Math. Z. (2010) 265:417–435 DOI 10.1007/s00209-009-0522-y Mathematische Zeitschrift New results on restriction of Fourier multipliers María Carro · Salvador Rodríguez Received: 23 September 2008 / Accepted: 28 October 2008 / Published online: 15 April 2009 © Springer-Verlag 2009 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. 1 Introduction A bounded function m is said to be a Fourier multiplier on L p (R) if the operator defined on functions in S (R d ) by T m ( f )(x ) =  R d  f (ξ)m(ξ)e 2π ix ξ d ξ, extends to a bounded operator on L p (R). Also, a bounded sequence (m k ) k∈Z is said to be a Fourier multiplier on L p (T) if the operator defined on trigonometric polynomials by T m ( f )(x ) =  k∈Z  f (k )m k e 2π ixk , extends to a bounded operator on L p (T). In this setting, de Leeuw’s restriction theorem (see [22]) asserts that if 1 ≤ p < ∞ and m is a continuous, bounded function on R which is a Fourier multiplier on L p (R) then, m| Z , the restriction of m to Z, is a Fourier multiplier on L p (T) with norm not exceeding the norm This work has been partially supported by MTM2007-60500 and by CURE 2005SGR00556. M. Carro · S. Rodríguez (B ) Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, 08007 Barcelona, Spain e-mail: salvarodriguez@ub.edu M. Carro e-mail: carro@ub.edu 123