Exact norm of convolution from L1 into a homogeneous Banach space
= Exact norm of convolution from L1 into a homogeneous Banach space
{title2=$\|g\mapsto f*g\|_{L^1\to B}=\|f\|_B$}
The <convolution> estimate gives the upper bound. Apply the operator to the <Fejér kernel>, whose $L^1$ norm is one. The <summability kernel on the circle> property implies $f*K_n\to f$ in $B$, giving the reverse bound. This identifies the operator norm even when no single input attains it.