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 :

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



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