Continuity of translation operators

Athreya, Krishna; Peters, Justin

doi:10.1090/S0002-9939-2011-10862-1

Continuity of translation operators
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by Krishna B. Athreya and Justin R. Peters PDF

Proc. Amer. Math. Soc. 139 (2011), 4027-4040 Request permission

Abstract:

For a Radon measure $\mu$ on $\mathbb {R},$ we show that $L^{\infty }(\mu )$ is invariant under the group of translation operators $T_t(f)(x) = {f(x-t)}\ (t \in \mathbb {R})$ if and only if $\mu$ is equivalent to the Lebesgue measure $m$. We also give necessary and sufficient conditions for $L^p(\mu ),\ 1 \leq p < \infty ,$ to be invariant under the group $\{ T_t\}$ in terms of the Radon-Nikodým derivative w.r.t. $m$.

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