Homogeneous Banach space on the circle ($\Vert {\tau }_{t}f{\Vert }_B=\Vert f{\Vert }_B,\Vert {\tau }_{t}f-f{\Vert }_B⟶0$)

A Banach space $B\subseteq L^1(\mathbb{T})$ is homogeneous when its inclusion into $L^1$ is continuous, circle translations act isometrically, and each orbit is continuous in the $B$ norm. One commonly normalizes $\Vert f{\Vert }_1\le \Vert f{\Vert }_B$. The continuous functions with the supremum norm, and $L^p(\mathbb{T})$ for $1\le p<\infty$, are examples. Full $L^{\infty }$ fails translation continuity. If $g$ is continuous, the Bochner integral $f\ast g=\int g(t){\tau }_{t}fdt/(2\pi )$ belongs to $B$ and satisfies $\Vert f\ast g{\Vert }_B\le \Vert f{\Vert }_B\Vert g{\Vert }_1$. Approximation by continuous functions extends this to every $g\in L^1$.

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.