A Fourier Analysis Question related to fourier transform of $C_c(\mathbb{R}^n)$ functions $f\in L^p(\mathbb{R}^n),1\leq p\leq 2$ and $g\in C_c(\mathbb{R}^n), g$ is not identically $0$, such that $f*g\equiv 0.$ Prove that $f(x)=0$ for almost every $x\in\mathbb{R}^n$.
My strategy is to use that we have $\hat{f}\cdot \hat{g}\equiv 0.$ Now for $n=1$, I know that $\hat{g}$ is a restriction to an entire function and hence if $g$ isn't identically $0$, then the zero set of $\hat{g}$ is discrete. [We simply define $\hat{g}(z)=\int_{\mathbb{R}}g(t)e^{-2\pi i tz} dt$.] Hence $\hat{f}$ is $0$ almost everywhere and so $f$ is $0$ almost everywhere. But for $n>1$ how do I prove the similar statement? 

The strategy would be the same as $\widehat g$ would be holomorphic on $\mathbb{C}^n$ and its zero set would be of strictly lower dimension and hence of zero measure. Have in mind that the awkward part here is talking about the Fourier transform of $f \in L^p(\mathbb{R})$ as you would have to prove the result in a dense set or use distributions.
Alternatively you could try the following. If you replace $f$ with a Schwartz function using density (here I think that you need the hypothesis on $p$ to use one of young convolution inequalities), you can study $\hat{f}\cdot\hat{g}$ by fixing variables and working with just entire functions on one variable.
