convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?

Let $U$ and $E$ be the homogeneous and fundamental solutions for the differential operator $L$

Meaning that $LE=δ_0$ and $LU=0$

Now assume that $U$ and $E$ are convoluble so that $U*E$ makes sense.Appling $L$ to $U*E$ we get$L(U*E)=LU*E=U*LE$ by the basic property of convolution

$LU*E=0*E=0$ and $U*LE=U*δ_0=U$ because the Dirac delta is the neutral element for convolution

So $U=0$