New results on restriction of Fourier multipliers

Abstract

We develop an extension of the Transference methods introduced by R. Coifman and G. Weiss and apply it to study the problem of the restriction of Fourier multipliers between rearrangement invariant spaces, obtaining natural extensions of the classical de Leeuw's result and its further extension to maximal Fourier multipliers due to C. Kenig and P. Tomas.

Transference and restriction of Fourier multipliers on Orlicz spaces

“…Of course most of the theorems on multipliers have a similar formulation on the previously mentioned groups, and the results can actually be transferred from one group to another one using certain methods. In the setting of Lebesgue spaces, the reader is referred to [13, 35] for standard techniques on transference between $$\mathbb {R}^n, \mathbb {T}^n, \mathbb {Z}^n$$, to [9] for general locally compact abelian groups and to [8] for an admissible pair of rearrangement invariant spaces. We also refer to [2, 5, 7, 20] for the case of weighted Lebesgue spaces and some restriction results.…”

On restriction of maximal multipliers in weighted settings

“…The following result is the weighted version of [6,Lemma 2] and the weak type maximal counterpart of [7,Proposition 4.10].…”