If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$
Question: Let $\mu$ be a probability measure $\mu\in \mathcal{P}(\mathbb{R}^n)$ and $g\in C_c^\infty(\mathbb{R}^n)$. Then the convolution $g*\mu$ is defined as $$(g*\mu)(x)=\int_{\mathbb{R}^n}g(x-y)\mu(dy).$$ Is there any hope of getting an estimate of Young's type in this case? What I would like to have is an estimate of the type $$\|g*\mu\|_{L^p}\leq A_\mu \|g\|_{L^p}, \text{ where $A$ is a quantity dependent on $\mu$ which stays bounded.} $$
Remarks:
(1) Note that if $\mu$ is an absolutely continuous measure w.r.t. the lebesgue measure i.e. $\mu<<\mathcal{L}_{{R}^n}$ then the Young's inequality holds.
(2) If $\mu$ is a dirac measure (in the sense of Lebesgue decomposition theorem) say $\mu=\delta_0$, a similar estimate follows as well $$ \|g*\delta_0\|_{L^p}^p=\int_{\mathbb{R}^n}dx\left|\int_{\mathbb{R}^n}g(x-y)d\delta_0(dy)\right|=\|g\|_{L^p}.$$
By Lebesgue's decomposition theorem $\mu$ can be written as a sum of absolutely continuous part, discrete (delta) part and a cantor part. I do not know how to handle the Cantor part.
Possible Way (not too sure about this one): Since the estimate works for absolutely continuous measures (or $L^1$ densities) can we somehow arrive at a bound by some approximations?
Any help is much appreciated!
I would do it using approximations. First suppose that $g \in C_c(\mathbb R^n)$. Construct a sequence of measures $\mu_k$ that are absolutely continuous w.r.t. Lebesgue measure (or alternatively are finite linear combinations of dirac delta functions) such that $\mu_k$ converges weakly to $\mu$. Then for each $x \in \mathbb R^n$, we have $\mu_k*g(x) \to \mu*g(x)$. Now use the dominated convergence theorem to show that $\|\mu_k*g - \mu*g\|_p \to 0$, and go from there.