# Are they known properties for the self-convolution application $H : f \mapsto f \star f$?

I was wondering if the application $$H : L^1(\mathbb{R}) \rightarrow L^1(\mathbb{R})$$ defined by :

$$H(f) := f \star f, \quad \forall f \in L^1(\mathbb{R})$$ had some known properties ? For example we know that , using Young inequality, that $$||H(f)||_{L^1(\mathbb{R})} \leq ||f||^2_{L^1(\mathbb{R})}$$, but that's very common.

Can we caracterize $$H(L^1(\mathbb{R}))$$ or $$H^{-1}(\{0 \})$$ ? Is the application $$H$$ surjective ? Are they counter-example to the surjectivity ? I'm looking for any properties or references since I'm just curious about it. I guess using Fourier transform will probably be a very useful tool.

Apart from these first questions, I'm also wondering if there exists properties for $$G : L^p(\mathbb{R}) \rightarrow L^r(\mathbb{R})$$ defined by :

$$G(f) = g \star f\quad \forall f \in L^p(\mathbb{R})$$ for a fixed $$g$$ in $$L^p(\mathbb{R})$$ with $$1 \leq p,q,r \leq + \infty$$ verifying the young equality :

$$\frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r}.$$