Suppose G is unimodular. If f is in LP(G) and g is in Lq(G) where 1 < p,q < ∞ and 1/p+1/q=1, then f * g ∈Co(G) and ||f * g||sup ≤ Ilfllp llgllq. * is convolution f and g. I read the proof of this theorem.This is proof of the folland's book
The fact that |f * g(x) | ≤ ||f||p ||g||q for all x ∈ G follows from Holder's inequality and the invariance of Haar integrals under translations and inversions. If f,g ∈ Cc(G), it is easy to check that f * g ∈ Cc(G). But Cc(G) is dense in LP(G), and if f_n→f in LP and g_n→g ∈Lq then f_n*g_n→f*g and g_n*f_n→g*f uniformly
I have a few questions
To prove that f * g ∈Co(G) Must show: 1)f*g is continious and 2)has compact support.is this correct?
How we use it :Cc(G) is dense in LP(G).
In what follows, $X$ is a locally compact, Hausdorff topological space.
Review the definition of
$C_{0}(X)=\{f:X\rightarrow\mathbb C$: $f$ *is continuous and vanish at infinity*$\}$
i.e, for every $\epsilon\gt 0$, there exists some compact set $K\subset X$ such that $$|f(x)|\lt \epsilon \quad\quad\quad \forall x\notin K$$ The sketch of proof. As you mentioned, since $C_c (X)$ is dense in $L^p$, There are $f_n, g_n\subset C_c (X)$$$f_n\to f$$ and $$g_n\to g$$ Now to show that $f\star g\in C_{0}(X)$, it suffices to show that $$||f_n\star g_n -f\star g||_\infty\to 0$$ why? Because of the following fact which is a result of Urysohn's lemma.
Fact. With the uniform norm, $C_c (X)$ is dense in $C_{0}(X)$.
Apply these notes to $G$.