Extension of Fourier $L^{p}—L^{q}$ multipliers

Abstract:

By $M_p^q(\Gamma )$ we denote the space of Fourier ${L^p} - {L^q}$ multipliers on the LCA group $\Gamma$. K. de Leeuw [4] (for $\Gamma = {R^a}$), N. Lohoué [16] and S. Saeki [19] have shown that if ${\Gamma _0}$ is a closed subgroup of $\Gamma$, and $\phi$ is a continuous function in $M_p^p(\Gamma )$, then the restriction ${\phi _0}$ of $\phi$ to ${\Gamma _0}$ is in $M_p^p({\Gamma _0})$, and ${\left \| {{\phi _0}} \right \|_{M_p^p}} \leqslant {\left \| \phi \right \|_{M_p^p}}$. We answer here a natural question arising from this result: we show that every continuous function $\psi$ in $M_p^p(\Gamma )$ is the restriction to ${\Gamma _0}$ of a continuous $M_p^p(\Gamma )$ function whose norm is the same as that of $\psi$. A Figà-Talamanca and G. I. Gaudry [8] proved this with the extra condition that ${\Gamma _0}$ be discrete: our technique develops their ideas. An extension theorem for $M_p^q({\Gamma _0})$ is obtained: this complements work of Gaudry [11] on restrictions of $M_p^q(\Gamma )$-functions to ${\Gamma _0}$.