Support theorems on RnRn and non-compact symmetric spaces

We consider convolution equations of the type f∗T=gf∗T=g, where f,g∈Lp(Rn)f,g∈Lp(Rn) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T, we show that f is compactly supported, provided g is. Similar results are proved for non-compact symmetric spaces as well