Regularity with respect to the Lebesgue measure through dimensions

Let us consider two probability measures $mu in mathcal{P}(mathbb{R}^{p})$ and $nu in mathcal{P}(mathbb{R}^{q})$ with $p,q in mathbb{N}^{*}$. We note $#$ the push forward operator i.e for $f: mathbb{R}^{p} rightarrow mathbb{R}^{q}$ measurable $f#mu in mathcal{P}(mathbb{R}^{q})$ defined by
$f#mu(A)=mu(f^{-1}(A))=mu({x in mathbb{R}^{p} | f(x) in A})$ for $A$ measurable set of $mathbb{R}^{q}$.

Let $l: mathbb{R}^{p} rightarrow mathbb{R}^{q}$ be a linear application and assume that $mu$ has a density $g$ with respect to the Lebesgue measure on $mathbb{R}^{p}$.

My question is:

If $q <p$ (we go from higher to lower dimension) does $l#mu$ has a density with respect to the Lebesgue measure on $mathbb{R}^{q}$ ?

I think this is true using the coarea formula. More precisely if $Lin mathbb{R}^{qtimes p}$ the matrix associated to $l$ and $J_L=sqrt{det(LL^{T})}$ the density of $l#mu$ looks like:
begin{equation}
h(y)=int_{l^{-1}(y)} frac{g(x)}{J_L} dV_{l^{-1}(y)}(x)
end{equation}
where denotes $dV_{l^{-1}(y)}(x)$ the volume element. Is it correct ?

What could go wrong if $q>p$ (we go from lower to higher dimension) ?