Young Inequality:
Let $f \in L^p(\mathbb{R}^n)$, $g \in L^q(\mathbb{R}^n)$. Then, $h := f *g \in L^r(\mathbb{R}^n)$. Furthermore, \begin{equation} \| h \|_{L^r(\mathbb{R}^n)} \leq \| f \|_{L^p(\mathbb{R}^n)} \, \| g \|_{L^q(\mathbb{R}^n)}, \end{equation} where $\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1$.
Motivation:
I am interested to know if the function $f$ above can be extended to distribution $f$ (a linear functional)? Assume $g \in L^q(\mathbb{R}^n)$ is a function, but $f \in L^p_{\operatorname{loc}}(\mathbb{R}^n) \subset \mathcal{D}'(\mathbb{R}^n)$ defines a regular distribution $h_f: \phi \mapsto \int f \phi$.
One may view the convolution $h=f*g$ as a regular distribution $h_f(g)$ acting on a test function $g$. My guess is that in this case, I am not too far from using $f$ as distribution.
For instance, we know that if $f,g \in L^1_{\operatorname{loc}}(\mathbb{R}^n)$ and $g$ has compact support, then $f*g \in L^1_{\operatorname{loc}}(\mathbb{R}^n)$ (see section 1.4.4 here). This is indeed the same as the Young inequality with $p=q=r=1$, but $f$,$g$ are relaxed from being intgegrable to be locally integrable, whereas further condition is imposed on $g$ to have compact support.
Question 1:
What can we say if we generalize above argument to all $p,q,r$ satisfying $\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1$. That is, $f \in L^p_{\operatorname{loc}}(\mathbb{R}^n)$, $g \in L^q_{\operatorname{loc}}(\mathbb{R}^n)$, $g$ has compact support, then would $f*g \in L^r_{\operatorname{loc}}(\mathbb{R}^n)$? If not, what conditions should be made to make the inequality happen while $f$ is a distribution?
Issue: A possible issue that comes to mind is that in inequality above when $p \neq 1$, since I have to define the norm of distribution on $L^p(\mathbb{R}^n)$ space by
\begin{equation} \| f \|^p_{L^p(\mathbb{R}^n)} = \int_{\mathbb{R}^n} |f|^p \mathrm{d}\mathbf{x} \end{equation}
This requires the power $f^p$ to be defined. However, the product of (or here, the power of) distributions are not as well-behaved as functions. Is such operation on distributions allowed?
Question 2:
At least I guess, above generalization for $p=1$ is fine. Instead of regular distributions made by $f$, what can we say for singular distributions, either for only $p=1$ or for all $p$? The singular distribution is when $h_f(g)$ is not represented by any form $\int fg$ with some $f \in L^1_{\operatorname{loc}}(\mathbb{R}^n)$.
For example, an application in my mind is to set $f(x_1,\cdots,x_n) = y(x_1) \delta(x_2) \cdots \delta(x_n)$ in the Young inequality, where $\delta$ is Dirac distribution and $y \in L^1(\mathbb{R})$ is some function.