Suppose 1≤p≤∞ and $f∈L^1(G)$ and $g∈L^p(G)$.
I know this is true $‖f*g‖_p ≤‖f‖_1 ‖g‖_q$ , Where ∗ is convolution of f and g.
I want to prove that If G is not unimodular, we still have g∗f is in LP when f has compact support.
proof:
By definition convolution we have $$‖g*f‖_p=‖∫g(xy^{-1} )f(y)∆(y^{-1} )dy‖ ≤∫‖g(xy^{-1})‖_p |f(y)|∆(y^{-1} )dy $$
since the LP norm is left-invariant $$ =‖g‖_p ∫f(y)∆(y^{-1} )dy$$ I know that integral out of support f is zero. But I do not know how can prove that value of the integral is finite
