Weak restricted and very restricted operators on $L^{2}$

Abstract:

A battlement is a real function with values in $\{ 0,1\}$ that looks like a castle battlement. A commuting with translation linear operator $T$ mapping step functions on ${\mathbf {R}}$ into the set of all measurable functions on ${\mathbf {R}}$ and satisfying $\parallel Tb{\parallel _2} \leqslant C\parallel b{\parallel _2}$ for all battlements $b$ is bounded on ${L^2}({\mathbf {R}})$. This remains true if the underlying space is the circle but is demonstrably false if the underlying space is the integers. Michael Cowling’s theorem that linear commuting with translation operators are bounded on ${L^2}$ if they are weak restricted $(2,2)$ is reproved and an application of this result to sums of exponentials is given.