Nonexistence of Henkin type projections via a Wiener theorem for multipliers

Abstract

Let $d\ge 2$, $l\ge 0$ 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.

32 pages