem for multipliers

Abstract

Let $d\geq 2$, $l\geq 0$ and suppose $X$ is one of the function spaces $W^{l,1}(\mathbb{T}^{d})$, $W^{l,\infty }(\mathbb{T}^{d})$ or $C^{l}(\mathbb{T}^{d})$. We extend a result of Henkin (1967), showing that, for appropriate $N\times N$ matrix operators $A(D)$, the subspace of $X^{N}$ consisting of $A(D)-$free elements is noncomplemented. In order to prove this we establish a new property of the Fourier multipliers that are bounded on $X$: the kernel $k$ of any such multiplier obeys a weaker version of Wiener's theorem for the singularities of measures.