Restrictions of $L^{p}$ transforms

Abstract:

Let G be a locally compact abelian group with dual $\Gamma$, E a subset of $\Gamma$, and $\phi$ a complex-valued function defined on $\Gamma$. Assume $\phi$ has $\sigma$-compact support. In this paper we prove that $\phi$ is a multiplier of type $({L^1},{L^{{p_1}}} \cap {L^{{p_2}}},E)\;(1 \leqq {p_1} \leqq 2,1 < {p_2} \leqq \infty )$ if and only if $\phi = \hat f$ a.e. on E for some $f \in {L^{{p_1}}}(G) \cap {L^{{p_2}}}(G)$. We give applications of this result to the problems of restrictions, uniqueness, inversion and characterization of ${L^p}$ transforms.