Convolution Dynamics

Define $T$ on $L^1$ into $L^1$ by $(Tf)(x)= \int_{-\infty}^{\infty} g(x-y)f(y)dy$. With assumptions on $g(x)$ and its Fourier transform,$\hat{g}(t) $, requiring, among other things, that there be only one point,$t_{0}$, at which $|\hat{g}(t_{0})|= sup_{s\epsilon R}|\hat{g}(s)|$ and that $0