Exact norm of convolution from L1 into a homogeneous Banach space ($\Vert g\mapsto f\ast g{\Vert }_{L^1\to B}=\Vert f{\Vert }_B$)

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 $f\ast K_n\to f$ in $B$, giving the reverse bound. This identifies the operator norm even when no single input attains it.